KAN: Kolmogorov–Arnold Networks

## Abstract

Abstract Inspired by the Kolmogorov-Arnold representation theorem, we propose Kolmogorov-Arnold Networks (KANs) as promising alternatives to Multi-Layer Perceptrons (MLPs). While MLPs have fixed activation functions on nodes (“neurons”), KANs have learnable activation functions on edges (“weights”). KANs have no linear weights at all – every weight parameter is replaced by a univariate function parametrized as a spline. We show that this seemingly simple change makes KANs outperform MLPs in terms of accuracy and interpretability, on small-scale AI + Science tasks. For accuracy, smaller KANs can achieve comparable or better accuracy than larger MLPs in function fitting tasks. Theoretically and empirically, KANs possess faster neural scaling laws than MLPs. For interpretability, KANs can be intuitively visualized and can easily interact with human users. Through two examples in mathematics and physics, KANs are shown to be useful “collaborators” helping scientists (re)discover mathematical and physical laws. In summary, KANs are promising alternatives for MLPs, opening opportunities for further improving today’s deep learning models which rely heavily on MLPs. Figure 0.1: Multi-Layer Perceptrons (MLPs) vs. Kolmogorov-Arnold Networks (KANs)

## 1 Introduction

Multi-layer perceptrons (MLPs) [, , ], also known as fully-connected feedforward neural networks, are foundational building blocks of today’s deep learning models. The importance of MLPs can never be overstated, since they are the default models in machine learning for approximating nonlinear functions, due to their expressive power guaranteed by the universal approximation theorem []. However, are MLPs the best nonlinear regressors we can build? Despite the prevalent use of MLPs, they have significant drawbacks. In transformers [] for example, MLPs consume almost all non-embedding parameters and are typically less interpretable (relative to attention layers) without post-analysis tools [].

We propose a promising alternative to MLPs, called Kolmogorov-Arnold Networks (KANs). Whereas MLPs are inspired by the universal approximation theorem, KANs are inspired by the Kolmogorov-Arnold representation theorem [, , ]. Like MLPs, KANs have fully-connected structures. However, while MLPs place fixed activation functions on nodes (“neurons”), KANs place learnable activation functions on edges (“weights”), as illustrated in Figure 0.1. As a result, KANs have no linear weight matrices at all: instead, each weight parameter is replaced by a learnable 1D function parametrized as a spline. KANs’ nodes simply sum incoming signals without applying any non-linearities. One might worry that KANs are hopelessly expensive, since each MLP’s weight parameter becomes KAN’s spline function. Fortunately, KANs usually allow much smaller computation graphs than MLPs.

Unsurprisingly, the possibility of using Kolmogorov-Arnold representation theorem to build neural networks has been studied [, , , , , , , ]. However, most work has stuck with the original depth-2 width-($2n+1$) representation, and many did not have the chance to leverage more modern techniques (e.g., back propagation) to train the networks. In [], a depth-2 width-($2n+1$) representation was investigated, with breaking of the curse of dimensionality observed both empirically and with an approximation theory given compositional structures of the function. Our contribution lies in generalizing the original Kolmogorov-Arnold representation to arbitrary widths and depths, revitalizing and contextualizing it in today’s deep learning world, as well as using extensive empirical experiments to highlight its potential for AI + Science due to its accuracy and interpretability.

Despite their elegant mathematical interpretation, KANs are nothing more than combinations of splines and MLPs, leveraging their respective strengths and avoiding their respective weaknesses. Splines are accurate for low-dimensional functions, easy to adjust locally, and able to switch between different resolutions. However, splines have a serious curse of dimensionality (COD) problem, because of their inability to exploit compositional structures. MLPs, on the other hand, suffer less from COD thanks to their feature learning, but are less accurate than splines in low dimensions, because of their inability to optimize univariate functions. The link between MLPs using ReLU-k as activation functions and splines have been established in [, ]. To learn a function accurately, a model should not only learn the compositional structure (external degrees of freedom), but should also approximate well the univariate functions (internal degrees of freedom). KANs are such models since they have MLPs on the outside and splines on the inside. As a result, KANs can not only learn features (thanks to their external similarity to MLPs), but can also optimize these learned features to great accuracy (thanks to their internal similarity to splines). For example, given a high dimensional function

$$ $\displaystyle f(x_{1},\cdots,x_{N})=\exp\left(\frac{1}{N}\sum_{i=1}^{N}{\rm sin}^{2}(x_{i})\right),$ (1.1) $$

splines would fail for large $N$ due to COD; MLPs can potentially learn the the generalized additive structure, but they are very inefficient for approximating the exponential and sine functions with say, ReLU activations. In contrast, KANs can learn both the compositional structure and the univariate functions quite well, hence outperforming MLPs by a large margin (see Figure 3.1).

Throughout this paper, we will use extensive numerical experiments to show that KANs can lead to accuracy and interpretability improvement over MLPs, at least on small-scale AI + Science tasks. The organization of the paper is illustrated in Figure 2.1. In Section 2, we introduce the KAN architecture and its mathematical foundation, introduce network simplification techniques to make KANs interpretable, and introduce a grid extension technique to make KANs more accurate. In Section 3, we show that KANs are more accurate than MLPs for data fitting: KANs can beat the curse of dimensionality when there is a compositional structure in data, achieving better scaling laws than MLPs. We also demonstrate the potential of KANs in PDE solving via a simple example of the Poisson equation. In Section 4, we show that KANs are interpretable and can be used for scientific discoveries. We use two examples from mathematics (knot theory) and physics (Anderson localization) to demonstrate that KANs can be helpful “collaborators” for scientists to (re)discover math and physical laws. Section 5 summarizes related works. In Section 6, we conclude by discussing broad impacts and future directions. Codes are available at [https://github.com/KindXiaoming/pykan](https://github.com/KindXiaoming/pykan) and can also be installed via pip install pykan.

## 2 Kolmogorov–Arnold Networks (KAN)

Figure: Figure 2.1: Our proposed Kolmogorov-Arnold networks are in honor of two great late mathematicians, Andrey Kolmogorov and Vladimir Arnold. KANs are mathematically sound, accurate and interpretable.
Refer to caption: figs/flowchart.png

Multi-Layer Perceptrons (MLPs) are inspired by the universal approximation theorem. We instead focus on the Kolmogorov-Arnold representation theorem, which can be realized by a new type of neural network called Kolmogorov-Arnold networks (KAN). We review the Kolmogorov-Arnold theorem in Section 2.1, to inspire the design of Kolmogorov-Arnold Networks in Section 2.2. In Section 2.3, we provide theoretical guarantees for the expressive power of KANs and their neural scaling laws, relating them to existing approximation and generalization theories in the literature. In Section 2.4, we propose a grid extension technique to make KANs increasingly more accurate. In Section 2.5, we propose simplification techniques to make KANs interpretable.

### 2.1 Kolmogorov-Arnold Representation theorem

Vladimir Arnold and Andrey Kolmogorov established that if $f$ is a multivariate continuous function on a bounded domain, then $f$ can be written as a finite composition of continuous functions of a single variable and the binary operation of addition. More specifically, for a smooth $f:[0,1]^{n}\to\mathbb{R}$,

$$ $f(\mathbf{x})=f(x_{1},\cdots,x_{n})=\sum_{q=1}^{2n+1}\Phi_{q}\left(\sum_{p=1}^{n}\phi_{q,p}(x_{p})\right),$ (2.1) $$

where $\phi_{q,p}:[0,1]\to\mathbb{R}$ and $\Phi_{q}:\mathbb{R}\to\mathbb{R}$. In a sense, they showed that the only true multivariate function is addition, since every other function can be written using univariate functions and sum. One might naively consider this great news for machine learning: learning a high-dimensional function boils down to learning a polynomial number of 1D functions. However, these 1D functions can be non-smooth and even fractal, so they may not be learnable in practice [, ]. Because of this pathological behavior, the Kolmogorov-Arnold representation theorem was basically sentenced to death in machine learning, regarded as theoretically sound but practically useless [, ].

However, we are more optimistic about the usefulness of the Kolmogorov-Arnold theorem for machine learning. First of all, we need not stick to the original Eq. (2.1) which has only two-layer non-linearities and a small number of terms ($2n+1$) in the hidden layer: we will generalize the network to arbitrary widths and depths. Secondly, most functions in science and daily life are often smooth and have sparse compositional structures, potentially facilitating smooth Kolmogorov-Arnold representations. The philosophy here is close to the mindset of physicists, who often care more about typical cases rather than worst cases. After all, our physical world and machine learning tasks must have structures to make physics and machine learning useful or generalizable at all [].

### 2.2 KAN architecture

Figure: Figure 2.2: Left: Notations of activations that flow through the network. Right: an activation function is parameterized as a B-spline, which allows switching between coarse-grained and fine-grained grids.
Refer to caption: figs/spline_notation.png

Suppose we have a supervised learning task consisting of input-output pairs $\{\mathbf{x}_{i},y_{i}\}$, where we want to find $f$ such that $y_{i}\approx f(\mathbf{x}_{i})$ for all data points.
Eq. (2.1) implies that we are done if we can find appropriate univariate functions $\phi_{q,p}$ and $\Phi_{q}$. This inspires us to design a neural network which explicitly parametrizes Eq. (2.1). Since all functions to be learned are univariate functions, we can parametrize each 1D function as a B-spline curve, with learnable coefficients of local B-spline basis functions (see Figure 2.2 right). Now we have a prototype of KAN, whose computation graph is exactly specified by Eq. (2.1) and illustrated in Figure 0.1 (b) (with the input dimension $n=2$), appearing as a two-layer neural network with activation functions placed on edges instead of nodes (simple summation is performed on nodes), and with width $2n+1$ in the middle layer.

As mentioned, such a network is known to be too simple to approximate any function arbitrarily well in practice with smooth splines! We therefore generalize our KAN to be wider and deeper. It is not immediately clear how to make KANs deeper, since Kolmogorov-Arnold representations correspond to two-layer KANs. To the best of our knowledge, there is not yet a “generalized” version of the theorem that corresponds to deeper KANs.

The breakthrough occurs when we notice the analogy between MLPs and KANs. In MLPs, once we define a layer (which is composed of a linear transformation and nonlinearties), we can stack more layers to make the network deeper. To build deep KANs, we should first answer: “what is a KAN layer?” It turns out that a KAN layer with $n_{\rm in}$-dimensional inputs and $n_{\rm out}$-dimensional outputs can be defined as a matrix of 1D functions

$$ $\displaystyle{\mathbf{\Phi}}=\{\phi_{q,p}\},\qquad p=1,2,\cdots,n_{\rm in},\qquad q=1,2\cdots,n_{\rm out},$ (2.2) $$

where the functions $\phi_{q,p}$ have trainable parameters, as detaild below. In the Kolmogov-Arnold theorem, the inner functions form a KAN layer with $n_{\rm in}=n$ and $n_{\rm out}=2n+1$, and the outer functions form a KAN layer with $n_{\rm in}=2n+1$ and $n_{\rm out}=1$. So the Kolmogorov-Arnold representations in Eq. (2.1) are simply compositions of two KAN layers. Now it becomes clear what it means to have deeper Kolmogorov-Arnold representations: simply stack more KAN layers!

Let us introduce some notation. This paragraph will be a bit technical, but readers can refer to Figure 2.2 (left) for a concrete example and intuitive understanding. The shape of a KAN is represented by an integer array

$$ $\displaystyle[n_{0},n_{1},\cdots,n_{L}],$ (2.3) $$

where $n_{i}$ is the number of nodes in the $i^{\rm th}$ layer of the computational graph. We denote the $i^{\rm th}$ neuron in the $l^{\rm th}$ layer by $(l,i)$, and the activation value of the $(l,i)$-neuron by $x_{l,i}$. Between layer $l$ and layer $l+1$, there are $n_{l}n_{l+1}$ activation functions: the activation function that connects $(l,i)$ and $(l+1,j)$ is denoted by

$$ $\displaystyle\phi_{l,j,i},\quad l=0,\cdots,L-1,\quad i=1,\cdots,n_{l},\quad j=1,\cdots,n_{l+1}.$ (2.4) $$

The pre-activation of $\phi_{l,j,i}$ is simply $x_{l,i}$; the post-activation of $\phi_{l,j,i}$ is denoted by $\tilde{x}_{l,j,i}\equiv\phi_{l,j,i}(x_{l,i})$. The activation value of the $(l+1,j)$ neuron is simply the sum of all incoming post-activations:

$$ $x_{l+1,j}=\sum_{i=1}^{n_{l}}\tilde{x}_{l,j,i}=\sum_{i=1}^{n_{l}}\phi_{l,j,i}(x_{l,i}),\qquad j=1,\cdots,n_{l+1}.$ (2.5) $$

In matrix form, this reads

$$ $\mathbf{x}_{l+1}=\underbrace{\begin{pmatrix}\phi_{l,1,1}(\cdot)&\phi_{l,1,2}(\cdot)&\cdots&\phi_{l,1,n_{l}}(\cdot)\\ \phi_{l,2,1}(\cdot)&\phi_{l,2,2}(\cdot)&\cdots&\phi_{l,2,n_{l}}(\cdot)\\ \vdots&\vdots&&\vdots\\ \phi_{l,n_{l+1},1}(\cdot)&\phi_{l,n_{l+1},2}(\cdot)&\cdots&\phi_{l,n_{l+1},n_{l}}(\cdot)\\ \end{pmatrix}}_{\mathbf{\Phi}_{l}}\mathbf{x}_{l},$ (2.6) $$

where ${\mathbf{\Phi}}_{l}$ is the function matrix corresponding to the $l^{\rm th}$ KAN layer. A general KAN network is a composition of $L$ layers: given an input vector $\mathbf{x}_{0}\in\mathbb{R}^{n_{0}}$, the output of KAN is

$$ ${\rm KAN}(\mathbf{x})=(\mathbf{\Phi}_{L-1}\circ\mathbf{\Phi}_{L-2}\circ\cdots\circ\mathbf{\Phi}_{1}\circ\mathbf{\Phi}_{0})\mathbf{x}.$ (2.7) $$

We can also rewrite the above equation to make it more analogous to Eq. (2.1), assuming output dimension $n_{L}=1$, and define $f(\mathbf{x})\equiv{\rm KAN}(\mathbf{x})$:

$$ $f(\mathbf{x})=\sum_{i_{L-1}=1}^{n_{L-1}}\phi_{L-1,i_{L},i_{L-1}}\left(\sum_{i_{L-2}=1}^{n_{L-2}}\cdots\left(\sum_{i_{2}=1}^{n_{2}}\phi_{2,i_{3},i_{2}}\left(\sum_{i_{1}=1}^{n_{1}}\phi_{1,i_{2},i_{1}}\left(\sum_{i_{0}=1}^{n_{0}}\phi_{0,i_{1},i_{0}}(x_{i_{0}})\right)\right)\right)\cdots\right),$ (2.8) $$

which is quite cumbersome. In contrast, our abstraction of KAN layers and their visualizations are cleaner and intuitive. The original Kolmogorov-Arnold representation Eq. (2.1) corresponds to a 2-Layer KAN with shape $[n,2n+1,1]$. Notice that all the operations are differentiable, so we can train KANs with back propagation. For comparison, an MLP can be written as interleaving of affine transformations $\mathbf{W}$ and non-linearities $\sigma$:

$$ ${\rm MLP}(\mathbf{x})=(\mathbf{W}_{L-1}\circ\sigma\circ\mathbf{W}_{L-2}\circ\sigma\circ\cdots\circ\mathbf{W}_{1}\circ\sigma\circ\mathbf{W}_{0})\mathbf{x}.$ (2.9) $$

It is clear that MLPs treat linear transformations and nonlinearities separately as $\mathbf{W}$ and $\sigma$, while KANs treat them all together in $\mathbf{\Phi}$. In Figure 0.1 (c) and (d), we visualize a three-layer MLP and a three-layer KAN, to clarify their differences.

Implementation details.
Although a KAN layer Eq. (2.5) looks extremely simple, it is non-trivial to make it well optimizable. The key tricks are:

- (1)
Residual activation functions. We include a basis function $b(x)$ (similar to residual connections) such that the activation function $\phi(x)$ is the sum of the basis function $b(x)$ and the spline function:
$\displaystyle\phi(x)=w_{b}b(x)+w_{s}{\rm spline}(x).$
(2.10)
We set
$\displaystyle b(x)={\rm silu}(x)=x/(1+e^{-x})$
(2.11)
in most cases. ${\rm spline}(x)$ is parametrized as a linear combination of B-splines such that
$\displaystyle{\rm spline}(x)=\sum_{i}c_{i}B_{i}(x)$
(2.12)
where $c_{i}$s are trainable (see Figure 2.2 for an illustration). In principle $w_{b}$ and $w_{s}$ are redundant since it can be absorbed into $b(x)$ and ${\rm spline}(x)$. However, we still include these factors (which are by default trainable) to better control the overall magnitude of the activation function.
- (2)
Initialization scales. Each activation function is initialized to have $w_{s}=1$ and ${\rm spline}(x)\approx 0$ (^1^11This is done by drawing B-spline coefficients $c_{i}\sim\mathcal{N}(0,\sigma^{2})$ with a small $\sigma$, typically we set $\sigma=0.1$.). $w_{b}$ is initialized according to the Xavier initialization, which has been used to initialize linear layers in MLPs.
- (3)
Update of spline grids. We update each grid on the fly according to its input activations, to address the issue that splines are defined on bounded regions but activation values can evolve out of the fixed region during training (^2^22Other possibilities are: (a) the grid is learnable with gradient descent, e.g., []; (b) use normalization such that the input range is fixed. We tried (b) at first but its performance is inferior to our current approach.).

Parameter count. For simplicity, let us assume a network

- (1)
of depth $L$,
- (2)
with layers of equal width $n_{0}=n_{1}=\cdots=n_{L}=N$,
- (3)
with each spline of order $k$ (usually $k=3$) on $G$ intervals (for $G+1$ grid points).

Then there are in total $O(N^{2}L(G+k))\sim O(N^{2}LG)$ parameters. In contrast, an MLP with depth $L$ and width $N$ only needs $O(N^{2}L)$ parameters, which appears to be more efficient than KAN. Fortunately, KANs usually require much smaller $N$ than MLPs, which not only saves parameters, but also achieves better generalization (see e.g., Figure 3.1 and 3.3) and facilitates interpretability.
We remark that for 1D problems, we can take $N=L=1$ and the KAN network in our implementation is nothing but a spline approximation. For higher dimensions, we characterize the generalization behavior of KANs with a theorem below.

### 2.3 KAN’s Approximation Abilities and Scaling Laws

Recall that in Eq. (2.1), the 2-Layer width-$(2n+1)$ representation may be non-smooth. However, deeper representations may bring the advantages of smoother activations. For example, the 4-variable function

$$ $\displaystyle f(x_{1},x_{2},x_{3},x_{4})=\exp\left({\sin}(x_{1}^{2}+x_{2}^{2})+{\sin}(x_{3}^{2}+x_{4}^{2})\right)$ (2.13) $$

can be smoothly represented by a $[4,2,1,1]$ KAN which is 3-Layer, but may not admit a 2-Layer KAN with smooth activations. To facilitate an approximation analysis, we still assume smoothness of activations, but allow the representations to be arbitrarily wide and deep, as in Eq. (2.7).
To emphasize the dependence of our KAN on the finite set of grid points, we use $\mathbf{\Phi}_{l}^{G}$ and $\Phi_{l,i,j}^{G}$ below to replace the notation $\mathbf{\Phi}_{l}$ and $\Phi_{l,i,j}$ used in Eq. (2.5) and (2.6).

###### Theorem 2.1 (Approximation theory, KAT) .

Let $\mathbf{x}=(x_{1},x_{2},\cdots,x_{n})$.
Suppose that a function $f(\mathbf{x})$ admits a representation

$$ $f=(\mathbf{\Phi}_{L-1}\circ\mathbf{\Phi}_{L-2}\circ\cdots\circ\mathbf{\Phi}_{1}\circ\mathbf{\Phi}_{0})\mathbf{x}\,,$ (2.14) $$

as in Eq. (2.7), where each one of the $\Phi_{l,i,j}$ are $(k+1)$-times continuously differentiable. Then there exists a constant $C$ depending on $f$ and its representation, such that we have the following approximation bound in terms of the grid size $G$: there exist $k$-th order B-spline functions $\Phi_{l,i,j}^{G}$ such that for any $0\leq m\leq k$, we have the bound

$$ $\|f-(\mathbf{\Phi}^{G}_{L-1}\circ\mathbf{\Phi}^{G}_{L-2}\circ\cdots\circ\mathbf{\Phi}^{G}_{1}\circ\mathbf{\Phi}^{G}_{0})\mathbf{x}\|_{C^{m}}\leq CG^{-k-1+m}\,.$ (2.15) $$

Here we adopt the notation of $C^{m}$-norm measuring the magnitude of derivatives up to order $m$:

$$ $\|g\|_{C^{m}}=\max_{|\beta|\leq m}\sup_{x\in[0,1]^{n}}\left|D^{\beta}g(x)\right|.$ $$

###### Proof.

By the classical 1D B-spline theory [] and the fact that $\Phi_{l,i,j}$ as continuous functions can be uniformly bounded on a bounded domain, we know that there exist finite-grid B-spline functions $\Phi_{l,i,j}^{G}$ such that for any $0\leq m\leq k$,

$$ $\|(\Phi_{l,i,j}\circ\mathbf{\Phi}_{l-1}\circ\mathbf{\Phi}_{l-2}\circ\cdots\circ\mathbf{\Phi}_{1}\circ\mathbf{\Phi}_{0})\mathbf{x}-(\Phi_{l,i,j}^{G}\circ\mathbf{\Phi}_{l-1}\circ\mathbf{\Phi}_{l-2}\circ\cdots\circ\mathbf{\Phi}_{1}\circ\mathbf{\Phi}_{0})\mathbf{x}\|_{C^{m}}\leq CG^{-k-1+m}\,,$ $$

with a constant $C$ independent of $G$. We fix those B-spline approximations. Therefore we have that the residue $R_{l}$ defined via

$$ $R_{l}\coloneqq(\mathbf{\Phi}^{G}_{L-1}\circ\cdots\circ\mathbf{\Phi}^{G}_{l+1}\circ\mathbf{\Phi}_{l}\circ\mathbf{\Phi}_{l-1}\circ\cdots\circ\mathbf{\Phi}_{0})\mathbf{x}-(\mathbf{\Phi}_{L-1}^{G}\circ\cdots\circ\mathbf{\Phi}_{l+1}^{G}\circ\mathbf{\Phi}_{l}^{G}\circ\mathbf{\Phi}_{l-1}\circ\cdots\circ\mathbf{\Phi}_{0})\mathbf{x}$ $$

satisfies

$$ $\|R_{l}\|_{C^{m}}\leq CG^{-k-1+m}\,,$ $$

with a constant independent of $G$. Finally notice that

$$ $f-(\mathbf{\Phi}^{G}_{L-1}\circ\mathbf{\Phi}^{G}_{L-2}\circ\cdots\circ\mathbf{\Phi}^{G}_{1}\circ\mathbf{\Phi}^{G}_{0})\mathbf{x}=R_{L-1}+R_{L-2}+\cdots+R_{1}+R_{0}\,,$ $$

we know that (2.15) holds.
∎

We know that asymptotically, provided that the assumption in Theorem 2.1 holds, KANs with finite grid size can approximate the function well with a residue rate independent of the dimension, hence beating curse of dimensionality! This comes naturally since we only use splines to approximate 1D functions. In particular, for $m=0$, we recover the accuracy in $L^{\infty}$ norm, which in turn provides a bound of RMSE on the finite domain, which gives a scaling exponent $k+1$. Of course, the constant $C$ is dependent on the representation; hence it will depend on the dimension. We will leave the discussion of the dependence of the constant on the dimension as a future work.

We remark that although the Kolmogorov-Arnold theorem Eq. (2.1) corresponds to a KAN representation with shape $[d,2d+1,1]$, its functions are not necessarily smooth. On the other hand, if we are able to identify a smooth representation (maybe at the cost of extra layers or making the KAN wider than the theory prescribes), then Theorem 2.1 indicates that we can beat the curse of dimensionality (COD). This should not come as a surprise since we can inherently learn the structure of the function and make our finite-sample KAN approximation interpretable.

Neural scaling laws: comparison to other theories. Neural scaling laws are the phenomenon where test loss decreases with more model parameters, i.e., $\ell\propto N^{-\alpha}$ where $\ell$ is test RMSE, $N$ is the number of parameters, and $\alpha$ is the scaling exponent. A larger $\alpha$ promises more improvement by simply scaling up the model. Different theories have been proposed to predict $\alpha$. Sharma & Kaplan [] suggest that $\alpha$ comes from data fitting on an input manifold of intrinsic dimensionality $d$. If the model function class is piecewise polynomials of order $k$ ($k=1$ for ReLU), then the standard approximation theory implies $\alpha=(k+1)/d$ from the approximation theory. This bound suffers from the curse of dimensionality, so people have sought other bounds independent of $d$ by leveraging compositional structures. In particular, Michaud et al. [] considered computational graphs that only involve unary (e.g., squared, sine, exp) and binary ($+$ and $\times$) operations, finding $\alpha=(k+1)/d^{*}=(k+1)/2$, where $d^{*}=2$ is the maximum arity. Poggio et al. [] leveraged the idea of compositional sparsity and proved that given function class $W_{m}$ (function whose derivatives are continuous up to $m$-th order), one needs $N=O(\epsilon^{-\frac{2}{m}})$ number of parameters to achieve error $\epsilon$, which is equivalent to $\alpha=\frac{m}{2}$. Our approach, which assumes the existence of smooth Kolmogorov-Arnold representations, decomposes the high-dimensional function into several 1D functions, giving $\alpha=k+1$ (where $k$ is the piecewise polynomial order of the splines). We choose $k=3$ cubic splines so $\alpha=4$ which is the largest and best scaling exponent compared to other works. We will show in Section 3.1 that this bound $\alpha=4$ can in fact be achieved empirically with KANs, while previous work [] reported that MLPs have problems even saturating slower bounds (e.g., $\alpha=1$) and plateau quickly. Of course, we can increase $k$ to match the smoothness of functions, but too high $k$ might be too oscillatory, leading to optimization issues.

Comparison between KAT and UAT.
The power of fully-connected neural networks is justified by the universal approximation theorem (UAT), which states that given a function and error tolerance $\epsilon>0$, a two-layer network with $k>N(\epsilon)$ neurons can approximate the function within error $\epsilon$. However, the UAT guarantees no bound for how $N(\epsilon)$ scales with $\epsilon$. Indeed, it suffers from the COD, and $N$ has been shown to grow exponentially with $d$ in some cases []. The difference between KAT and UAT is a consequence that KANs take advantage of the intrinsically low-dimensional representation of the function while MLPs do not. In KAT, we highlight quantifying the approximation error in the compositional space. In the literature, generalization error bounds, taking into account finite samples of training data, for a similar space have been studied for regression problems; see [, ], and also specifically for MLPs with ReLU activations []. On the other hand, for general function spaces like Sobolev or Besov spaces, the nonlinear $n$-widths theory [, , ] indicates that we can never beat the curse of dimensionality, while MLPs with ReLU activations can achieve the tight rate [, , ]. This fact again motivates us to consider functions of compositional structure, the much "nicer" functions that we encounter in practice and in science, to overcome the COD. Compared with MLPs, we may use a smaller architecture in practice, since we learn general nonlinear activation functions; see also [] where the depth of the ReLU MLPs needs to reach at least $\log n$ to have the desired rate, where $n$ is the number of samples. Indeed, we will show that KANs are nicely aligned with symbolic functions while MLPs are not.

###### Theorem 2.1 (Approximation theory, KAT) .

###### Proof.

### 2.4 For accuracy: Grid Extension

Figure: Figure 2.3: We can make KANs more accurate by grid extension (fine-graining spline grids). Top left (right): training dynamics of a $[2,5,1]$ ($[2,1,1]$) KAN. Both models display staircases in their loss curves, i.e., loss suddently drops then plateaus after grid extension. Bottom left: test RMSE follows scaling laws against grid size $G$. Bottom right: training time scales favorably with grid size $G$.
Refer to caption: x2.png

In principle, a spline can be made arbitrarily accurate to a target function as the grid can be made arbitrarily fine-grained. This good feature is inherited by KANs. By contrast, MLPs do not have the notion of “fine-graining”. Admittedly, increasing the width and depth of MLPs can lead to improvement in performance (“neural scaling laws”). However, these neural scaling laws are slow (discussed in the last section). They are also expensive to obtain, because models of varying sizes are trained independently. By contrast, for KANs, one can first train a KAN with fewer parameters and then extend it to a KAN with more parameters by simply making its spline grids finer, without the need to retraining the larger model from scratch.

We next describe how to perform grid extension (illustrated in Figure 2.2 right), which is basically fitting a new fine-grained spline to an old coarse-grained spline. Suppose we want to approximate a 1D function $f$ in a bounded region $[a,b]$ with B-splines of order $k$. A coarse-grained grid with $G_{1}$ intervals has grid points at $\{t_{0}=a,t_{1},t_{2},\cdots,t_{G_{1}}=b\}$, which is augmented to $\{t_{-k},\cdots,t_{-1},t_{0},\cdots,t_{G_{1}},t_{G_{1}+1},\cdots,t_{G_{1}+k}\}$. There are $G_{1}+k$ B-spline basis functions, with the $i^{\rm th}$ B-spline $B_{i}(x)$ being non-zero only on $[t_{-k+i},t_{i+1}]$ $(i=0,\cdots,G_{1}+k-1)$. Then $f$ on the coarse grid is expressed in terms of linear combination of these B-splines basis functions $f_{\rm coarse}(x)=\sum_{i=0}^{G_{1}+k-1}c_{i}B_{i}(x)$. Given a finer grid with $G_{2}$ intervals, $f$ on the fine grid is correspondingly $f_{\rm fine}(x)=\sum_{j=0}^{G_{2}+k-1}c_{j}^{\prime}B_{j}^{\prime}(x)$. The parameters $c^{\prime}_{j}$s can be initialized from the parameters $c_{i}$ by minimizing the distance between $f_{\rm fine}(x)$ to $f_{\rm coarse}(x)$ (over some distribution of $x$):

$$ $\{c_{j}^{\prime}\}=\underset{\{c_{j}^{\prime}\}}{\rm argmin}\ \mathop{\mathbb{E}}_{x\sim p(x)}\left(\sum_{j=0}^{G_{2}+k-1}c_{j}^{\prime}B_{j}^{\prime}(x)-\sum_{i=0}^{G_{1}+k-1}c_{i}B_{i}(x)\right)^{2},$ (2.16) $$

which can be implemented by the least squares algorithm. We perform grid extension for all splines in a KAN independently.

Toy example: staricase-like loss curves. We use a toy example $f(x,y)={\rm exp}({\rm sin}(\pi x)+y^{2})$ to demonstrate the effect of grid extension. In Figure 2.3 (top left), we show the train and test RMSE for a $[2,5,1]$ KAN. The number of grid points starts as 3, increases to a higher value every 200 LBFGS steps, ending up with 1000 grid points. It is clear that every time fine graining happens, the training loss drops faster than before (except for the finest grid with 1000 points, where optimization ceases to work probably due to bad loss landscapes). However, the test losses first go down then go up, displaying a U-shape, due to the bias-variance tradeoff (underfitting vs. overfitting). We conjecture that the optimal test loss is achieved at the interpolation threshold when the number of parameters match the number of data points. Since our training samples are 1000 and the total parameters of a $[2,5,1]$ KAN is $15G$ ($G$ is the number of grid intervals), we expect the interpolation threshold to be $G=1000/15\approx 67$, which roughly agrees with our experimentally observed value $G\sim 50$.

Small KANs generalize better. Is this the best test performance we can achieve? Notice that the synthetic task can be represented exactly by a $[2,1,1]$ KAN, so we train a $[2,1,1]$ KAN and present the training dynamics in Figure 2.3 top right. Interestingly, it can achieve even lower test losses than the $[2,5,1]$ KAN, with clearer staircase structures and the interpolation threshold is delayed to a larger grid size as a result of fewer parameters. This highlights a subtlety of choosing KAN architectures. If we do not know the problem structure, how can we determine the minimal KAN shape? In Section 2.5, we will propose a method to auto-discover such minimal KAN architecture via regularization and pruning.

Scaling laws: comparison with theory. We are also interested in how the test loss decreases as the number of grid parameters increases. In Figure 2.3 (bottom left), a [2,1,1] KAN scales roughly as ${\rm test\ RMSE}\propto G^{-3}$. However, according to the Theorem 2.1, we would expect ${\rm test\ RMSE}\propto G^{-4}$. We found that the errors across samples are not uniform. This is probably attributed to boundary effects []. In fact, there are a few samples that have significantly larger errors than others, making the overall scaling slow down. If we plot the square root of the median (not mean) of the squared losses, we get a scaling closer to $G^{-4}$. Despite this suboptimality (probably due to optimization), KANs still have much better scaling laws than MLPs, for data fitting (Figure 3.1) and PDE solving (Figure 3.3). In addition, the training time scales favorably with the number of grid points $G$, shown in Figure 2.3 bottom right (^3^33When $G=1000$, training becomes significantly slower, which is specific to the use of the LBFGS optimizer with line search. We conjecture that the loss landscape becomes bad for $G=1000$, so line search with trying to find an optimal step size within maximal iterations without early stopping.).

External vs Internal degrees of freedom. A new concept that KANs highlights is a distinction between external versus internal degrees of freedom (parameters). The computational graph of how nodes are connected represents external degrees of freedom (“dofs”), while the grid points inside an activation function are internal degrees of freedom. KANs benefit from the fact that they have both external dofs and internal dofs. External dofs (that MLPs also have but splines do not) are responsible for learning compositional structures of multiple variables. Internal dofs (that splines also have but MLPs do not) are responsible for learning univariate functions.

### 2.5 For Interpretability: Simplifying KANs and Making them interactive

One loose end from the last subsection is that we do not know how to choose the KAN shape that best matches the structure of a dataset. For example, if we know that the dataset is generated via the symbolic formula $f(x,y)={\rm exp}({\rm sin}(\pi x)+y^{2})$, then we know that a $[2,1,1]$ KAN is able to express this function. However, in practice we do not know the information a priori, so it would be nice to have approaches to determine this shape automatically. The idea is to start from a large enough KAN and train it with sparsity regularization followed by pruning. We will show that these pruned KANs are much more interpretable than non-pruned ones. To make KANs maximally interpretable, we propose a few simplification techniques in Section 2.5.1, and an example of how users can interact with KANs to make them more interpretable in Section 2.5.2.

#### 2.5.1 Simplification techniques

1. Sparsification. For MLPs, L1 regularization of linear weights is used to favor sparsity. KANs can adapt this high-level idea, but need two modifications:

- (1)
There is no linear “weight” in KANs. Linear weights are replaced by learnable activation functions, so we should define the L1 norm of these activation functions.
- (2)
We find L1 to be insufficient for sparsification of KANs; instead an additional entropy regularization is necessary (see Appendix C for more details).

We define the L1 norm of an activation function $\phi$ to be its average magnitude over its $N_{p}$ inputs, i.e.,

$$ $\left|\phi\right|_{1}\equiv\frac{1}{N_{p}}\sum_{s=1}^{N_{p}}\left|\phi(x^{(s)})\right|.$ (2.17) $$

Then for a KAN layer $\mathbf{\Phi}$ with $n_{\rm in}$ inputs and $n_{\rm out}$ outputs, we define the L1 norm of $\mathbf{\Phi}$ to be the sum of L1 norms of all activation functions, i.e.,

$$ $\left|\mathbf{\Phi}\right|_{1}\equiv\sum_{i=1}^{n_{\rm in}}\sum_{j=1}^{n_{\rm out}}\left|\phi_{i,j}\right|_{1}.$ (2.18) $$

In addition, we define the entropy of $\mathbf{\Phi}$ to be

$$ $S(\mathbf{\Phi})\equiv-\sum_{i=1}^{n_{\rm in}}\sum_{j=1}^{n_{\rm out}}\frac{\left|\phi_{i,j}\right|_{1}}{\left|\mathbf{\Phi}\right|_{1}}{\rm log}\left(\frac{\left|\phi_{i,j}\right|_{1}}{\left|\mathbf{\Phi}\right|_{1}}\right).$ (2.19) $$

The total training objective $\ell_{\rm total}$ is the prediction loss $\ell_{\rm pred}$ plus L1 and entropy regularization of all KAN layers:

$$ $\ell_{\rm total}=\ell_{\rm pred}+\lambda\left(\mu_{1}\sum_{l=0}^{L-1}\left|\mathbf{\Phi}_{l}\right|_{1}+\mu_{2}\sum_{l=0}^{L-1}S(\mathbf{\Phi}_{l})\right),$ (2.20) $$

where $\mu_{1},\mu_{2}$ are relative magnitudes usually set to $\mu_{1}=\mu_{2}=1$, and $\lambda$ controls overall regularization magnitude.

2. Visualization. When we visualize a KAN, to get a sense of magnitudes, we set the transparency of an activation function $\phi_{l,i,j}$ proportional to ${\rm tanh}(\beta A_{l,i,j})$ where $\beta=3$ . Hence, functions with small magnitude appear faded out to allow us to focus on important ones.

3. Pruning. After training with sparsification penalty, we may also want to prune the network to a smaller subnetwork. We sparsify KANs on the node level (rather than on the edge level). For each node (say the $i^{\rm th}$ neuron in the $l^{\rm th}$ layer), we define its incoming and outgoing score as

$$ $I_{l,i}=\underset{k}{\rm max}(\left|\phi_{l-1,i,k}\right|_{1}),\qquad O_{l,i}=\underset{j}{\rm max}(\left|\phi_{l+1,j,i}\right|_{1}),$ (2.21) $$

and consider a node to be important if both incoming and outgoing scores are greater than a threshold hyperparameter $\theta=10^{-2}$ by default. All unimportant neurons are pruned.

4. Symbolification. In cases where we suspect that some activation functions are in fact symbolic (e.g., ${\rm cos}$ or ${\rm log}$), we provide an interface to set them to be a specified symbolic form, fix_symbolic(l,i,j,f) can set the $(l,i,j)$ activation to be $f$. However, we cannot simply set the activation function to be the exact symbolic formula, since its inputs and outputs may have shifts and scalings. So, we obtain preactivations $x$ and postactivations $y$ from samples, and fit affine parameters $(a,b,c,d)$ such that
$y\approx cf(ax+b)+d$. The fitting is done by iterative grid search of $a,b$ and linear regression.

Besides these techniques, we provide additional tools that allow users to apply more fine-grained control to KANs, listed in Appendix A.

#### 2.5.2 A toy example: how humans can interact with KANs

Figure: Figure 2.4: An example of how to do symbolic regression with KAN.
Refer to caption: figs/sr.png

Above we have proposed a number of simplification techniques for KANs. We can view these simplification choices as buttons one can click on. A user interacting with these buttons can decide which button is most promising to click next to make KANs more interpretable. We use an example below to showcase how a user could interact with a KAN to obtain maximally interpretable results.

Let us again consider the regression task

$$ $\displaystyle f(x,y)=\exp\left({\sin}(\pi x)+y^{2}\right).$ (2.22) $$

Given data points $(x_{i},y_{i},f_{i})$, $i=1,2,\cdots,N_{p}$, a hypothetical user Alice is interested in figuring out the symbolic formula. The steps of Alice’s interaction with the KANs are described below (illustrated in Figure 2.4):

Step 1: Training with sparsification. Starting from a fully-connected $[2,5,1]$ KAN, training with sparsification regularization can make it quite sparse. 4 out of 5 neurons in the hidden layer appear useless, hence we want to prune them away.

Step 2: Pruning. Automatic pruning is seen to discard all hidden neurons except the last one, leaving a $[2,1,1]$ KAN. The activation functions appear to be known symbolic functions.

Step 3: Setting symbolic functions. Assuming that the user can correctly guess these symbolic formulas from staring at the KAN plot, they can set

$$ fix_symbolic(0,0,0,‘sin’) (2.23) fix_symbolic(0,1,0,‘xˆ2’) $\displaystyle\texttt{fix_symbolic(1,0,0,`exp')}.$ $$

In case the user has no domain knowledge or no idea which symbolic functions these activation functions might be, we provide a function suggest_symbolic to suggest symbolic candidates.

Step 4: Further training. After symbolifying all the activation functions in the network, the only remaining parameters are the affine parameters. We continue training these affine parameters, and when we see the loss dropping to machine precision, we know that we have found the correct symbolic expression.

Step 5: Output the symbolic formula. Sympy is used to compute the symbolic formula of the output node. The user obtains $1.0e^{1.0y^{2}+1.0{\rm sin}(3.14x)}$, which is the true answer (we only displayed two decimals for $\pi$).

Remark: Why not symbolic regression (SR)?
It is reasonable to use symbolic regression for this example. However, symbolic regression methods are in general brittle and hard to debug. They either return a success or a failure in the end without outputting interpretable intermediate results. In contrast, KANs do continuous search (with gradient descent) in function space, so their results are more continuous and hence more robust. Moreover, users have more control over KANs as compared to SR due to KANs’ transparency. The way we visualize KANs is like displaying KANs’ “brain” to users, and users can perform “surgery” (debugging) on KANs. This level of control is typically unavailable for SR. We will show examples of this in Section 4.4. More generally, when the target function is not symbolic, symbolic regression will fail but KANs can still provide something meaningful. For example, a special function (e.g., a Bessel function) is impossible to SR to learn unless it is provided in advance, but KANs can use splines to approximate it numerically anyway (see Figure 4.1 (d)).

## 3 KANs are accurate

In this section, we demonstrate that KANs are more effective at representing functions than MLPs in various tasks (regression and PDE solving). When comparing two families of models, it is fair to compare both their accuracy (loss) and their complexity (number of parameters). We will show that KANs display more favorable Pareto Frontiers than MLPs. Moreover, in Section 3.5, we show that KANs can naturally work in continual learning without catastrophic forgetting.

### 3.1 Toy datasets

Figure: Figure 3.1: Compare KANs to MLPs on five toy examples. KANs can almost saturate the fastest scaling law predicted by our theory $(\alpha=4)$, while MLPs scales slowly and plateau quickly.
Refer to caption: x3.png

In Section 2.3, our theory suggested that test RMSE loss $\ell$ scales as $\ell\propto N^{-4}$ with model parameters $N$. However, this relies on the existence of a Kolmogorov-Arnold representation. As a sanity check, we construct five examples we know have smooth KA representations:

- (1)
$f(x)=J_{0}(20x)$, which is the Bessel function. Since it is a univariate function, it can be represented by a spline, which is a $[1,1]$ KAN.
- (2)
$f(x,y)={\rm exp}({\rm sin}(\pi x)+y^{2})$. We know that it can be exactly represented by a $[2,1,1]$ KAN.
- (3)
$f(x,y)=xy$. We know from Figure 4.1 that it can be exactly represented by a $[2,2,1]$ KAN.
- (4)
A high-dimensional example $f(x_{1},\cdots,x_{100})={\rm exp}(\frac{1}{100}\sum_{i=1}^{100}{\rm sin}^{2}(\frac{\pi x_{i}}{2}))$ which can be represented by a $[100,1,1]$ KAN.
- (5)
A four-dimensional example $f(x_{1},x_{2},x_{3},x_{4})={\rm exp}(\frac{1}{2}({\rm sin}(\pi(x_{1}^{2}+x_{2}^{2}))+{\rm sin}(\pi(x_{3}^{2}+x_{4}^{2}))))$ which can be represented by a $[4,4,2,1]$ KAN.

We train these KANs by increasing grid points every 200 steps, in total covering $G=\{3,5,10,20,50,100,200,500,1000\}$. We train MLPs with different depths and widths as baselines. Both MLPs and KANs are trained with LBFGS for 1800 steps in total. We plot test RMSE as a function of the number of parameters for KANs and MLPs in Figure 3.1, showing that KANs have better scaling curves than MLPs, especially for the high-dimensional example. For comparison, we plot the lines predicted from our KAN theory as red dashed ($\alpha=k+1=4$), and the lines predicted from Sharma & Kaplan [] as black-dashed ($\alpha=(k+1)/d=4/d$). KANs can almost saturate the steeper red lines, while MLPs struggle to converge even as fast as the slower black lines and plateau quickly. We also note that for the last example, the 2-Layer KAN $[4,9,1]$ behaves much worse than the 3-Layer KAN (shape $[4,2,2,1]$). This highlights the greater expressive power of deeper KANs, which is the same for MLPs: deeper MLPs have more expressive power than shallower ones. Note that we have adopted the vanilla setup where both KANs and MLPs are trained with LBFGS without advanced techniques, e.g., switching between Adam and LBFGS, or boosting []. We leave the comparison of KANs and MLPs in advanced setups for future work.

### 3.2 Special functions

Figure: Figure 3.2: Fitting special functions. We show the Pareto Frontier of KANs and MLPs in the plane spanned by the number of model parameters and RMSE loss. Consistently accross all special functions, KANs have better Pareto Frontiers than MLPs. The definitions of these special functions are in Table 1.
Refer to caption: x4.png

One caveat for the above results is that we assume knowledge of the “true” KAN shape. In practice, we do not know the existence of KA representations. Even when we are promised that such a KA representation exists, we do not know the KAN shape a priori. Special functions in more than one variables are such cases, because it would be (mathematically) surprising if multivariate special functions (e.g., a Bessel function $f(\nu,x)=J_{\nu}(x)$) could be written in KA represenations, involving only univariate functions and sums). We show below that:

- (1)
Finding (approximate) compact KA representations of special functions is possible, revealing novel mathematical properties of special functions from the perspective of Kolmogorov-Arnold representations.
- (2)
KANs are more efficient and accurate in representing special functions than MLPs.

We collect 15 special functions common in math and physics, summarized in Table 1. We choose MLPs with fixed width 5 or 100 and depths swept in $\{2,3,4,5,6\}$. We run KANs both with and without pruning. KANs without pruning: We fix the shape of KAN, whose width are set to 5 and depths are swept in {2,3,4,5,6}.
KAN with pruning. We use the sparsification $(\lambda=10^{-2}\ {\rm or}\ 10^{-3})$ and pruning technique in Section 2.5.1 to obtain a smaller KAN pruned from a fixed-shape KAN. Each KAN is initialized to have $G=3$, trained with LBFGS, with increasing number of grid points every 200 steps to cover $G=\{3,5,10,20,50,100,200\}$. For each hyperparameter combination, we run 3 random seeds.

For each dataset and each model family (KANs or MLPs), we plot the Pareto frontier (^4^44Pareto frontier is defined as fits that are optimal in the sense of no other fit being both simpler and more accurate.), in the (number of parameters, RMSE) plane, shown in Figure 3.2. KANs’ performance is shown to be consistently better than MLPs, i.e., KANs can achieve lower training/test losses than MLPs, given the same number of parameters. Moreover, we report the (surprisingly compact) shapes of our auto-discovered KANs for special functions in Table 1. On one hand, it is interesting to interpret what these compact representations mean mathematically (we include the KAN illustrations in Figure F.1 and F.2 in Appendix F). On the other hand, these compact representations imply the possibility of breaking down a high-dimensional lookup table into several 1D lookup tables, which can potentially save a lot of memory, with the (almost negligible) overhead to perform a few additions at inference time.

**Table 1: Special functions**
| Name | scipy.special API | Minimal KAN shape<br>test RMSE $<10^{-2}$ | Minimal KAN test RMSE | Best KAN shape | Best KAN test RMSE | MLP test RMSE |
| --- | --- | --- | --- | --- | --- | --- |
| Jacobian elliptic functions | ${\rm ellipj}(x,y)$ | [2,2,1] | $7.29\times 10^{-3}$ | [2,3,2,1,1,1] | ${\bf 1.33\times 10^{-4}}$ | $6.48\times 10^{-4}$ |
| Incomplete elliptic integral of the first kind | ${\rm ellipkinc}(x,y)$ | [2,2,1,1] | $1.00\times 10^{-3}$ | [2,2,1,1,1] | ${\bf 1.24\times 10^{-4}}$ | $5.52\times 10^{-4}$ |
| Incomplete elliptic integral of the second kind | ${\rm ellipeinc}(x,y)$ | [2,2,1,1] | $8.36\times 10^{-5}$ | [2,2,1,1] | ${\bf 8.26\times 10^{-5}}$ | $3.04\times 10^{-4}$ |
| Bessel function of the first kind | ${\rm jv}(x,y)$ | [2,2,1] | $4.93\times 10^{-3}$ | [2,3,1,1,1] | ${\bf 1.64\times 10^{-3}}$ | $5.52\times 10^{-3}$ |
| Bessel function of the second kind | ${\rm yv}(x,y)$ | [2,3,1] | $1.89\times 10^{-3}$ | [2,2,2,1] | ${\bf 1.49\times 10^{-5}}$ | $3.45\times 10^{-4}$ |
| Modified Bessel function of the second kind | ${\rm kv}(x,y)$ | [2,1,1] | $4.89\times 10^{-3}$ | [2,2,1] | ${\bf 2.52\times 10^{-5}}$ | $1.67\times 10^{-4}$ |
| Modified Bessel function of the first kind | ${\rm iv}(x,y)$ | [2,4,3,2,1,1] | $9.28\times 10^{-3}$ | [2,4,3,2,1,1] | ${\bf 9.28\times 10^{-3}}$ | $1.07\times 10^{-2}$ |
| Associated Legendre function $(m=0)$ | ${\rm lpmv}(0,x,y)$ | [2,2,1] | $5.25\times 10^{-5}$ | [2,2,1] | ${\bf 5.25\times 10^{-5}}$ | $1.74\times 10^{-2}$ |
| Associated Legendre function $(m=1)$ | ${\rm lpmv}(1,x,y)$ | [2,4,1] | $6.90\times 10^{-4}$ | [2,4,1] | ${\bf 6.90\times 10^{-4}}$ | $1.50\times 10^{-3}$ |
| Associated Legendre function $(m=2)$ | ${\rm lpmv}(2,x,y)$ | [2,2,1] | $4.88\times 10^{-3}$ | [2,3,2,1] | ${\bf 2.26\times 10^{-4}}$ | $9.43\times 10^{-4}$ |
| spherical harmonics $(m=0,n=1)$ | ${\rm sph_harm}(0,1,x,y)$ | [2,1,1] | $2.21\times 10^{-7}$ | [2,1,1] | ${\bf 2.21\times 10^{-7}}$ | $1.25\times 10^{-6}$ |
| spherical harmonics $(m=1,n=1)$ | ${\rm sph_harm}(1,1,x,y)$ | [2,2,1] | $7.86\times 10^{-4}$ | [2,3,2,1] | ${\bf 1.22\times 10^{-4}}$ | $6.70\times 10^{-4}$ |
| spherical harmonics $(m=0,n=2)$ | ${\rm sph_harm}(0,2,x,y)$ | [2,1,1] | $1.95\times 10^{-7}$ | [2,1,1] | ${\bf 1.95\times 10^{-7}}$ | $2.85\times 10^{-6}$ |
| spherical harmonics $(m=1,n=2)$ | ${\rm sph_harm}(1,2,x,y)$ | [2,2,1] | $4.70\times 10^{-4}$ | [2,2,1,1] | ${\bf 1.50\times 10^{-5}}$ | $1.84\times 10^{-3}$ |
| spherical harmonics $(m=2,n=2)$ | ${\rm sph_harm}(2,2,x,y)$ | [2,2,1] | $1.12\times 10^{-3}$ | [2,2,3,2,1] | ${\bf 9.45\times 10^{-5}}$ | $6.21\times 10^{-4}$ |

### 3.3 Feynman datasets

The setup in Section 3.1 is when we clearly know “true” KAN shapes. The setup in Section 3.2 is when we clearly do not know “true” KAN shapes. This part investigates a setup lying in the middle: Given the structure of the dataset, we may construct KANs by hand, but we are not sure if they are optimal. In this regime, it is interesting to compare human-constructed KANs and auto-discovered KANs via pruning (techniques in Section 2.5.1).

**Table 2: Feynman dataset**
| Feynman Eq. | Original Formula | Dimensionless formula | Variables | Human-constructed<br>KAN shape | Pruned<br>KAN shape<br>(smallest shape<br>that achieves<br>RMSE < $10^{-2}$) | Pruned<br>KAN shape<br>(lowest loss) | Human-constructed<br>KAN loss<br>(lowest test RMSE) | Pruned<br>KAN loss<br>(lowest test RMSE) | Unpruned<br>KAN loss<br>(lowest test RMSE) | MLP<br>loss<br>(lowest test RMSE) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| I.6.2 | ${\rm exp}(-\frac{\theta^{2}}{2\sigma^{2}})/\sqrt{2\pi\sigma^{2}}$ | ${\rm exp}(-\frac{\theta^{2}}{2\sigma^{2}})/\sqrt{2\pi\sigma^{2}}$ | $\theta,\sigma$ | [2,2,1,1] | [2,2,1] | [2,2,1,1] | $7.66\times 10^{-5}$ | ${\bf 2.86\times 10^{-5}}$ | $4.60\times 10^{-5}$ | $1.45\times 10^{-4}$ |
| I.6.2b | ${\rm exp}(-\frac{(\theta-\theta_{1})^{2}}{2\sigma^{2}})/\sqrt{2\pi\sigma^{2}}$ | ${\rm exp}(-\frac{(\theta-\theta_{1})^{2}}{2\sigma^{2}})/\sqrt{2\pi\sigma^{2}}$ | $\theta,\theta_{1},\sigma$ | [3,2,2,1,1] | [3,4,1] | [3,2,2,1,1] | $1.22\times 10^{-3}$ | ${\bf 4.45\times 10^{-4}}$ | $1.25\times 10^{-3}$ | $7.40\times 10^{-4}$ |
| I.9.18 | $\frac{Gm_{1}m_{2}}{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}$ | $\frac{a}{(b-1)^{2}+(c-d)^{2}+(e-f)^{2}}$ | $a,b,c,d,e,f$ | [6,4,2,1,1] | [6,4,1,1] | [6,4,1,1] | ${\bf 1.48\times 10^{-3}}$ | $8.62\times 10^{-3}$ | $6.56\times 10^{-3}$ | $1.59\times 10^{-3}$ |
| I.12.11 | $q(E_{f}+Bv{\rm sin}\theta)$ | $1+a{\rm sin}\theta$ | $a,\theta$ | [2,2,2,1] | [2,2,1] | [2,2,1] | $2.07\times 10^{-3}$ | $1.39\times 10^{-3}$ | $9.13\times 10^{-4}$ | ${\bf 6.71\times 10^{-4}}$ |
| I.13.12 | $Gm_{1}m_{2}(\frac{1}{r_{2}}-\frac{1}{r_{1}})$ | $a(\frac{1}{b}-1)$ | $a,b$ | [2,2,1] | [2,2,1] | [2,2,1] | $7.22\times 10^{-3}$ | $4.81\times 10^{-3}$ | $2.72\times 10^{-3}$ | ${\bf 1.42\times 10^{-3}}$ |
| I.15.3x | $\frac{x-ut}{\sqrt{1-(\frac{u}{c})^{2}}}$ | $\frac{1-a}{\sqrt{1-b^{2}}}$ | $a,b$ | [2,2,1,1] | [2,1,1] | [2,2,1,1,1] | $7.35\times 10^{-3}$ | $1.58\times 10^{-3}$ | $1.14\times 10^{-3}$ | ${\bf 8.54\times 10^{-4}}$ |
| I.16.6 | $\frac{u+v}{1+\frac{uv}{c^{2}}}$ | $\frac{a+b}{1+ab}$ | $a,b$ | [2,2,2,2,2,1] | [2,2,1] | [2,2,1] | $1.06\times 10^{-3}$ | $1.19\times 10^{-3}$ | $1.53\times 10^{-3}$ | ${\bf 6.20\times 10^{-4}}$ |
| I.18.4 | $\frac{m_{1}r_{1}+m_{2}r_{2}}{m_{1}+m_{2}}$ | $\frac{1+ab}{1+a}$ | $a,b$ | [2,2,2,1,1] | [2,2,1] | [2,2,1] | $3.92\times 10^{-4}$ | ${\bf 1.50\times 10^{-4}}$ | $1.32\times 10^{-3}$ | $3.68\times 10^{-4}$ |
| I.26.2 | ${\rm arcsin}(n{\rm sin}\theta_{2})$ | ${\rm arcsin}(n{\rm sin}\theta_{2})$ | $n,\theta_{2}$ | [2,2,2,1,1] | [2,2,1] | [2,2,2,1,1] | $1.22\times 10^{-1}$ | ${\bf 7.90\times 10^{-4}}$ | $8.63\times 10^{-4}$ | $1.24\times 10^{-3}$ |
| I.27.6 | $\frac{1}{\frac{1}{d_{1}}+\frac{n}{d_{2}}}$ | $\frac{1}{1+ab}$ | $a,b$ | [2,2,1,1] | [2,1,1] | [2,1,1] | $2.22\times 10^{-4}$ | ${\bf 1.94\times 10^{-4}}$ | $2.14\times 10^{-4}$ | $2.46\times 10^{-4}$ |
| I.29.16 | $\sqrt{x_{1}^{2}+x_{2}^{2}-2x_{1}x_{2}{\rm cos}(\theta_{1}-\theta_{2})}$ | $\sqrt{1+a^{2}-2a{\rm cos}(\theta_{1}-\theta_{2})}$ | $a,\theta_{1},\theta_{2}$ | [3,2,2,3,2,1,1] | [3,2,2,1] | [3,2,3,1] | $2.36\times 10^{-1}$ | $3.99\times 10^{-3}$ | ${\bf 3.20\times 10^{-3}}$ | $4.64\times 10^{-3}$ |
| I.30.3 | $I_{*,0}\frac{{\rm sin}^{2}(\frac{n\theta}{2})}{{\rm sin}^{2}(\frac{\theta}{2})}$ | $\frac{{\rm sin}^{2}(\frac{n\theta}{2})}{{\rm sin}^{2}(\frac{\theta}{2})}$ | $n,\theta$ | [2,3,2,2,1,1] | [2,4,3,1] | [2,3,2,3,1,1] | $3.85\times 10^{-1}$ | ${\bf 1.03\times 10^{-3}}$ | $1.11\times 10^{-2}$ | $1.50\times 10^{-2}$ |
| I.30.5 | ${\rm arcsin}(\frac{\lambda}{nd})$ | ${\rm arcsin}(\frac{a}{n})$ | $a,n$ | [2,1,1] | [2,1,1] | [2,1,1,1,1,1] | $2.23\times 10^{-4}$ | ${\bf 3.49\times 10^{-5}}$ | $6.92\times 10^{-5}$ | $9.45\times 10^{-5}$ |
| I.37.4 | $I_{*}=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}{\rm cos}\delta$ | $1+a+2\sqrt{a}{\rm cos}\delta$ | $a,\delta$ | [2,3,2,1] | [2,2,1] | [2,2,1] | $7.57\times 10^{-5}$ | ${\bf 4.91\times 10^{-6}}$ | $3.41\times 10^{-4}$ | $5.67\times 10^{-4}$ |
| I.40.1 | $n_{0}{\rm exp}(-\frac{mgx}{k_{b}T})$ | $n_{0}e^{-a}$ | $n_{0},a$ | [2,1,1] | [2,2,1] | [2,2,1,1,1,2,1] | $3.45\times 10^{-3}$ | $5.01\times 10^{-4}$ | ${\bf 3.12\times 10^{-4}}$ | $3.99\times 10^{-4}$ |
| I.44.4 | $nk_{b}T{\rm ln}(\frac{V_{2}}{V_{1}})$ | $n{\rm ln}a$ | $n,a$ | [2,2,1] | [2,2,1] | [2,2,1] | ${\bf 2.30\times 10^{-5}}$ | $2.43\times 10^{-5}$ | $1.10\times 10^{-4}$ | $3.99\times 10^{-4}$ |
| I.50.26 | $x_{1}({\rm cos}(\omega t)+\alpha{\rm cos}^{2}(wt))$ | ${\rm cos}a+\alpha{\rm cos}^{2}a$ | $a,\alpha$ | [2,2,3,1] | [2,3,1] | [2,3,2,1] | ${\bf 1.52\times 10^{-4}}$ | $5.82\times 10^{-4}$ | $4.90\times 10^{-4}$ | $1.53\times 10^{-3}$ |
| II.2.42 | $\frac{k(T_{2}-T_{1})A}{d}$ | $(a-1)b$ | $a,b$ | [2,2,1] | [2,2,1] | [2,2,2,1] | $8.54\times 10^{-4}$ | $7.22\times 10^{-4}$ | $1.22\times 10^{-3}$ | ${\bf 1.81\times 10^{-4}}$ |
| II.6.15a | $\frac{3}{4\pi\epsilon}\frac{p_{d}z}{r^{5}}\sqrt{x^{2}+y^{2}}$ | $\frac{1}{4\pi}c\sqrt{a^{2}+b^{2}}$ | $a,b,c$ | [3,2,2,2,1] | [3,2,1,1] | [3,2,1,1] | $2.61\times 10^{-3}$ | $3.28\times 10^{-3}$ | $1.35\times 10^{-3}$ | ${\bf 5.92\times 10^{-4}}$ |
| II.11.7 | $n_{0}(1+\frac{p_{d}E_{f}{\rm cos}\theta}{k_{b}T})$ | $n_{0}(1+a{\rm cos}\theta)$ | $n_{0},a,\theta$ | [3,3,3,2,2,1] | [3,3,1,1] | [3,3,1,1] | $7.10\times 10^{-3}$ | $8.52\times 10^{-3}$ | $5.03\times 10^{-3}$ | ${\bf 5.92\times 10^{-4}}$ |
| II.11.27 | $\frac{n\alpha}{1-\frac{n\alpha}{3}}\epsilon E_{f}$ | $\frac{n\alpha}{1-\frac{n\alpha}{3}}$ | $n,\alpha$ | [2,2,1,2,1] | [2,1,1] | [2,2,1] | $2.67\times 10^{-5}$ | $4.40\times 10^{-5}$ | ${\bf 1.43\times 10^{-5}}$ | $7.18\times 10^{-5}$ |
| II.35.18 | $\frac{n_{0}}{{\rm exp}(\frac{\mu_{m}B}{k_{b}T})+{\rm exp}(-\frac{\mu_{m}B}{k_{b}T})}$ | $\frac{n_{0}}{{\rm exp}(a)+{\rm exp}(-a)}$ | $n_{0},a$ | [2,1,1] | [2,1,1] | [2,1,1,1] | $4.13\times 10^{-4}$ | $1.58\times 10^{-4}$ | ${\bf 7.71\times 10^{-5}}$ | $7.92\times 10^{-5}$ |
| II.36.38 | $\frac{\mu_{m}B}{k_{b}T}+\frac{\mu_{m}\alpha M}{\epsilon c^{2}k_{b}T}$ | $a+\alpha b$ | $a,\alpha,b$ | [3,3,1] | [3,2,1] | [3,2,1] | $2.85\times 10^{-3}$ | ${\bf 1.15\times 10^{-3}}$ | $3.03\times 10^{-3}$ | $2.15\times 10^{-3}$ |
| II.38.3 | $\frac{YAx}{d}$ | $\frac{a}{b}$ | $a,b$ | [2,1,1] | [2,1,1] | [2,2,1,1,1] | $1.47\times 10^{-4}$ | ${\bf 8.78\times 10^{-5}}$ | $6.43\times 10^{-4}$ | $5.26\times 10^{-4}$ |
| III.9.52 | $\frac{p_{d}E_{f}}{h}\frac{{\rm sin}^{2}((\omega-\omega_{0})t/2)}{((\omega-\omega_{0})t/2)^{2}}$ | $a\frac{{\rm sin}^{2}(\frac{b-c}{2})}{(\frac{b-c}{2})^{2}}$ | $a,b,c$ | [3,2,3,1,1] | [3,3,2,1] | [3,3,2,1,1,1] | $4.43\times 10^{-2}$ | $3.90\times 10^{-3}$ | $2.11\times 10^{-2}$ | ${\bf 9.07\times 10^{-4}}$ |
| III.10.19 | $\mu_{m}\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}$ | $\sqrt{1+a^{2}+b^{2}}$ | $a,b$ | [2,1,1] | [2,1,1] | [2,1,2,1] | $2.54\times 10^{-3}$ | $1.18\times 10^{-3}$ | $8.16\times 10^{-4}$ | ${\bf 1.67\times 10^{-4}}$ |
| III.17.37 | $\beta(1+\alpha{\rm cos}\theta)$ | $\beta(1+\alpha{\rm cos}\theta)$ | $\alpha,\beta,\theta$ | [3,3,3,2,2,1] | [3,3,1] | [3,3,1] | $1.10\times 10^{-3}$ | $5.03\times 10^{-4}$ | ${\bf 4.12\times 10^{-4}}$ | $6.80\times 10^{-4}$ |

Feynman dataset. The Feynman dataset collects many physics equations from Feynman’s textbooks [, ]. For our purpose, we are interested in problems in the Feynman_no_units dataset that have at least 2 variables, since univariate problems are trivial for KANs (they simplify to 1D splines). A sample equation from the Feynman dataset is the relativisic velocity addition formula

$$ $\displaystyle f(u,v)=(u+v)/(1+uv).$ (3.1) $$

The dataset can be constructed by randomly drawing $u_{i}\in(-1,1)$, $v_{i}\in(-1,1)$, and computing $f_{i}=f(u_{i},v_{i})$. Given many tuples $(u_{i},v_{i},f_{i})$, a neural network is trained and aims to predict $f$ from $u$ and $v$. We are interested in (1) how well a neural network can perform on test samples; (2) how much we can learn about the structure of the problem from neural networks.

We compare four kinds of neural networks:

- (1)
Human-constructued KAN. Given a symbolic formula, we rewrite it in Kolmogorov-Arnold representations. For example, to multiply two numbers $x$ and $y$, we can use the identity $xy=\frac{(x+y)^{2}}{4}-\frac{(x-y)^{2}}{4}$, which corresponds to a $[2,2,1]$ KAN. The constructued shapes are listed in the “Human-constructed KAN shape” in Table 2.
- (2)
KANs without pruning. We fix the KAN shape to width 5 and depths are swept over {2,3,4,5,6}.
- (3)
KAN with pruning. We use the sparsification $(\lambda=10^{-2}\ {\rm or}\ 10^{-3})$ and the pruning technique from Section 2.5.1 to obtain a smaller KAN from a fixed-shape KAN from (2).
- (4)
MLPs with fixed width 5, depths swept in $\{2,3,4,5,6\}$, and activations chosen from $\{{\rm Tanh},{\rm ReLU},{\rm SiLU}\}$.

Each KAN is initialized to have $G=3$, trained with LBFGS, with increasing number of grid points every 200 steps to cover $G=\{3,5,10,20,50,100,200\}$. For each hyperparameter combination, we try 3 random seeds. For each dataset (equation) and each method, we report the results of the best model (minimal KAN shape, or lowest test loss) over random seeds and depths in Table 2. We find that MLPs and KANs behave comparably on average. For each dataset and each model family (KANs or MLPs), we plot the Pareto frontier in the plane spanned by the number of parameters and RMSE losses, shown in Figure D.1 in Appendix D. We conjecture that the Feynman datasets are too simple to let KANs make further improvements, in the sense that variable dependence is usually smooth or monotonic, which is in contrast to the complexity of special functions which often demonstrate oscillatory behavior.

Auto-discovered KANs are smaller than human-constructed ones. We report the pruned KAN shape in two columns of Table 2; one column is for the minimal pruned KAN shape that can achieve reasonable loss (i.e., test RMSE smaller than $10^{-2}$); the other column is for the pruned KAN that achieves lowest test loss. For completeness, we visualize all 54 pruned KANs in Appendix D (Figure D.2 and D.3). It is interesting to observe that auto-discovered KAN shapes (for both minimal and best) are usually smaller than our human constructions. This means that KA representations can be more efficient than we imagine. At the same time, this may make interpretability subtle because information is being squashed into a smaller space than what we are comfortable with.

Consider the relativistic velocity composition $f(u,v)=\frac{u+v}{1+uv}$, for example. Our construction is quite deep because we were assuming that multiplication of $u,v$ would use two layers (see Figure 4.1 (a)), inversion of $1+uv$ would use one layer, and multiplication of $u+v$ and $1/(1+uv)$ would use another two layers(^5^55Note that we cannot use the logarithmic construction for division, because $u$ and $v$ here might be negative numbers.), resulting a total of 5 layers. However, the auto-discovered KANs are only 2 layers deep! In hindsight, this is actually expected if we recall the rapidity trick in relativity: define the two “rapidities” $a\equiv{\rm arctanh}\ u$ and $b\equiv{\rm arctanh}\ v$. The relativistic composition of velocities are simple additions in rapidity space, i.e., $\frac{u+v}{1+uv}={\rm tanh}({\rm arctanh}\ u+{\rm arctanh}\ v)$, which can be realized by a two-layer KAN. Pretending we do not know the notion of rapidity in physics, we could potentially discover this concept right from KANs without trial-and-error symbolic manipulations. The interpretability of KANs which can facilitate scientific discovery is the main topic in Section 4.

### 3.4 Solving partial differential equations

Figure: Figure 3.3: The PDE example. We plot L2 squared and H1 squared losses between the predicted solution and ground truth solution. First and second: training dynamics of losses. Third and fourth: scaling laws of losses against the number of parameters. KANs converge faster, achieve lower losses, and have steeper scaling laws than MLPs.
Refer to caption: x5.png

We consider a Poisson equation with zero Dirichlet boundary data. For $\Omega=[-1,1]^{2}$, consider the PDE

$$ $\displaystyle u_{xx}+u_{yy}$ $\displaystyle=f\quad\text{in}\,\,\Omega\,,$ (3.2) $\displaystyle u$ $\displaystyle=0\quad\text{on}\,\,\partial\Omega\,.$ $$

We consider the data $f=-\pi^{2}(1+4y^{2})\sin(\pi x)\sin(\pi y^{2})+2\pi\sin(\pi x)\cos(\pi y^{2})$ for which $u=\sin(\pi x)\sin(\pi y^{2})$ is the true solution. We use the framework of physics-informed neural networks (PINNs) [, ] to solve this PDE, with the loss function given by

$$ $\text{loss}_{\text{pde}}=\alpha\text{loss}_{i}+\text{loss}_{b}\coloneqq\alpha\frac{1}{n_{i}}\sum_{i=1}^{n_{i}}|u_{xx}(z_{i})+u_{yy}(z_{i})-f(z_{i})|^{2}+\frac{1}{n_{b}}\sum_{i=1}^{n_{b}}u^{2}\,,$ $$

where we use $\text{loss}_{i}$ to denote the interior loss, discretized and evaluated by a uniform sampling of $n_{i}$ points $z_{i}=(x_{i},y_{i})$ inside the domain, and similarly we use $\text{loss}_{b}$ to denote the boundary loss, discretized and evaluated by a uniform sampling of $n_{b}$ points on the boundary. $\alpha$ is the hyperparameter balancing the effect of the two terms.

We compare the KAN architecture with that of MLPs using the same hyperparameters $n_{i}=10000$, $n_{b}=800$, and $\alpha=0.01$. We measure both the error in the $L^{2}$ norm and energy ($H^{1}$) norm and see that KAN achieves a much better scaling law with a smaller error, using smaller networks and fewer parameters; see Figure 3.3. A 2-Layer width-10 KAN is 100 times more accurate than a 4-Layer width-100 MLP ($10^{-7}$ vs $10^{-5}$ MSE) and 100 times more parameter efficient ($10^{2}$ vs $10^{4}$ parameters). Therefore we speculate that KANs might have the potential of serving as a good neural network representation for model reduction of PDEs. However, we want to note that our implementation of KANs are typically 10x slower than MLPs to train. The ground truth being a symbolic formula might be an unfair comparison for MLPs since KANs are good at representing symbolic formulas. In general, KANs and MLPs are good at representing different function classes of PDE solutions, which needs detailed future study to understand their respective boundaries.

### 3.5 Continual Learning

Figure: Figure 3.4: A toy continual learning problem. The dataset is a 1D regression task with 5 Gaussian peaks (top row). Data around each peak is presented sequentially (instead of all at once) to KANs and MLPs. KANs (middle row) can perfectly avoid catastrophic forgetting, while MLPs (bottom row) display severe catastrophic forgetting.
Refer to caption: x6.png

Catastrophic forgetting is a serious problem in current machine learning []. When a human masters a task and switches to another task, they do not forget how to perform the first task. Unfortunately, this is not the case for neural networks. When a neural network is trained on task 1 and then shifted to being trained on task 2, the network will soon forget about how to perform task 1. A key difference between artificial neural networks and human brains is that human brains have functionally distinct modules placed locally in space. When a new task is learned, structure re-organization only occurs in local regions responsible for relevant skills [, ], leaving other regions intact. Most artificial neural networks, including MLPs, do not have this notion of locality, which is probably the reason for catastrophic forgetting.

We show that KANs have local plasticity and can avoid catastrophic forgetting by leveraging the locality of splines. The idea is simple: since spline bases are local, a sample will only affect a few nearby spline coefficients, leaving far-away coefficients intact (which is desirable since far-away regions may have already stored information that we want to preserve). By contrast, since MLPs usually use global activations, e.g., ReLU/Tanh/SiLU etc., any local change may propagate uncontrollably to regions far away, destroying the information being stored there.

We use a toy example to validate this intuition. The 1D regression task is composed of 5 Gaussian peaks. Data around each peak is presented sequentially (instead of all at once) to KANs and MLPs, as shown in Figure 3.4 top row. KAN and MLP predictions after each training phase are shown in the middle and bottom rows. As expected, KAN only remodels regions where data is present on in the current phase, leaving previous regions unchanged. By contrast, MLPs remodels the whole region after seeing new data samples, leading to catastrophic forgetting.

Here we simply present our preliminary results on an extremely simple example, to demonstrate how one could possibly leverage locality in KANs (thanks to spline parametrizations) to reduce catastrophic forgetting. However, it remains unclear whether our method can generalize to more realistic setups, especially in high-dimensional cases where it is unclear how to define “locality”. In future work, We would also like to study how our method can be connected to and combined with SOTA methods in continual learning [, ].

## 4 KANs are interpretable

In this section, we show that KANs are interpretable and interactive thanks to the techniques we developed in Section 2.5. We want to test the use of KANs not only on synthetic tasks (Section 4.1 and 4.2), but also in real-life scientific research. We demonstrate that KANs can (re)discover both highly non-trivial relations in knot theory (Section 4.3) and phase transition boundaries in condensed matter physics (Section 4.4). KANs could potentially be the foundation model for AI + Science due to their accuracy (last section) and interpretability (this section).

### 4.1 Supervised toy datasets

Figure: Figure 4.1: KANs are interepretable for simple symbolic tasks
Refer to caption: figs/interpretable_examples.png

We first examine KANs’ ability to reveal the compositional structures in symbolic formulas. Six examples are listed below and their KANs are visualized in Figure 4.1. KANs are able to reveal the compositional structures present in these formulas, as well as learn the correct univariate functions.

- (a)
Multiplication $f(x,y)=xy$. A $[2,5,1]$ KAN is pruned to a $[2,2,1]$ KAN. The learned activation functions are linear and quadratic. From the computation graph, we see that the way it computes $xy$ is leveraging $2xy=(x+y)^{2}-(x^{2}+y^{2})$.
- (b)
Division of positive numbers $f(x,y)=x/y$. A $[2,5,1]$ KAN is pruned to a $[2,1,1]$ KAN. The learned activation functions are logarithmic and exponential functions, and the KAN is computing $x/y$ by leveraging the identity $x/y={\exp}({\log}x-{\log}y)$.
- (c)
Numerical to categorical. The task is to convert a real number in $[0,1]$ to its first decimal digit (as one hots), e.g., $0.0618\to[1,0,0,0,0,\cdots]$, $0.314\to[0,0,0,1,0,\cdots]$. Notice that activation functions are learned to be spikes located around the corresponding decimal digits.
- (d)
Special function $f(x,y)={\rm exp}(J_{0}(20x)+y^{2})$. One limitation of symbolic regression is that it will never find the correct formula of a special function if the special function is not provided as prior knowledge. KANs can learn special functions – the highly wiggly Bessel function $J_{0}(20x)$ is learned (numerically) by KAN.
- (e)
Phase transition $f(x_{1},x_{2},x_{3})={\rm tanh}(5(x_{1}^{4}+x_{2}^{4}+x_{3}^{4}-1))$. Phase transitions are of great interest in physics, so we want KANs to be able to detect phase transitions and to identify the correct order parameters. We use the tanh function to simulate the phase transition behavior, and the order parameter is the combination of the quartic terms of $x_{1},x_{2},x_{3}$. Both the quartic dependence and tanh dependence emerge after KAN training. This is a simplified case of a localization phase transition discussed in Section 4.4.
- (f)
Deeper compositions $f(x_{1},x_{2},x_{3},x_{4})=\sqrt{(x_{1}-x_{2})^{2}+(x_{3}-x_{4})^{2}}$. To compute this, we would need the identity function, squared function, and square root, which requires at least a three-layer KAN. Indeed, we find that a $[4,3,3,1]$ KAN can be auto-pruned to a $[4,2,1,1]$ KAN, which exactly corresponds to the computation graph we would expect.

More examples from the Feynman dataset and the special function dataset are visualized in Figure D.2, D.3, F.1, F.2 in Appendices D and F.

### 4.2 Unsupervised toy dataset

Often, scientific discoveries are formulated as supervised learning problems, i.e., given input variables $x_{1},x_{2},\cdots,x_{d}$ and output variable(s) $y$, we want to find an interpretable function $f$ such that $y\approx f(x_{1},x_{2},\cdots,x_{d})$. However, another type of scientific discovery can be formulated as unsupervised learning, i.e., given a set of variables $(x_{1},x_{2},\cdots,x_{d})$, we want to discover a structural relationship between the variables. Specifically, we want to find a non-zero $f$ such that

$$ $\displaystyle f(x_{1},x_{2},\cdots,x_{d})\approx 0.$ (4.1) $$

For example, consider a set of features $(x_{1},x_{2},x_{3})$ that satisfies $x_{3}={\rm exp}({\rm sin}(\pi x_{1})+x_{2}^{2})$. Then a valid $f$ is $f(x_{1},x_{2},x_{3})={\rm sin}(\pi x_{1})+x_{2}^{2}-{\rm log}(x_{3})=0$, implying that points of $(x_{1},x_{2},x_{3})$ form a 2D submanifold specified by $f=0$ instead of filling the whole 3D space.

If an algorithm for solving the unsupervised problem can be devised, it has a considerable advantage over the supervised problem, since it requires only the sets of features $S=(x_{1},x_{2},\cdots,x_{d})$. The supervised problem, on the other hand, tries to predict subsets of features in terms of the others, i.e. it splits $S=S_{\text{in}}\cup S_{\text{out}}$ into input and output features of the function to be learned. Without domain expertise to advise the splitting, there are $2^{d}-2$ possibilities such that $|S_{\text{in}}|>0$ and $|S_{\text{out}}|>0$. This exponentially large space of supervised problems can be avoided by using the unsupervised approach.
This unsupervised learning approach will be valuable to the knot dataset in Section 4.3. A Google Deepmind team [] manually chose signature to be the target variable, otherwise they would face this combinatorial problem described above.
This raises the question whether we can instead tackle the unsupervised learning directly. We present our method and a toy example below.

We tackle the unsupervised learning problem by turning it into a supervised learning problem on all of the $d$ features, without requiring the choice of a splitting. The essential idea is to learn a function $f(x_{1},\dots,x_{d})=0$ such that $f$ is not the 0-function. To do this, similar to contrastive learning, we define positive samples and negative samples: positive samples are feature vectors of real data. Negative samples are constructed by feature corruption. To ensure that the overall feature distribution for each topological invariant stays the same, we perform feature corruption by random permutation of each feature across the entire training set. Now we want to train a network $g$ such that $g(\mathbf{x}_{\rm real})=1$ and $g(\mathbf{x}_{\rm fake})=0$ which turns the problem into a supervised problem. However, remember that we originally want $f(\mathbf{x}_{\rm real})=0$ and $f(\mathbf{x}_{\rm fake})\neq 0$. We can achieve this by having $g=\sigma\circ f$ where $\sigma(x)={\rm exp}(-\frac{x^{2}}{2w^{2}})$ is a Gaussian function with a small width $w$, which can be conveniently realized by a KAN with shape $[...,1,1]$ whose last activation is set to be the Gaussian function $\sigma$ and all previous layers form $f$. Except for the modifications mentioned above, everything else is the same for supervised training.

Figure: Figure 4.2: Unsupervised learning of a toy task. KANs can identify groups of dependent variables, i.e., $(x_{1},x_{2},x_{3})$ and $(x_{4},x_{5})$ in this case.
Refer to caption: figs/unsupervised_toy.png

Now we demonstrate that the unsupervised paradigm works for a synthetic example. Let us consider a 6D dataset, where $(x_{1},x_{2},x_{3})$ are dependent variables such that $x_{3}=\exp(\sin(x_{1})+x_{2}^{2})$; $(x_{4},x_{5})$ are dependent variables with $x_{5}=x_{4}^{3}$; $x_{6}$ is independent of the other variables. In Figure 4.2, we show that for seed = 0, KAN reveals the functional dependence among $x_{1}$,$x_{2}$, and $x_{3}$; for another seed = 2024, KAN reveals the functional dependence between $x_{4}$ and $x_{5}$. Our preliminary results rely on randomness (different seeds) to discover different relations; in the future we would like to investigate a more systematic and more controlled way to discover a complete set of relations. Even so, our tool in its current status can provide insights for scientific tasks. We present our results with the knot dataset in Section 4.3.

### 4.3 Application to Mathematics: Knot Theory

Knot theory is a subject in low-dimensional topology that sheds light on topological aspects of three-manifolds and four-manifolds and has a variety of applications, including in biology and topological quantum computing. Mathematically, a knot $K$ is an embedding of $S^{1}$ into $S^{3}$. Two knots $K$ and $K^{\prime}$ are topologically equivalent if one can be deformed into the other via deformation of the ambient space $S^{3}$, in which case we write $[K]=[K^{\prime}]$. Some knots are topologically trivial, meaning that they can be smoothly deformed to a standard circle. Knots have a variety of deformation-invariant features $f$ called topological invariants, which may be used to show that two knots are topologically inequivalent, $[K]\neq[K^{\prime}]$ if $f(K)\neq f(K^{\prime})$. In some cases the topological invariants are geometric in nature. For instance, a hyperbolic knot $K$ has a knot complement $S^{3}\setminus K$ that admits a canonical hyperbolic metric $g$ such that $\text{vol}_{g}(K)$ is a topological invariant known as the hyperbolic volume. Other topological invariants are algebraic in nature, such as the Jones polynomial.

Given the fundamental nature of knots in mathematics and the importance of its applications, it is interesting to study whether ML can lead to new results. For instance, in [] reinforcement learning was utilized to establish ribbonness of certain knots, which ruled out many potential counterexamples to the smooth 4d Poincaré conjecture.

Supervised learning In [], supervised learning and human domain experts were utilized to arrive at a new theorem relating algebraic and geometric knot invariants. In this case, gradient saliency identified key invariants for the supervised problem, which led the domain experts to make a conjecture that was subsequently refined and proven. We study whether a KAN can achieve good interpretable results on the same problem, which predicts the signature of a knot. Their main results from studying the knot theory dataset are:

- (1)
They use network attribution methods to find that the signature $\sigma$ is mostly dependent on meridinal distance $\mu$ (real $\mu_{r}$, imag $\mu_{i}$) and longitudinal distance $\lambda$.
- (2)
Human scientists later identified that $\sigma$ has high correlation with the ${\rm slope}\equiv{\rm Re}(\frac{\lambda}{\mu})=\frac{\lambda\mu_{r}}{\mu_{r}^{2}+\mu_{i}^{2}}$ and derived a bound for $|2\sigma-{\rm slope}|$.

We show below that KANs not only rediscover these results with much smaller networks and much more automation, but also present some interesting new results and insights.

Figure: Figure 4.3: Knot dataset, supervised mode. With KANs, we rediscover Deepmind’s results that signature is mainly dependent on meridinal translation (real and imaginary parts).
Refer to caption: figs/math.png

To investigate (1), we treat 17 knot invariants as inputs and signature as outputs. Similar to the setup in [], signatures (which are even numbers) are encoded as one-hot vectors and networks are trained with cross-entropy loss. We find that an extremely small $[17,1,14]$ KAN is able to achieve $81.6\%$ test accuracy (while Deepmind’s 4-layer width-300 MLP achieves 78% test accuracy). The $[17,1,14]$ KAN ($G=3$, $k=3$) has $\approx 200$ parameters, while the MLP has $\approx 3\times 10^{5}$ parameters, shown in Table 3. It is remarkable that KANs can be both more accurate and much more parameter efficient than MLPs at the same time. In terms of interpretability, we scale the transparency of each activation according to its magnitude, so it becomes immediately clear which input variables are important without the need for feature attribution (see Figure 4.3 left): signature is mostly dependent on $\mu_{r}$, and slightly dependent on $\mu_{i}$ and $\lambda$, while dependence on other variables is small. We then train a $[3,1,14]$ KAN on the three important variables, obtaining test accuracy $78.2\%$. Our results have one subtle difference from results in []: they find that signature is mostly dependent on $\mu_{i}$, while we find that signature is mostly dependent on $\mu_{r}$. This difference could be due to subtle algorithmic choices, but has led us to carry out the following experiments: (a) ablation studies. We show that $\mu_{r}$ contributes more to accuracy than $\mu_{i}$ (see Figure 4.3): for example, $\mu_{r}$ alone can achieve $65.0\%$ accuracy, while $\mu_{i}$ alone can only achieve $43.8\%$ accuracy. (b) We find a symbolic formula (in Table 4) which only involves $\mu_{r}$ and $\lambda$, but can achieve $77.8\%$ test accuracy.

**Table 3: KANs can achieve better accuracy than MLPs with much fewer parameters in the signature classification problem. Soon after our preprint was first released, Prof. Shi Lab from Georgia tech discovered that an MLP with only 60 parameters is sufficient to achieve 80% accuracy (public but unpublished results). This is good news for AI + Science because this means perhaps many AI + Science tasks are not that computationally demanding than we might think (either with MLPs or with KANs), hence many new scientific discoveries are possible even on personal laptops.**
| Method | Architecture | Parameter Count | Accuracy |
| --- | --- | --- | --- |
| Deepmind’s MLP | 4 layer, width-300 | $3\times 10^{5}$ | $78.0\%$ |
| KANs | 2 layer, $[17,1,14]$ ($G=3$, $k=3$) | $2\times 10^{2}$ | $81.6\%$ |

To investigate (2), i.e., obtain the symbolic form of $\sigma$, we formulate the problem as a regression task. Using auto-symbolic regression introduced in Section 2.5.1, we can convert a trained KAN into symbolic formulas. We train KANs with shapes $[3,1]$, $[3,1,1]$, $[3,2,1]$, whose corresponding symbolic formulas are displayed in Table 4 B-D. It is clear that by having a larger KAN, both accuracy and complexity increase. So KANs provide not just a single symbolic formula, but a whole Pareto frontier of formulas, trading off simplicity and accuracy. However, KANs need additional inductive biases to further simplify these equations to rediscover the formula from [] (Table 4 A). We have tested two scenarios: (1) in the first scenario, we assume the ground truth formula has a multi-variate Pade representation (division of two multi-variate Taylor series). We first train $[3,2,1]$ and then fit it to a Pade representation. We can obtain Formula E in Table 4, which bears similarity with Deepmind’s formula. (2) We hypothesize that the division is not very interpretable for KANs, so we train two KANs (one for the numerator and the other for the denominator) and divide them manually. Surprisingly, we end up with the formula F (in Table 4) which only involves $\mu_{r}$ and $\lambda$, although $\mu_{i}$ is also provided but ignored by KANs.

**Table 4: Symbolic formulas of signature as a function of meridinal translation $\mu$ (real $\mu_{r}$, imag $\mu_{i}$) and longitudinal translation $\lambda$. In [], formula A was discovered by human scientists inspired by neural network attribution results. Formulas B-F are auto-discovered by KANs. KANs can trade-off between simplicity and accuracy (B, C, D). By adding more inductive biases, KAN is able to discover formula E which is not too dissimilar from formula A. KANs also discovered a formula F which only involves two variables ($\mu_{r}$ and $\lambda$) instead of all three variables, with little sacrifice in accuracy.**
| Id | Formula | Discovered by | test<br>acc | $r^{2}$ with<br>Signature | $r^{2}$ with DM<br>formula |
| --- | --- | --- | --- | --- | --- |
| A | $\frac{\lambda\mu_{r}}{(\mu_{r}^{2}+\mu_{i}^{2})}$ | Human (DM) | 83.1% | 0.946 | 1 |
| B | $-0.02{\rm sin}(4.98\mu_{i}+0.85)+0.08|4.02\mu_{r}+6.28|-0.52-0.04e^{-0.88(1-0.45\lambda)^{2}}$ | $[3,1]$ KAN | 62.6% | 0.837 | 0.897 |
| C | $0.17{\rm tan}(-1.51+0.1e^{-1.43(1-0.4\mu_{i})^{2}+0.09e^{-0.06(1-0.21\lambda)^{2}}}+1.32e^{-3.18(1-0.43\mu_{r})^{2}})$ | $[3,1,1]$ KAN | 71.9% | 0.871 | 0.934 |
| D | $-0.09+1.04{\rm exp}(-9.59(-0.62{\rm sin}(0.61\mu_{r}+7.26))-0.32{\rm tan}(0.03\lambda-6.59)+1-0.11e^{-1.77(0.31-\mu_{i})^{2})^{2}}-1.09e^{-7.6(0.65(1-0.01\lambda)^{3}}+0.27{\rm atan}(0.53\mu_{i}-0.6)+0.09+{\rm exp}(-2.58(1-0.36\mu_{r})^{2}))$ | $[3,2,1]$ KAN | 84.0% | 0.947 | 0.997 |
| E | $\frac{4.76\lambda\mu_{r}}{3.09\mu_{i}+6.05\mu_{r}^{2}+3.54\mu_{i}^{2}}$ | [3,2,1] KAN<br>+ Pade approx | $82.8\%$ | 0.946 | 0.997 |
| F | $\frac{2.94-2.92(1-0.10\mu_{r})^{2}}{0.32(0.18-\mu_{r})^{2}+5.36(1-0.04\lambda)^{2}+0.50}$ | $[3,1]$ KAN/$[3,1]$ KAN | 77.8% | 0.925 | 0.977 |

So far, we have rediscovered the main results from []. It is remarkable to see that KANs made this discovery very intuitive and convenient. Instead of using feature attribution methods (which are great methods), one can instead simply stare at visualizations of KANs. Moreover, automatic symbolic regression also makes the discovery of symbolic formulas much easier.

In the next part, we propose a new paradigm of “AI for Math” not included in the Deepmind paper, where we aim to use KANs’ unsupervised learning mode to discover more relations (besides signature) in knot invariants.

Figure: Figure 4.4: Knot dataset, unsupervised mode. With KANs, we rediscover three mathematical relations in the knot dataset.
Refer to caption: figs/knot_unsupervised.png

Unsupervised learning As we mentioned in Section 4.2, unsupervised learning is the setup that is more promising since it avoids manual partition of input and output variables which have combinatorially many possibilities. In the unsupervised learning mode, we treat all 18 variables (including signature) as inputs such that they are on the same footing. Knot data are positive samples, and we randomly shuffle features to obtain negative samples.
An $[18,1,1]$ KAN is trained to classify whether a given feature vector belongs to a positive sample (1) or a negative sample (0). We manually set the second layer activation to be the Gaussian function with a peak one centered at zero, so positive samples will have activations at (around) zero, implicitly giving a relation among knot invariants $\sum_{i=1}^{18}g_{i}(x_{i})=0$ where $x_{i}$ stands for a feature (invariant), and $g_{i}$ is the corresponding activation function which can be readily read off from KAN diagrams. We train the KANs with $\lambda=\{10^{-2},10^{-3}\}$ to favor sparse combination of inputs, and ${\rm seed}=\{0,1,\cdots,99\}$. All 200 networks can be grouped into three clusters, with representative KANs displayed in Figure 4.4. These three groups of dependent variables are:

- (1)
The first group of dependent variables is signature, real part of meridinal distance, and longitudinal distance (plus two other variables which can be removed because of (3)). This is the signature dependence studied above, so it is very interesting to see that this dependence relation is rediscovered again in the unsupervised mode.
- (2)
The second group of variables involve cusp volume $V$, real part of meridinal translation $\mu_{r}$ and longitudinal translation $\lambda$. Their activations all look like logarithmic functions (which can be verified by the implied symbolic functionality in Section 2.5.1). So the relation is $-\log V+\log\mu_{r}+\log\lambda=0$ which is equivalent to $V=\mu_{r}\lambda$, which is true by definition. It is, however, reassuring that we discover this relation without any prior knowledge.
- (3)
The third group of variables includes the real part of short geodesic $g_{r}$ and injectivity radius. Their activations look qualitatively the same but differ by a minus sign, so it is conjectured that these two variables have a linear correlation. We plot 2D scatters, finding that $2r$ upper bounds $g_{r}$, which is also a well-known relation [].

It is interesting that KANs’ unsupervised mode can rediscover several known mathematical relations. The good news is that the results discovered by KANs are probably reliable; the bad news is that we have not discovered anything new yet. It is worth noting that we have chosen a shallow KAN for simple visualization, but deeper KANs can probably find more relations if they exist. We would like to investigate how to discover more complicated relations with deeper KANs in future work.

### 4.4 Application to Physics: Anderson localization

Anderson localization is the fundamental phenomenon in which disorder in a quantum system leads to the localization of electronic wave functions, causing all transport to be ceased []. In one and two dimensions, scaling arguments show that all electronic eigenstates are exponentially localized for an infinitesimal amount of random disorder [, ]. In contrast, in three dimensions, a critical energy forms a phase boundary that separates the extended states from the localized states, known as a mobility edge. The understanding of these mobility edges is crucial for explaining various fundamental phenomena such as the metal-insulator transition in solids [], as well as localization effects of light in photonic devices [, , , , ]. It is therefore necessary to develop microscopic models that exhibit mobility edges to enable detailed investigations. Developing such models is often more practical in lower dimensions, where introducing quasiperiodicity instead of random disorder can also result in mobility edges that separate localized and extended phases. Furthermore, experimental realizations of analytical mobility edges can help resolve the debate on localization in interacting systems [, ]. Indeed, several recent studies have focused on identifying such models and deriving exact analytic expressions for their mobility edges [, , , , , , ].

Here, we apply KANs to numerical data generated from quasiperiodic tight-binding models to extract their mobility edges. In particular, we examine three classes of models: the Mosaic model (MM) [], the generalized Aubry-André model (GAAM) [] and the modified Aubry-André model (MAAM) []. For the MM, we testify KAN’s ability to accurately extract mobility edge as a 1D function of energy. For the GAAM, we find that the formula obtained from a KAN closely matches the ground truth. For the more complicated MAAM, we demonstrate yet another example of the symbolic interpretability of this framework. A user can simplify the complex expression obtained from KANs (and corresponding symbolic formulas) by means of a “collaboration” where the human generates hypotheses to obtain a better match (e.g., making an assumption of the form of certain activation function), after which KANs can carry out quick hypotheses testing.

To quantify the localization of states in these models, the inverse participation ratio (IPR) is commonly used. The IPR for the $k^{th}$ eigenstate, $\psi^{(k)}$, is given by

$$ $\displaystyle\text{IPR}_{k}=\frac{\sum_{n}|\psi^{(k)}_{n}|^{4}}{\left(\sum_{n}|\psi^{(k)}_{n}|^{2}\right)^{2}}$ (4.2) $$

where the sum runs over the site index. Here, we use the related measure of localization – the fractal dimension of the states, given by

$$ $\displaystyle D_{k}=-\frac{\log(\text{IPR}_{k})}{\log(N)}$ (4.3) $$

where $N$ is the system size. $D_{k}=0(1)$ indicates localized (extended) states.

Mosaic Model (MM) We first consider a class of tight-binding models defined by the Hamiltonian []

$$ $\displaystyle H=t\sum_{n}\left(c^{\dagger}_{n+1}c_{n}+\text{H.c.}\right)+\sum_{n}V_{n}(\lambda,\phi)c^{\dagger}_{n}c_{n},$ (4.4) $$

where $t$ is the nearest-neighbor coupling, $c_{n}(c^{\dagger}_{n})$ is the annihilation (creation) operator at site $n$ and the potential energy $V_{n}$ is given by

$$ $\displaystyle V_{n}(\lambda,\phi)=\begin{cases}\lambda\cos(2\pi nb+\phi)&j=m\kappa\\ 0,&\text{otherwise,}\end{cases}$ (4.5) $$

To introduce quasiperiodicity, we set $b$ to be irrational (in particular, we choose $b$ to be the golden ratio $\frac{1+\sqrt{5}}{2}$). $\kappa$ is an integer and the quasiperiodic potential occurs with interval $\kappa$. The energy ($E$) spectrum for this model generically contains extended and localized regimes separated by a mobility edge. Interestingly, a unique feature found here is that the mobility edges are present for an arbitrarily strong quasiperiodic potential (i.e. there are always extended states present in the system that co-exist with localized ones).

The mobility edge can be described by $g(\lambda,E)\equiv\lambda-|f_{\kappa}(E)|=0$. $g(\lambda,E)$ > 0 and $g(\lambda,E)$ < 0 correspond to localized and extended phases, respectively. Learning the mobility edge therefore hinges on learning the “order parameter” $g(\lambda,E)$. Admittedly, this problem can be tackled by many other theoretical methods for this class of models [], but we will demonstrate below that our KAN framework is ready and convenient to take in assumptions and inductive biases from human users.

Let us assume a hypothetical user Alice, who is a new PhD student in condensed matter physics, and she is provided with a $[2,1]$ KAN as an assistant for the task. Firstly, she understands that this is a classification task, so it is wise to set the activation function in the second layer to be sigmoid by using the fix_symbolic functionality. Secondly, she realizes that learning the whole 2D function $g(\lambda,E)$ is unnecessary because in the end she only cares about $\lambda=\lambda(E)$ determined by $g(\lambda,E)=0$. In so doing, it is reasonable to assume $g(\lambda,E)=\lambda-h(E)=0$. Alice simply sets the activation function of $\lambda$ to be linear by again using the fix_symbolic functionality. Now Alice trains the KAN network and conveniently obtains the mobility edge, as shown in Figure 4.5. Alice can get both intuitive qualitative understanding (bottom) and quantitative results (middle), which well match the ground truth (top).

Figure: Figure 4.5: Results for the Mosaic Model. Top: phase diagram. Middle and Bottom: KANs can obtain both qualitative intuition (bottom) and extract quantitative results (middle). $\varphi=\frac{1+\sqrt{5}}{2}$ is the golden ratio.
Refer to caption: figs/mosaic_results.png

Generalized Andre-Aubry Model (GAAM) We next consider a class of tight-binding models defined by the Hamiltonian []

$$ $\displaystyle H=t\sum_{n}\left(c^{\dagger}_{n+1}c_{n}+\text{H.c.}\right)+\sum_{n}V_{n}(\alpha,\lambda,\phi)c^{\dagger}_{n}c_{n},$ (4.6) $$

where $t$ is the nearest-neighbor coupling, $c_{n}(c^{\dagger}_{n})$ is the annihilation (creation) operator at site $n$ and the potential energy $V_{n}$ is given by

$$ $\displaystyle V_{n}(\alpha,\lambda,\phi)=2\lambda\frac{\cos(2\pi nb+\phi)}{1-\alpha\cos(2\pi nb+\phi)},$ (4.7) $$

which is smooth for $\alpha\in(-1,1)$. To introduce quasiperiodicity, we again set $b$ to be irrational (in particular, we choose $b$ to be the golden ratio). As before, we would like to obtain an expression for the mobility edge. For these models, the mobility edge is given by the closed form expression [, ],

$$ $\displaystyle\alpha E=2(t-\lambda).$ (4.8) $$

We randomly sample the model parameters: $\phi$, $\alpha$ and $\lambda$ (setting the energy scale $t=1$) and calculate the energy eigenvalues as well as the fractal dimension of the corresponding eigenstates, which forms our training dataset.

**Table 5: Symbolic formulas for two systems GAAM and MAAM, ground truth ones and KAN-discovered ones.**
| System | Origin | Mobility Edge Formula | Accuracy |
| --- | --- | --- | --- |
| GAAM | Theory | $\alpha E+2\lambda-2=0$ | 99.2% |
| KAN auto | $\cancel{1.52E^{2}}+21.06\alpha E+\cancel{0.66E}+\cancel{3.55\alpha^{2}}+\cancel{0.91\alpha}+45.13\lambda-54.45=0$ | 99.0% |  |
| MAAM | Theory | $E+{\rm exp}(p)-\lambda{\rm cosh}p=0$ | 98.6% |
| KAN auto | $13.99{\rm sin}(0.28{\rm sin}(0.87\lambda+2.22)-0.84{\rm arctan}(0.58E-0.26)+0.85{\rm arctan}(0.94p+0.13)-8.14)-16.74+43.08{\rm exp}(-0.93(0.06(0.13-p)^{2}-0.27{\rm tanh}(0.65E+0.25)+0.63{\rm arctan}(0.54\lambda-0.62)+1)^{2})=0$ | 97.1% |  |
| KAN man (step 2) + auto | $4.19(0.28{\rm sin}(0.97\lambda+2.17)-0.77{\rm arctan}(0.83E-0.19)+{\rm arctan}(0.97p+0.15)-0.35)^{2}-28.93+39.27{\rm exp}(-0.6(0.28{\rm cosh}^{2}(0.49p-0.16)-0.34{\rm arctan}(0.65E+0.51)+0.83{\rm arctan}(0.54\lambda-0.62)+1)^{2})=0$ | 97.7% |  |
| KAN man (step 3) + auto | $-4.63E-10.25(-0.94{\rm sin}(0.97\lambda-6.81)+{\rm tanh}(0.8p-0.45)+0.09)^{2}+11.78{\rm sin}(0.76p-1.41)+22.49{\rm arctan}(1.08\lambda-1.32)+31.72=0$ | 97.7% |  |
| KAN man (step 4A) | $6.92E-6.23(-0.92\lambda-1)^{2}+2572.45(-0.05\lambda+0.95{\rm cosh}(0.11p+0.4)-1)^{2}-12.96{\rm cosh}^{2}(0.53p+0.16)+19.89=0$ | 96.6% |  |
| KAN man (step 4B) | $7.25E-8.81(-0.83\lambda-1)^{2}-4.08(-p-0.04)^{2}+12.71(-0.71\lambda+(0.3p+1)^{2}-0.86)^{2}+10.29=0$ | 95.4% |  |

Here the “order parameter” to be learned is $g(\alpha,E,\lambda,\phi)=\alpha E+2(\lambda-1)$ and mobility edge corresponds to $g=0$. Let us again assume that Alice wants to figure out the mobility edge but only has access to IPR or fractal dimension data, so she decides to use KAN to help her with the task. Alice wants the model to be as small as possible, so she could either start from a large model and use auto-pruning to get a small model, or she could guess a reasonable small model based on her understanding of the complexity of the given problem. Either way, let us assume she arrives at a $[4,2,1,1]$ KAN. First, she sets the last activation to be sigmoid because this is a classification problem. She trains her KAN with some sparsity regularization to accuracy 98.7% and visualizes the trained KAN in Figure 4.6 (a) step 1. She observes that $\phi$ is not picked up on at all, which makes her realize that the mobility edge is independent of $\phi$ (agreeing with Eq. (4.8)). In addition, she observes that almost all other activation functions are linear or quadratic, so she turns on automatic symbolic snapping, constraining the library to be only linear or quadratic. After that, she immediately gets a network which is already symbolic (shown in Figure 4.6 (a) step 2), with comparable (even slightly better) accuracy 98.9%. By using symbolic_formula functionality, Alice conveniently gets the symbolic form of $g$, shown in Table 5 GAAM-KAN auto (row three). Perhaps she wants to cross out some small terms and snap coefficient to small integers, which takes her close to the true answer.

This hypothetical story for Alice would be completely different if she is using a symbolic regression method. If she is lucky, SR can return the exact correct formula. However, the vast majority of the time SR does not return useful results and it is impossible for Alice to “debug” or interact with the underlying process of symbolic regression. Furthermore, Alice may feel uncomfortable/inexperienced to provide a library of symbolic terms as prior knowledge to SR before SR is run. By constrast in KANs, Alice does not need to put any prior information to KANs. She can first get some clues by staring at a trained KAN and only then it is her job to decide which hypothesis she wants to make (e.g., “all activations are linear or quadratic”) and implement her hypothesis in KANs. Although it is not likely for KANs to return the correct answer immediately, KANs will always return something useful, and Alice can collaborate with it to refine the results.

Modified Andre-Aubry Model (MAAM) The last class of models we consider is defined by the Hamiltonian []

$$ $\displaystyle H=\sum_{n\neq n^{\prime}}te^{-p|n-n^{\prime}|}\left(c^{\dagger}_{n}c_{n^{\prime}}+\text{H.c.}\right)+\sum_{n}V_{n}(\lambda,\phi)c^{\dagger}_{n}c_{n},$ (4.9) $$

where $t$ is the strength of the exponentially decaying coupling in space, $c_{n}(c^{\dagger}_{n})$ is the annihilation (creation) operator at site $n$ and the potential energy $V_{n}$ is given by

$$ $\displaystyle V_{n}(\lambda,\phi)=\lambda\cos(2\pi nb+\phi),$ (4.10) $$

As before, to introduce quasiperiodicity, we set $b$ to be irrational (the golden ratio). For these models, the mobility edge is given by the closed form expression [],

$$ $\displaystyle\lambda\cosh(p)=E+t=E+t_{1}{\rm exp}(p)$ (4.11) $$

where we define $t_{1}\equiv t{\rm exp}(-p)$ as the nearest neighbor hopping strength, and we set $t_{1}=1$ below.

Let us assume Alice wants to figure out the mobility edge for MAAM. This task is more complicated and requires more human wisdom. As in the last example, Alice starts from a $[4,2,1,1]$ KAN and trains it but gets an accuracy around 75% which is less than acceptable. She then chooses a larger $[4,3,1,1]$ KAN and successfully gets 98.4% which is acceptable (Figure 4.6 (b) step 1). Alice notices that $\phi$ is not picked up on by KANs, which means that the mobility edge is independent of the phase factor $\phi$ (agreeing with Eq. (4.11)). If Alice turns on the automatic symbolic regression (using a large library consisting of exp, tanh etc.), she would get a complicated formula in Tabel 5-MAAM-KAN auto, which has 97.1% accuracy. However, if Alice wants to find a simpler symbolic formula, she will want to use the manual mode where she does the symbolic snapping by herself. Before that she finds that the $[4,3,1,1]$ KAN after training can then be pruned to be $[4,2,1,1]$, while maintaining $97.7\%$ accuracy (Figure 4.6 (b)). Alice may think that all activation functions except those dependent on $p$ are linear or quadratic and snap them to be either linear or quadratic manually by using fix_symbolic. After snapping and retraining, the updated KAN is shown in Figure 4.6 (c) step 3, maintaining $97.7\%$ accuracy. From now on, Alice may make two different choices based on her prior knowledge. In one case, Alice may have guessed that the dependence on $p$ is ${\rm cosh}$, so she sets the activations of $p$ to be ${\rm cosh}$ function. She retrains KAN and gets 96.9% accuracy (Figure 4.6 (c) Step 4A). In another case, Alice does not know the ${\rm cosh}\ p$ dependence, so she pursues simplicity and again assumes the functions of $p$ to be quadratic. She retrains KAN and gets 95.4% accuracy (Figure 4.6 (c) Step 4B). If she tried both, she would realize that ${\rm cosh}$ is better in terms of accuracy, while quadratic is better in terms of simplicity. The formulas corresponding to these steps are listed in Table 5. It is clear that the more manual operations are done by Alice, the simpler the symbolic formula is (which slight sacrifice in accuracy). KANs have a “knob" that a user can tune to trade-off between simplicity and accuracy (sometimes simplicity can even lead to better accuracy, as in the GAAM case).

Figure: Figure 4.6: Human-KAN collaboration to discover mobility edges of GAAM and MAAM. The human user can choose to be lazy (using the auto mode) or more involved (using the manual mode). More details in text.
Refer to caption: figs/al_complex.png

## 5 Related works

Kolmogorov-Arnold theorem and neural networks. The connection between the Kolmogorov-Arnold theorem (KAT) and neural networks is not new in the literature  [, , , , , , , , , ], but the pathological behavior of inner functions makes KAT appear unpromising in practice []. Most of these prior works stick to the original 2-layer width-($2n+1$) networks, which were limited in expressive power and many of them are even predating back-propagation. Therefore, most studies were built on theories with rather limited or artificial toy experiments. More broadly speaking, KANs are also somewhat related to generalized additive models (GAMs) [], graph neural networks [] and kernel machines []. The connections are intriguing and fundamental but might be out of the scope of the current paper. Our contribution lies in generalizing the Kolmogorov network to arbitrary widths and depths, revitalizing and contexualizing them in today’s deep learning stream, as well as highlighting its potential role as a foundation model for AI + Science.

Neural Scaling Laws (NSLs). NSLs are the phenomena where test losses behave as power laws against model size, data, compute etc [, , , , , , , ]. The origin of NSLs still remains mysterious, but competitive theories include intrinsic dimensionality [], quantization of tasks [], resource theory [], random features [], compositional sparsity [], and maximu arity []. This paper contributes to this space by showing that a high-dimensional function can surprisingly scale as a 1D function (which is the best possible bound one can hope for) if it has a smooth Kolmogorov-Arnold representation. Our paper brings fresh optimism to neural scaling laws, since it promises the fastest scaling exponent ever. We have shown in our experiments that this fast neural scaling law can be achieved on synthetic datasets, but future research is required to address the question whether this fast scaling is achievable for more complicated tasks (e.g., language modeling): Do KA representations exist for general tasks? If so, does our training find these representations in practice?

Mechanistic Interpretability (MI). MI is an emerging field that aims to mechanistically understand the inner workings of neural networks [, , , , , , , , ]. MI research can be roughly divided into passive and active MI research. Most MI research is passive in focusing on understanding existing neural networks trained with standard methods. Active MI research attempts to achieve interpretability by designing intrinsically interpretable architectures or developing training methods to explicitly encourage interpretability [, ]. Our work lies in the second category, where the model and training method are by design interpretable.

Learnable activations. The idea of learnable activations in neural networks is not new in machine learning. Trainable activations functions are learned in a differentiable way [, , , ] or searched in a discrete way []. Activation function are parametrized as polynomials [], splines [, , ], sigmoid linear unit [], or neural networks []. KANs use B-splines to parametrize their activation functions. We also present our preliminary results on learnable activation networks (LANs), whose properties lie between KANs and MLPs and their results are deferred to Appendix B to focus on KANs in the main paper.

Symbolic Regression. There are many off-the-shelf symbolic regression methods based on genetic algorithms (Eureka [], GPLearn [], PySR []), neural-network based methods (EQL [], OccamNet []), physics-inspired method (AI Feynman [, ]), and reinforcement learning-based methods []. KANs are most similar to neural network-based methods, but differ from previous works in that our activation functions are continuously learned before symbolic snapping rather than manually fixed [, ].

Physics-Informed Neural Networks (PINNs) and Physics-Informed Neural Operators (PINOs).
In Subsection 3.4, we demonstrate that KANs can replace the paradigm of using MLPs for imposing PDE loss when solving PDEs. We refer to Deep Ritz Method [], PINNs [, , ] for PDE solving, and Fourier Neural operator [], PINOs [, , ], DeepONet [] for operator learning methods learning the solution map. There is potential to replace MLPs with KANs in all the aforementioned networks.

AI for Mathematics. As we saw in Subsection 4.3, AI has recently been applied to several problems in Knot theory, including detecting whether a knot is the unknot [, ] or a ribbon knot [], and predicting knot invariants and uncovering relations among them [, , , ]. For a summary of data science applications to datasets in mathematics and theoretical physics see e.g. [, ], and for ideas how to obtain rigorous results from ML techniques in these fields, see [].

## 6 Discussion

In this section, we discuss KANs’ limitations and future directions from the perspective of mathematical foundation, algorithms and applications.

Mathematical aspects: Although we have presented preliminary mathematical analysis of KANs (Theorem 2.1), our mathematical understanding of them is still very limited. The Kolmogorov-Arnold representation theorem has been studied thoroughly in mathematics, but the theorem corresponds to KANs with shape $[n,2n+1,1]$, which is a very restricted subclass of KANs. Does our empirical success with deeper KANs imply something fundamental in mathematics? An appealing generalized Kolmogorov-Arnold theorem could define “deeper” Kolmogorov-Arnold representations beyond depth-2 compositions, and potentially relate smoothness of activation functions to depth. Hypothetically, there exist functions which cannot be represented smoothly in the original (depth-2) Kolmogorov-Arnold representations, but might be smoothly represented with depth-3 or beyond. Can we use this notion of “Kolmogorov-Arnold depth” to characterize function classes?

Algorithmic aspects: We discuss the following:

- (1)
Accuracy. Multiple choices in architecture design and training are not fully investigated so alternatives can potentially further improve accuracy. For example, spline activation functions might be replaced by radial basis functions or other local kernels. Adaptive grid strategies can be used.
- (2)
Efficiency. One major reason why KANs run slowly is because different activation functions cannot leverage batch computation (large data through the same function). Actually, one can interpolate between activation functions being all the same (MLPs) and all different (KANs), by grouping activation functions into multiple groups (“multi-head”), where members within a group share the same activation function.
- (3)
Hybrid of KANs and MLPs. KANs have two major differences compared to MLPs:
(i)
activation functions are on edges instead of on nodes,
(ii)
activation functions are learnable instead of fixed.
Which change is more essential to explain KAN’s advantage? We present our preliminary results in Appendix B where we study a model which has (ii), i.e., activation functions are learnable (like KANs), but not (i), i.e., activation functions are on nodes (like MLPs). Moreover, one can also construct another model with fixed activations (like MLPs) but on edges (like KANs).
- (4)
Adaptivity. Thanks to the intrinsic locality of spline basis functions, we can introduce adaptivity in the design and training of KANs to enhance both accuracy and efficiency: see the idea of multi-level training like multigrid methods as in [, ], or domain-dependent basis functions like multiscale methods as in [].

Application aspects: We have presented some preliminary evidences that KANs are more effective than MLPs in science-related tasks, e.g., fitting physical equations and PDE solving. We would like to apply KANs to solve Navier-Stokes equations, density functional theory, or any other tasks that can be formulated as regression or PDE solving. We would also like to apply KANs to machine-learning-related tasks, which would require integrating KANs into current architectures, e.g., transformers – one may propose “kansformers” which replace MLPs by KANs in transformers.

KAN as a “language model” for AI + Science The reason why large language models are so transformative is because they are useful to anyone who can speak natural language. The language of science is functions. KANs are composed of interpretable functions, so when a human user stares at a KAN, it is like communicating with it using the language of functions. This paragraph aims to promote the AI-Scientist-Collaboration paradigm rather than our specific tool KANs. Just like people use different languages to communicate, we expect that in the future KANs will be just one of the languages for AI + Science, although KANs will be one of the very first languages that would enable AI and human to communicate. However, enabled by KANs, the AI-Scientist-Collaboration paradigm has never been this easy and convenient, which leads us to rethink the paradigm of how we want to approach AI + Science: Do we want AI scientists, or do we want AI that helps scientists? The intrinsic difficulty of (fully automated) AI scientists is that it is hard to make human preferences quantitative, which would codify human preferences into AI objectives. In fact, scientists in different fields may feel differently about which functions are simple or interpretable. As a result, it is more desirable for scientists to have an AI that can speak the scientific language (functions) and can conveniently interact with inductive biases of individual scientist(s) to adapt to a specific scientific domain.

Final takeaway: Should I use KANs or MLPs?

Currently, the biggest bottleneck of KANs lies in its slow training. KANs are usually 10x slower than MLPs, given the same number of parameters. We should be honest that we did not try hard to optimize KANs’ efficiency though, so we deem KANs’ slow training more as an engineering problem to be improved in the future rather than a fundamental limitation. If one wants to train a model fast, one should use MLPs. In other cases, however, KANs should be comparable or better than MLPs, which makes them worth trying. The decision tree in Figure 6.1 can help decide when to use a KAN. In short, if you care about interpretability and/or accuracy, and slow training is not a major concern, we suggest trying KANs, at least for small-scale AI + Science problems.

Figure: Figure 6.1: Should I use KANs or MLPs?
Refer to caption: figs/decision_tree.png

## Acknowledgement

We would like to thank Mikail Khona, Tomaso Poggio, Pingchuan Ma, Rui Wang, Di Luo, Sara Beery, Catherine Liang, Yiping Lu, Nicholas H. Nelsen, Nikola Kovachki, Jonathan W. Siegel, Hongkai Zhao, Juncai He, Shi Lab (Humphrey Shi, Steven Walton, Chuanhao Yan) and Matthieu Darcy for fruitful discussion and constructive suggestions. Z.L., F.R., J.H., M.S. and M.T. are supported by IAIFI through NSF grant PHY-2019786. The work of FR is in addition supported by the NSF grant PHY-2210333 and by startup funding from Northeastern University. Y.W and T.H are supported by the NSF Grant DMS-2205590 and the Choi Family Gift Fund. S. V. and M. S. acknowledge support from the U.S. Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) under Grant No. N00014-20-1-2325 on Robust Photonic Materials with Higher-Order Topological Protection.

## Appendix A KAN Functionalities

Table 6 includes common functionalities that users may find useful.

**Table 6: KAN functionalities**
| Functionality | Descriptions |
| --- | --- |
| model.train(dataset) | training model on dataset |
| model.plot() | plotting |
| model.prune() | pruning |
| model.fix_symbolic(l,i,j,fun) | fix the activation function $\phi_{l,i,j}$<br>to be the symbolic function fun |
| model.suggest_symbolic(l,i,j) | suggest symbolic functions that match<br>the numerical value of $\phi_{l,i,j}$ |
| model.auto_symbolic() | use top 1 symbolic suggestions from suggest_symbolic<br>to replace all activation functions |
| model.symbolic_formula() | return the symbolic formula |

## Appendix B Learnable activation networks (LANs)

### B.1 Architecture

Besides KAN, we also proposed another type of learnable activation networks (LAN), which are almost MLPs but with learnable activation functions parametrized as splines. KANs have two main changes to standard MLPs: (1) the activation functions become learnable rather than being fixed; (2) the activation functions are placed on edges rather than nodes. To disentangle these two factors, we also propose learnable activation networks (LAN) which only has learnable activations but still on nodes, illustrated in Figure B.1.

For a LAN with width $N$, depth $L$, and grid point number $G$, the number of parameters is $N^{2}L+NLG$ where $N^{2}L$ is the number of parameters for weight matrices and $NLG$ is the number of parameters for spline activations, which causes little overhead in addition to MLP since usually $G\ll N$ so $NLG\ll N^{2}L$. LANs are similar to MLPs so they can be initialized from pretrained MLPs and fine-tuned by allowing learnable activation functions. An example is to use LAN to improve SIREN, presented in Section  B.3.

Comparison of LAN and KAN.
Pros of LANs:

- (1)
LANs are conceptually simpler than KANs. They are closer to standard MLPs (the only change is that activation functions become learnable).
- (2)
LANs scale better than KANs. LANs/KANs have learnable activation functions on nodes/edges, respectively. So activation parameters in LANs/KANs scale as $N$/$N^{2}$, where $N$ is model width.

Cons of LANs:

- (1)
LANs seem to be less interpretable (weight matrices are hard to interpret, just like in MLPs);
- (2)
LANs also seem to be less accurate than KANs, but still more accurate than MLPs. Like KANs, LANs also admit grid extension if theLANs’ activation functions are parametrized by splines.

### B.2 LAN interpretability results

Figure: Figure B.1: Training of a learnable activation network (LAN) on the toy example $f(x,y)={\rm exp}({\rm sin}(\pi x)+y^{2})$.
Refer to caption: figs/lan_toy_interpretability_evolution.png

Figure: Figure B.2: LANs on synthetic examples. LANs do not appear to be very interpretable. We conjecture that the weight matrices leave too many degree of freedoms.
Refer to caption: figs/lan_interpretable_examples.png

We present preliminary interpretabilty results of LANs in Figure B.2. With the same examples in Figure 4.1 for which KANs are perfectly interpretable, LANs seem much less interpretable due to the existence of weight matrices. First, weight matrices are less readily interpretable than learnable activation functions. Second, weight matrices bring in too many degrees of freedom, making learnable activation functions too unconstrained. Our preliminary results with LANs seem to imply that getting rid of linear weight matrices (by having learnable activations on edges, like KANs) is necessary for interpretability.

### B.3 Fitting Images (LAN)

Figure: Figure B.3: A SIREN network (fixed sine activations) can be adapted to LANs (learnable activations) to improve image representations.
Refer to caption: figs/siren.png

Implicit neural representations view images as 2D functions $f(x,y)$, where the pixel value $f$ is a function of two coordinates of the pixel $x$ and $y$. To compress an image, such an implicit neural representation ($f$ is a neural network) can achieve impressive compression of parameters while maintaining almost original image quality. SIREN [] proposed to use MLPs with periodic activation functions to fit the function $f$. It is natural to consider other activation functions, which are allowed in LANs. However, since we initialize LAN activations to be smooth but SIREN requires high-frequency features, LAN does not work immediately. Note that each activation function in LANs is a sum of the base function and the spline function, i.e., $\phi(x)=b(x)+{\rm spline}(x)$, we set $b(x)$ to sine functions, the same setup as in SIREN but let ${\rm spline}(x)$ be trainable. For both MLP and LAN, the shape is [2,128,128,128,128,128,1]. We train them with the Adam optimizer, batch size 4096, for 5000 steps with learning rate $10^{-3}$ and 5000 steps with learning rate $10^{-4}$. As shown in Figure B.3, the LAN (orange) can achieve higher PSNR than the MLP (blue) due to the LAN’s flexibility to fine tune activation functions. We show that it is also possible to initialize a LAN from an MLP and further fine tune the LAN (green) for better PSNR. We have chosen $G=5$ in our experiments, so the additional parameter increase is roughly $G/N=5/128\approx 4\%$ over the original parameters.

## Appendix C Dependence on hyperparameters

We show the effects of hyperparamters on the $f(x,y)={\rm exp}({\rm sin}(\pi x)+y^{2})$ case in Figure C.1. To get an interpretable graph, we want the number of active activation functions to be as small (ideally 3) as possible.

- (1)
We need entropy penalty to reduce the number of active activation functions. Without entropy penalty, there are many duplicate functions.
- (2)
Results can depend on random seeds. With some unlucky seed, the pruned network could be larger than needed.
- (3)
The overall penalty strength $\lambda$ effectively controls the sparsity.
- (4)
The grid number $G$ also has a subtle effect on interpretability. When $G$ is too small, because each one of activation function is not very expressive, the network tends to use the ensembling strategy, making interpretation harder.
- (5)
The piecewise polynomial order $k$ only has a subtle effect on interpretability. However, it behaves a bit like the random seeds which do not display any visible pattern in this toy example.

Figure: Figure C.1: Effects of hyperparameters on interpretability results.
Refer to caption: figs/interpretability_hyperparameters.png

## Appendix D Feynman KANs

We include more results on the Feynman dataset (Section 3.3). Figure D.1 shows the pareto frontiers of KANs and MLPs for each Feynman dataset. Figure D.3 and D.2 visualize minimal KANs (under the constraint test RMSE $<10^{-2}$) and best KANs (with the lowest test RMSE loss) for each Feynman equation fitting task.

Figure: Figure D.1: The Pareto Frontiers of KANs and MLPs for Feynman datasets.
Refer to caption: x7.png

Figure: Figure D.2: Best Feynman KANs
Refer to caption: x8.png

Figure: Figure D.3: Minimal Feynman KANs
Refer to caption: x9.png

## Appendix E Remark on grid size

For both PDE and regression tasks, when we choose the training data on uniform grids, we witness a sudden increase in training loss (i.e., sudden drop in performance) when the grid size is updated to a large level, comparable to the different training points in one spatial direction. This could be due to implementation of B-spline in higher dimensions and needs further investigation.

## Appendix F KANs for special functions

We include more results on the special function dataset (Section 3.2). Figure F.2 and F.1 visualize minimal KANs (under the constraint test RMSE $<10^{-2}$) and best KANs (with the lowest test RMSE loss) for each special function fitting task.

Figure: Figure F.1: Best special KANs
Refer to caption: x10.png

Figure: Figure F.2: Minimal special KANs
Refer to caption: x11.png

## Peer Reviews

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: Are the constants C in the proof in appendix B and the statement of Theorem 2.1 the same? If not, different notation would be better.  

Did you compare the symbolic regression results with other methods? Some baselines would be useful.

As asked above, could you please clarify how the residual error bound in the proof of Theorem 2.1 is obtained?
soundness: 2
strengths: The main contribution of the paper is a practical instantiation of the Arnold-Kolmogorov theorem as a deep network which can avoid the difficulties plaguing shallow versions of the theorem. There are a number of experiments validating the method ranging from replicating discovery results from knot theory, automatic and interactive symbolic regression and solution of partial differential equations. The paper is generally well-written and clear.
summary: The paper proposes a replacement for multilayer perceptrons based on the Arnold-Kolmogorov theorem which gives a two layer network for function approximation as a sum of univariate functions of individual features followed by a function of the sum. The Arnold-Kolmogorov theorem has previously been declared irrelevant for building learning machines since the involved functions can be nonsmooth [cite].
The paper reconsiders the Arnold-Kolmogorov theorem for building learning models and proposes that deep versions of networks using the Arnold-Kolmogorov theorem might not be limited by the non-smoothness that appears in shallow networks. The paper proposes Kolmogorov Arnold networks which are stacked layers of the univariate function+sum layers that appear in the 2-layer network.
The paper further gives an approximation theorem which bounds the error in the derivatives up to some order for a function approximated by a KAN network with B-splines.
The advantages of KANs over MLPs are said to be that KANs can be more interpretable than MLPs which makes them useful for symbolic regression tasks and are useful in data-driven scientific discovery.
weaknesses: I have a few concerns, listed in the following.

1. In light of previous concerns in the literature with the Kolmogorov-Arnold theorem as a basis for learning machines, I think the paper should focus more on establishing whether the problem of non-smoothness is in fact avoided in deeper versions. Since this would establish whether the proposed model is general enough and can scale to high dimensions.

2. The paper presents Theorem 2.1 which gives a bound on the supremum of the error in the derivative. However, it is not clear whether the bound is actually useful since it depends on f and the network in an unknown way through the constant C.
If a general theoretical argument demonstrating smoothness is too difficult, perhaps the paper could investigate the smoothness of the learned functions (or the gradients during training) to provide empirical evidence that the model learns smooth functions. 

3. Regarding the proof of Theorem 2.1 in Appendix B. It is not clear to me how the bound on the residual is obtained in line 894.
If I understand correctly, the bound on B-spline approximation in line 887 is for a single univariate function, phi_{l,i,j}, whose inputs are exact. However when the inputs from the layer below are also approximated and then summed, one would expect the bound of sup of error in the derivatives to also be summed? Whereas the error in line 887 for exact inputs and that in line 894 for approximate inputs is exactly the same. Could you clarify how the error for the residual is obtained?

4. I don’t quite understand how in the symbolic regression tasks the units are automatically converted into symbolic functions. I also couldn’t find the metric used for accuracy in the SR experiments.

5. I think the motivation described in lines 62-64 should be rephrased. Regardless of the aim of interpretability, smoothness is still quite essential for learning which is the point of Giros & Poggio [1989]. 

Minor corrections.

Line 16. Interpretabe -> interpretable

205. KAT -> KAN

473. pareton -> pareto

---

confidence: 3
contribution: 4
initial_rating: 8
presentation: 4
questions: 1.	About the experiment in Figure 5. It is known that standard MLPs (e.g., with ReLU activations) perform poorly in fitting periodic functions such as sine. Is it possible to also try MLPs with other activation functions (e.g., the sine function) to see if it can get better or even comparable performance with KAN?

2.	About Table 1. Does the line labeled by “Theory” refer to the ground-truth formula and the other lines the formulae predicted by KAN? If so, why does the ground-truth formula also have an “accuracy”?
soundness: 3
strengths: 1.	The design of KAN is well motivated and rooted in previous literature, i.e., the Kolmogorov-Arnold representation theorem. Why this theorem was not widely applied for machine learning previously is also discussed.

2.	The presentation of the paper is clear and easy to follow.

3.	The paper covers comprehensive aspects of the proposed architecture, including its theoretical foundation, detailed architectural design, guarantee on approximation ability, implementation details and tricks, etc. 

4.	The paper clearly states the “comfort zone” and limitations of KAN, and points out potential future directions.

5.	The view of external and internal degrees of freedom is interesting.
summary: This paper proposes a novel network architecture named KAN based on the Kolmogorov-Arnold representation theorem. Unlike MLPs, the KAN architecture is composed of univariate but learnable activation functions and the summation operation. It extends the Kolmogorov-Arnold representation of a function to deeper layers, which partially addresses the issue of non-smooth functions in the original two-layer Kolmogorov-Arnold representation. KAN shows good interpretability and performance in tasks that deal with symbolic formulae (e.g., applications in scientific discovery).
weaknesses: One major concern is that the selection of architectural hyperparameters of KAN (e.g., the number of layers, the number of nodes in each layer) is not discussed in detail, and it seems that we need some prior knowledge of the target function when selecting these hyperparameters. More detailed concerns and questions are listed below.

**1. About the theoretical guarantee of the approximation ability in Theorem 2.1.** Theorem 2.1 states that if the target function admits a structure of the composition of $L$ KAN layers with smooth activation functions, then we can use an $L$-layer KAN (with B-splines) to approximate it well. However, it does not discuss the scenario where the structure (in particular, the number of layers $L$) of the target function is unknown, which is common in real applications. I wonder if we use a shallower KAN (less than $L$ layers) or a deeper KAN (more than $L$ layers) to approximate the target function (which is assumed to have $L$ layers), how will the result in Theorem 2.1 change? It seems that a deeper KAN can still approximate the target function by setting higher layers as identity mappings, but things become more nuanced for a shallower KAN.

**2. About how to select the number of layers in experiments.** Do we start from training a deep and wide KAN, and then use the sparsification trick in Appendix E to prune redundant nodes or even layers? Or do we simply apply a grid search over different number of layers and then select the best performing combination?  It is encouraged to provide a step-by-step description of the process for determining KAN architectures in the experiments.

There are also some subquestions in this part:

(a)	If we are to first train a deep KAN whose number of layers is probably more than the number of layers of the target function, what will this KAN look like after being trained to convergence? Let us assume the target function $f$ is a composition of $L$ symbolic formulae, and the KAN we use has $M>L$ layers. Would the last $M-L$ layers behave as identity mappings, and the first $L$ layers capture exactly $f$’s structure?

(b)	(As a continuation of question (a)) Although a deeper KAN may have better performance, would the increasing depth degrade its interpretability?

(c)	If we use a KAN that is too deep, would there be difficulties in optimization? If so, how many layers would you recommend starting with?

(d)	Are there advanced and task-dependent strategies (maybe heuristics) to select these hyperparameters, especially for cases where prior knowledge of the target function is limited?

**3. About how to select the number of nodes per layer in experiments.** I find it surprising that in Line 300, a KAN with only one internal node can perform very well in the task of knot theory. That being said, it is unclear how to determine that we only need one node in the intermediate layer. Is it achieved by some prior knowledge of the task or by training a wide KAN and then pruning out redundant nodes?

---

Another concern is about the necessity of the sparsity loss introduced in Appendix E. Is there any ablation study of this sparsity loss? By ablation study, I refer not only to the analysis of performance changes with and without the sparsity loss but also to how the structure of the learned KAN changes. For example, if there is no sparsity loss, is it possible for a layer to learn multiple identical functions “distributed” over different nodes, such as $ x^{2}/3, x^{2}/3, x^{2}/3$, instead of a single $ x^{2}$?

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: Here are some questions and suggestions for the authors to consider during the rebuttal and discussion phase:

1. **Baselines for symbolic regression:** In the symbolic regression experiments, the authors do not include comparisons to state-of-the-art approaches like PySR (Cranmer, 2023). Such approaches have demonstrated reliable performance on a wide range of symbolic regression tasks and have been successfully applied to real-world scientific discovery problems. To better assess the effectiveness of KANs for symbolic regression, it is important to compare them against the best-performing methods in this domain. Can the authors include experiments comparing KANs to PySR and other leading symbolic regression techniques, and discuss how KANs perform relative to these state-of-the-art baselines in terms of accuracy, efficiency, and interpretability?

2. **Baselines for function regression:** In the function regression experiments, to ensure a fair and meaningful comparison, the authors should consider employing best practices for MLP training, such as using appropriate network sizes, optimization algorithms (e.g., Adam or SGD with learning rate scheduling), and regularization techniques (e.g., dropout or weight decay). Can the authors update their function regression experiments to include MLP baselines trained using best practices and discuss how KANs compare to these more robustly trained MLP models?

3. **Baselines for PDE solving:** In the PDE solving experiments, the authors compare KANs against vanilla MLPs with Glorot initialization, which is known to be a weak baseline in the current literature on Physics-Informed Neural Networks (PINNs). To better demonstrate the effectiveness of KANs in this context, the authors should consider comparing against more advanced and well-established PINN architectures and initialization schemes. Can the authors include experiments comparing KANs to state-of-the-art PINN methods like PirateNets and discuss how KANs perform relative to these stronger baselines?

4. **Continual learning comparison:** The continual learning results presented in the paper are not entirely surprising, given that the spline activations used in KANs are compactly supported, and their local behavior can be modulated by neighboring knots without altering their global structure. For a more fair and informative comparison, the authors should consider evaluating KANs against MLPs with compactly supported activations, such as Radial Basis Functions (RBF), as networks with activations like ReLU or Tanh are inherently prone to catastrophic forgetting. Can the authors provide additional experiments comparing KANs to MLPs with compactly supported activations in the continual learning setting and discuss the implications of these results?

5. **Appendix content and organization:** The Appendix contains a substantial amount of supplementary results that may not significantly contribute to the main messages of the paper and could easily be overlooked by readers. For example, the LAN approach and results presented in Appendix P are interesting but could potentially be the topic of a separate paper (although LANs appear to only introduce a small tweak to the SIREN architecture). This also raises the question of how KANs would perform for an image regression task and why such an evaluation was not provided. In general, to enhance the overall clarity and focus of the manuscript, the authors should carefully evaluate which key results from the Appendix are essential to strengthen the main messages of the paper and consider removing or reorganizing the remaining content. Can the authors provide a rationale for the current organization of the Appendix and discuss how they plan to refine it to better support the core ideas of the paper?

6. **Training details and data normalization:** Some important training details, such as data normalization protocols, are omitted from the text. It appears that KANs require inputs to be in the range [0, 1]. Are the inputs to the MLP networks normalized in the same way? Inconsistencies in data normalization can lead to unfair comparisons and affect the interpretation of the results. Can the authors clarify their data normalization procedures for both KANs and MLPs and discuss any potential impact on the reported findings? 

7. **Computational complexity of grid extension and refinement:** The paper introduces the concepts of grid extension and grid refinement for KANs but does not provide a detailed discussion on their computational complexity. Understanding the computational overhead associated with these operations is crucial for assessing the practical applicability of KANs in various settings. Can the authors provide a more in-depth analysis of the computational complexity of grid extension and refinement, including theoretical bounds and empirical measurements, and discuss the implications for the scalability and efficiency of KANs? Moreover, the update of spline grids mentioned in p. 16 seems to be a very important step of the algorithm but it is not adequately explained.

8. **Code availability:**  The authors did not provide any code with this submission and I was unable to assess their implementation. This may have helped with resolving some of the above questions.
soundness: 2
strengths: The paper presents a novel neural network architecture inspired by the Kolmogorov-Arnold representation theorem (KST). The authors creatively combine the insights from the KST with modern deep learning techniques, generalizing the original two-layer representation to multiple layers through the introduction of KAN layers. This novel approach to network design demonstrates originality and has the potential to open new avenues for research at the intersection of deep learning and approximation theory.


The authors provide extensive numerical evaluations across a diverse collection of experiments. There they aim to showcase the effectiveness of KANs in various domains, including synthetic toy datasets, special functions, physics equations (Feynman datasets), and partial differential equations. These experiments aim to provide a comprehensive assessment of KANs' performance, interpretability, and scaling laws compared to traditional MLPs. 


The paper is well-structured and clearly written, making it accessible to a broad audience. The authors provide sufficient background information on the KST and its relation to neural networks, ensuring that readers can understand the motivation behind KANs. The mathematical formulation of KANs and the approximation theory are presented in a rigorous and understandable manner.


The significance of this work lies in its potential to bridge the gap between deep learning and approximation theory, as well as its implications for interpretable and accurate learning in scientific applications. The authors demonstrate that KANs can be effective in discovering mathematical and physical laws, showcasing some early promise as a tool for scientific discovery. Moreover, the scaling laws exhibited by KANs suggest that they may be a promising alternative to MLPs in certain domains.


I would also like to acknowledge that this research has already generated considerable interest in the recent literature, with several studies exploring the possibilities brought by combining deep learning with the KST. The paper has opened new avenues for research, as the community seeks to assess and enhance the capabilities of KANs and related architectures.
summary: This paper introduces Kolmogorov-Arnold Networks (KANs) as an alternative to Multi-Layer Perceptrons (MLPs), particularly when interpretability is desired, such as for extracting symbolic formulas from datasets in scientific applications. The authors contextualize the Kolmogorov-Arnold representation theorem within modern machine learning, relating it to MLPs and generalizing the representation from two layers to multiple layers via introduced KAN layers, thereby enhancing expressive power while addressing some pathological behavior of Kolmogorov-Arnold representations. 

The paper aims to demonstrate the interpretability of KANs and their potential to serve as a useful tool for scientific discoveries, presenting examples from mathematics (knot theory) and physics (Anderson localization) where KANs can assist scientists in (re)discovering mathematical and physical laws. Furthermore, the authors show through theory and experiments that KANs are accurate and have favorable scaling laws, while MLPs scale more slowly and plateau quickly.

The focus of the paper is on relatively small-scale numerical examples from various scientific domains, with the scalability and extensibility of KANs to large-scale machine learning tasks left as future work. Overall, the paper introduces KANs as a promising alternative to MLPs, especially for interpretable and accurate learning in scientific applications, and lays the groundwork for further exploration of their potential.
weaknesses: While I believe that this paper puts forth a creative idea that is worthwhile publishing, the current presentation makes it very difficult to assess whether KANs are indeed a good and competitive method to employ for all the different tasks presented in this work: symbolic regression, function regression, PDE solving, etc. The main reason is that the vast majority of numerical evaluations are poorly designed, either by disadvantaging competing methods, or by comparing against very weak baselines, or by providing no comparison to competing methods (for example, in the symbolic regression benchmarks).

1. **Comparison to state-of-the-art methods:** While the paper presents promising results for KANs in various application settings, such as symbolic regression and solving PDEs, the effectiveness of KANs compared to state-of-the-art methods in each domain remains unclear. The authors primarily compare KANs against basic MLP architectures, which may not represent the best-performing approaches in these specific areas. To provide a more comprehensive assessment of KANs' capabilities, it would be beneficial to compare them against well-established and state-of-the-art methods that are commonly used in practice for each application. This would help to better understand the relative strengths and weaknesses of KANs and their potential to advance the state of the art in these domains.

2. **Overstated claims and conclusions:** Due to the lack of comparisons against state-of-the-art methods in each application area, the current claims and conclusions in the paper may be somewhat overstated. While the results demonstrate the potential of KANs, it is difficult to assess their true effectiveness and superiority without a more comprehensive evaluation against well-established benchmarks and leading approaches. The authors should consider tempering their claims and conclusions to reflect the limitations of their current experimental setup and the need for further validation against state-of-the-art methods.

3. **Interpretability claims and symbolic regression protocol:** The authors emphasize the interpretability of KANs as one of their key strengths. However, the symbolic regression protocol presented in Section E.2 and Figure 10 appears to be largely heuristic and sensitive to various hyperparameter choices. The effectiveness of this protocol in recovering interpretable models from real-world data remains questionable, as the current results focus on rediscovering known scientific formulas. Moreover, the authors do not provide a comparison against state-of-the-art symbolic regression alternatives, making it difficult to assess the relative interpretability of KANs. The arguments against symbolic regression methods in the appendix are not entirely convincing, given the reported success of such methods in challenging settings [1]. To strengthen the interpretability claims, the authors should consider evaluating KANs against leading symbolic regression approaches and demonstrating their effectiveness in recovering interpretable models from real-world data.

4. **Unconventional protocols for training MLPs:** The authors employ unconventional protocols for training MLPs, such as using networks with small widths and full-batch L-BFGS optimization for a limited number of training steps. While these choices may be made to match the parameter complexity and training cost of KANs, they could potentially disadvantage the effectiveness of MLPs and render the comparisons and conclusions questionable. Recent results [3,4] suggest that a fair comparison between KANs and MLPs may lead to different conclusions than those reported in this paper. To address this concern, the authors should consider using more standard and well-established training protocols for MLPs, ensuring a fair and unbiased comparison between the two architectures.

5. **Comparison against plain MLPs in PINNs:** In the context of Physics-Informed Neural Networks (PINNs), the authors compare KANs against plain MLP networks initialized using the Glorot scheme. However, this setup is known to yield a poor baseline in the current PINNs literature, as the derivatives of such networks have limited expressive power to adequately minimize PDE residuals [citation needed]. To provide a more meaningful comparison, the authors should consider evaluating KANs against state-of-the-art PINN frameworks such as PirateNets [5] that have been specifically designed to address the challenges of PDE solving. This would help to better understand the potential advantages of KANs in the context of PINNs and their ability to advance the state of the art in this domain.

[1] Cranmer, M. (2023). Interpretable machine learning for science with PySR and SymbolicRegression. jl. arXiv preprint arXiv:2305.01582.

[2] Cranmer, M., Sanchez Gonzalez, A., Battaglia, P., Xu, R., Cranmer, K., Spergel, D., & Ho, S. (2020). Discovering symbolic models from deep learning with inductive biases. Advances in neural information processing systems, 33, 17429-17442.

[3] Yu, R., Yu, W., & Wang, X. (2024). Kan or mlp: A fairer comparison. arXiv preprint arXiv:2407.16674.

[4] Shukla, K., Toscano, J. D., Wang, Z., Zou, Z., & Karniadakis, G. E. (2024). A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks. arXiv preprint arXiv:2406.02917.

[5] Wang, S., Li, B., Chen, Y., & Perdikaris, P. (2024). PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks. arXiv preprint arXiv:2402.00326.

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 2
questions: I detail below some questions/suggestions to the authors, in order of how important they would be toward increasing the review score of their submission.

- **[More Challenging Problems For Testing Accuracy]** As mentioned above, I think the paper would greatly benefit from more challenging benchmarks, as the ones reported are too simple and largely not relevant to scientific computing. Seeing KANs outperform realistic MLPs (using higher widths and Fourier features, for example) on more challenging tasks could highly increase my score for the submission, depending on the results. Some potential tasks are detailed above in the previous section.

- **[Comparison Against Symbolic Regression Baselines]** The authors highlight discovery of close-form expressions from data as a key feature of KANs, but experiments lack comparison against other symbolic regression baselines, as the ones detailed in lines 2115-2121. Seeing KANs outperform these baselines could make for a much stronger paper.

- **[Spline Grid Boundary]** Using B-splines require all inputs to fall within a specified interval, which may not be the case for intermediate layers of a deep KAN. On line 852, the authors mention they "update each grid on the fly according to its input activations," but no further information is given as to how this is done, to the best of my knowledge. I feel like this is an important aspect to detail in the paper, maybe even in the main text. Related to this, how would the authors deal with problems where the inputs are defined in an unbounded (or very large) domain, such as when inputs follow a distribution with a 'fat' tail?

- **[Skip Connections]** What is the effect of removing the silu 'basis function' used in the KAN layer? Are there other suitable functions to choose instead of silu? This would be a good ablation to consider reporting.

- **[Grid Extension]** The authors propose progressively extending the grid used for computing B-splines, which improves the predictive accuracy of a given model. Is there an advantage to progressively increasing this grid as opposed to starting with a large one? That would be an interesting ablation to include in the paper.

- **[Continual Learning]** For the continual learning experiment, what happens if a deeper KAN is used? It makes sense that the local nature of B-spline computations yields this nice feature, but it seems to me that once more compositions are at play this property would be lost.
soundness: 3
strengths: ### Originality/Significance: 
The paper presents several interesting original thoughts, and helps bring attention to new forms of computation to scientific deep learning. Although the connections between KAT and neural networks are quite old, dating back to the 1960s and 1970s, their work helps renew attention to this field, and applies modern knowledge about neural networks to improve on previous failed attempts of using KAT for applied problems. The authors also present interesting ideas about interpretability for deep neural networks, by using KANs to guide the discovery of closed form expressions for unknown functions.

### Quality:
The main idea (exploiting deeper/wider forms of KAT) is well-motivated and appears theoretically promising. The authors provide valuable arguments on section 2 about why such an idea could work. The paper also comes with many experiments being conducted in the main text, as well as in the appendix. Overall, I believe this paper contains very interesting ideas, and helps spark discussion about an old topic using modern lenses.


### Clarity:
The ideas are presented in a mostly clear manner, with helpful figures to illustrate the main arguments. Their main theorem (Theorem 2.1) is clearly stated in the main text, including all of its assumptions.
summary: The authors propose a novel neural network architecture based on the Kolmogorov-Arnold Representation Theorem (KAT), called Kolmogorov-Arnold Networks (KANs). Although the KAT has been studied in the context of neural networks, KANs differ from most other works by allowing for wider/deeper representations, in addition to implementing functions implied by the theorem via B-splines. The authors formally describe the main component of their new architecture, the KAN-layer, which draws inspiration from the representation formula implied by the KAT. In essence, a KAN-layer works by approximating the functions implied by KAT via B-splines, as well as adding a 'residual' connection using a silu activation. These KAN-layers can then be composed together to create a deeper KAN architecture. The authors also state a theorem regarding the approximation capabilities of KANs, assuming a specific smooth representation of the target function.

The authors then go on to detail experiments using their new architecture on two main goals: a) providing interpretable results from scientific data; and b) creating accurate predictions for regression tasks and learning the solution of a PDE. On their first goal, they evaluate KANs on tasks from mathematics (knot theory) and physics (Anderson localization), as well as some toy problems on symbolic regression. For their second goal, they compare KANs against certain MLPs on regression for several smooth functions with closed form solutions, as well as a physics-informed problem where they learn the solution of a 2D Poisson equation. 

Finally, the authors summarize their results and argue that KANs offer significant advantages over traditional MLPs for scientific tasks that are "small-scale", leaving more thorough experimentation on larger-scale problems for future work.
weaknesses: ## Summary of weaknesses:
In summary, although the main idea of the paper seems very interesting, my main criticism of this submission is regarding their experiments, which are not conclusive to me as a reviewer. Despite presenting a very large number of different experiments, they are all very simple, and comparisons against MLPs appear to be using very weak baselines. The authors do recognize that they mainly consider small-scale experiments [line 485], but even for scientific tasks there are more realistic and interesting problems to be considered, as will be detailed later. In addition to this, their main theorem regarding approximation power of KANs (theorem 2.1), which implies very strong scaling laws, use an assumption which is known to be wrong for KAT functions (ie: inner functions being $(k+1)$-times continuously differentiable). I further detail these criticisms below, and how the authors might address them.

## Experimental Setup:
I will mainly focus on the results reported in section 4 here, which is an area I am more familiar with. My main criticisms regarding experiments is that a) all functions considered are very well-behaved and not something for which deep learning would be used in practice, with other regression tools generally being better suited; b) I am not confident that their comparisons against MLPs are done in a consistent and fair way; c) there is not enough information to fully reproduce experiments. I expand on each of these points below.

a) For regression tasks (Toy datasets + Special Functions + Feynman Datasets), all the target functions are very simple, beyond what is typical for scientific machine learning. The functions from the "Special Functions" dataset are slightly more complicated than from the other two, but still a very simple task given enough training data. A better testbed for examining approximation properties and accuracy would be for example: 1) regression against complicated PDE solutions from publicly available dataset such as PDEBench and/or PDEArena, which are more relevant to scientific computing; 2) regression against images, which was in fact considered in the appendix for their "LAN" architecture, but not KANs; 3) weather modeling, using public datasets such as ERA5 (very challenging) or NOAA's SST-V2, for example. Regarding the experiment on PINNs, it would also be more comprehensive to study more than just the Poisson equation, which is again very simple for tasks of this nature. Suggestions for other PDEs to consider, which are not unrealistic for the first iteration of a new architecture include the Allen-Cahn equation and the Darcy problem. In short, although it is true that some problems in scientific computing are simpler or lower dimensional than ones in computer vision and NLP, which are big topics in deep learning, there is still no shortage of more realistic problems in science, which would provide a better testbed for their architecture.

b) Many aspects of the experiments provided are not likely to yield a fair comparison against MLP architectures. In some cases, only MLPs of very small width (such as 5 or 20) are considered, which is not realistic of what someone would use in practice, even if the number of parameters is similar to that of KANs used. For some of the regression tasks, the grid definition of KAN's B-splines can go as high as 1000, which means that a KAN of comparable width/depth still presents far more parameters (and likely flops) than an MLP. To address this issue, I would recommend keeping a fixed number of trainable parameters (or flops/training time), comparing different hyperparameter combinations, both for KANs and MLPs. Ideally, MLPs should have widths of at least 32 or 64 for any real scenario. In addition to this, MLPs are more efficiently trained using the Adam optimizer, whereas all experiments used the L-BFGS algorithm, which is not as common and can often lead to MLPs being stuck on local minima.

c) There are many missing details for reproducing the experiments in the paper, and no code was provided as supplementary material. For example, on many of the regression tasks, the target functions are reported, but not the total number of points in the training/testing set, nor what the domain for these functions is. Other missing information for reproducibility include: 1) the batch size used during training for different benchmarks; 2) the learning rate of the optimizer; 3) the architectures used in the continual learning example. The best way of solving all this missing information issue would be to **provide the code used for each experiment as a supplementary material**, organized neatly for each benchmark.


## Theoretical Assumptions:
A main assumption of theorem 2.1 of the paper is that the functions $\phi_{l,i,j}$ are $(k+1)$-times continuously differentiable. However, the KAT only guarantees that these functions are continuous, as and the authors themselves mention, it has been shown in the literature that these functions can provably be very chaotic, with infinite set of non-differentiability. The authors argue that by using wider/deeper representations, these irregularities may disappear, but this would require further theoretical evidence. I believe there are more reasonable ways of proving universal approximation guarantees for KANs, albeit without the strong scaling laws implied by the $k$-order spline functions. This main theorem therefore can likely be stated differently, in order to obtain a weaker but more theoretically reasonable statement.




## Other minor considerations that did not affect my score:
- [line 15 + many other mentions] The analogy of referring to computations on 'edges' vs 'nodes' is not very clear or compelling to me. Depending on how you chose to represent any given architecture as a graph, it could be argued either way whether an operation happens at an 'edge' or at a 'node'.
- [line 54] Although it is true that "In science, symbolic functions are prevalent," that is not generally the case in scientific computing, where solutions are often defined on complicated domains without reasonable closed-form expressions. In fact, for tasks where numerical solutions are needed (such as solutions to ODEs, PDEs, etc), it is almost always the case that the solutions are not symbolic.
- [line 260] There are only three examples in Figure 2, not six.
- [Figure 7] It is more common for the y-axis of the plots to show the error itself, not the error squared, as is done in the figure.
- [line 875] It would also be nice to see a comparison on the computational complexity of KANs vs MLPs, in addition to parameter count.
- [line 1819] In the case of PINNs, it is almost always better to sample points randomly and independently at each iteration, as opposed to maintaining a fixed grid. I would be interested to see how KANs perform in this setting.

---

confidence: 3
contribution: 3
initial_rating: 5
presentation: 3
questions: * How do KANs compare to other interpretable models, like Neural Additive Models or symbolic regression methods, in terms of accuracy, interpretability, and computational cost, especially on larger datasets?

* Can you provide a more detailed analysis of the computational complexity of KANs, both for training and inference? Have you explored any optimizations beyond the "multi-head" approach?

* Can you elaborate on the dependence of the constant C in Theorem 2.1 on the dimension? How does this impact the scalability of KANs to higher-dimensional problems?

* Can you provide empirical evidence supporting the claim that KANs can naturally work in continual learning without catastrophic forgetting, beyond the toy example in Appendix J?

* How sensitive are KANs to the choice of hyperparameters, such as the grid size $G$ and the spline order $k$? Have you explored techniques for automatically selecting these hyperparameters?

**Additional Feedback** (this does not affect my score)

* In my opinion, some formulations, e.g. "the Kolmogorov-Arnold representation theorem was basically sentenced to death" (l.119), or "the fastest scaling exponent ever" (l.1888) appear overly "dramatic" for a scientific paper. I suggest the authors scan their paper for cases like these and try to use more neutral/grounded language.

* There are several typos, e.g. "altnertives" (abstract), "On" (l.084), "staricase-like" (l.965), "theLANs" (l.1705), "maximu" (l.1885). I suggest the authors use an automated spell-checker to find problems like these and correct them (I likely didn't find all typos).

* Some terms are spelled inconsistently in different places of the text, for example "Deepmind" vs. "deepmind" (I actually would recommend to use "DeepMind" instead), or "ReLU" vs. "ReLu". The authors should decide on one spelling and keep it throughout the paper.

**In summary**
I believe this is a very interesting paper that should be published, but I feel major revisions are necessary (see above). If the authors implement these revisions (or at least most of them), I will gladly raise my score.
soundness: 3
strengths: + The paper presents an interesting and original neural network architecture inspired by the Kolmogorov-Arnold representation theorem, which has very appealing properties for some applications (e.g. interpetability).

+ The paper is generally well-written and easy to follow. The architecture of KANs is clearly explained, and the figures effectively illustrate the differences between KANs and MLPs.

+ The emphasis on interpretability (and the ability to extract symbolic formulas) is relevant for scientific applications, where understanding the underlying mechanisms learned by the model is crucial.
summary: This paper proposes Kolmogorov-Arnold networks (KANs) as a more interpretable alternative to multi-layer perceptrons (MLPs), particularly for scientific applications. KANs replace the fixed activation functions on nodes in MLPs with learnable activation functions on edges, represented by B-splines. The paper explores the theoretical approximation capabilities of KANs, proposes training techniques for interpretability and accuracy (pruning, symbolifying, grid extension), and demonstrates their application in mathematics (knot theory), physics (Anderson localization), and other (toy) examples.
weaknesses: - Technically, the paper could be considered to be over the page limit. While the main text does keep to 9 pages, this is only the case because extremely important sections, e.g., related works and the discussion, are moved to the appendix. I feel important sections like these have to be included in the main text, even if this means other parts of the text need to be shortened to make space for them. I am sure many authors of other submissions have faced similar "space issues" and consequently had to cut parts from their main text. This paper is the only submission I am reviewing that "misuses" the appendix to circumvent the page limit restrictions (I think the appendix should be reserved for things like detailed proofs or additional, but non-essential, information). Sorry for bringing up a technicality like this, but I feel it is necessary in the interest of fairness.

- The paper contains very strong claims, e.g. that KANs with finite grid size beat the curse of dimensionality. At the same time, the authors admit that the constant $C$ in Eq.11 (Theorem 2.1) implicitly depends on the dimension (but details are left to future work). As the saying goes, extraordinary claims require extraordinary evidence, which unfortunately is missing here. The authors should either weaken their claims or provide strong empirical evidence.

- While reading the paper, I couldn't help but feel that the name "Kolmogorov-Arnold network", at least to some degree, overstates the difference to MLPs: KANs and MLPs have more in common than a superficial reading of the paper would suggest. Comparing the Kolmogorov-Arnold Theorem (KAT) with the Universal Approximation Theorem (UAT), a very important aspect is that KAT requires a width of only $2n+1$, whereas the UAT only holds in the limit of infinite width. I understand that relaxing the width-restriction from KAT may be practically useful (and the authors motivate this choice well), but it also blurs the lines between KAT and UAT. To me, KANs seem more like a KAT-inspired re-interpretation of MLPs (rather than something entirely new), which is useful for tasks where a symbolic interpretation of the learned function is desirable. For example, one distinguishing feature of KANs pointed out in the abstract are the learnable activation functions, wheareas MLPs have a fixed activation function. But if I re-write an MLP $W_2\sigma(W_1\mathbf{x})$ as $W_2\phi(\mathbf{x})$ with $phi(\mathbf{x}) = W_1\mathbf{x}$, I can also consider $phi$ to be a "learnable activation function". Similarly, the basis functions of the B-splines in KANs could be equivalently considered simply as additional input features, similar to Fourier features (https://arxiv.org/abs/2006.10739), or other tricks that were sometimes used in older works (e.g. feeding the square of an input to a layer, see e.g., "Neural Networks: Tricks of the Trade"). I think the authors should try to "de-mistify" KANs by pointing out and discussing these similarities between MLPs and KANs more transparently.

- While the examples in mathematics and physics are interesting, the scale of these problems is relatively small. The paper lacks evaluation on larger, more complex datasets. This makes it difficult to assess the true scalability and practical utility of KANs for many real-world applications. For example, a very simple test could be to compare KANs vs. MLPs on something like MNIST. Even if KANs perform worse than MLPs here, this does not mean at all that they are not useful. However, it would certainly help readers to assess whether KANs (in their current form) are likely to be directly relevant for their work or not. I encourage the authors to include at least one example like this.

- The example in Appendix J on continual learning is practically not relevant and too trivial. Of course, the B-splines in KANs are "local" and thus the corresponding weights/coefficients are not affected by catastrophic forgetting in this setting, but the toy example is also clearly constructed to take advantage of this fact. The authors should either demonstrate that KANs are less affected by catastrophic forgetting than MLPs in a more realistic/"less constructed" setting, or remove this claim from the paper.

- The paper mentions that KANs are slower to train than MLPs by roughly a factor of 10, and that KANs cannot leverage batch computation in the same way as MLPs. However, these discussion are hidden in the appendix. I think this is a highly important and practically relevant aspect, which should be discussed more prominently and in greater detail in the main text. An analysis of the computational cost, including training time and memory usage, is needed to fully understand the trade-off between interpretability and efficiency.

- The appendix is very long with >25 additional pages, not all of which feel strictly necessary to me. I think the authors should consider cutting some parts of the appendix in the interest of brevity. For example, the secion on LANs, while interesting, feels better suited as a separate submission (after doing more experiments/refining the idea).

Meta-review: This paper proposes Kolmogorov-Arnold networks (KANs) as a more interpretable alternative to multi-layer perceptrons (MLPs), particularly for scientific applications. The authors have made significant improvements to the paper in response to reviewer feedback, including adding new experiments, toning down strong claims, and providing a more detailed analysis of computational complexity.

The reviewers have raised several points, including the need for more empirical evidence to support the claims, the similarity between KANs and MLPs, and the lack of evaluation on larger, more complex datasets. The authors have addressed these concerns by adding new experiments, including comparisons to MLPs on MNIST and image-fitting tasks, and providing a more detailed analysis of the computational complexity of KANs.

While the reviewers have noted that KANs are slower to train than MLPs and may not have a clear advantage over MLPs on all tasks, they have also acknowledged the potential benefits of KANs, including their interpretability and ability to discover new functions. The authors have been responsive to feedback and have made significant improvements to the paper.

Hence, after discussion with the authors and among themselves, the reviewers find the paper much improved, with all reviewers leaning towards acceptance. We are therefore happy to accept the paper. We would still like to encourage the authors to address all the reviewers' comments in the camera-ready version.

Decision: Accept (Oral)