KAN versus MLP on Irregular or Noisy Functions

## Abstract

Abstract In this paper, we compare the performance of Kolmogorov-Arnold Networks (KAN) and Multi-Layer Perceptron (MLP) networks on irregular or noisy functions. We control the number of parameters and the size of the training samples to ensure a fair comparison. For clarity, we categorize the functions into six types: regular functions, continuous functions with local non-differentiable points, functions with jump discontinuities, functions with singularities, functions with coherent oscillations, and noisy functions. Our experimental results indicate that KAN does not always perform best. For some types of functions, MLP outperforms or performs comparably to KAN. Furthermore, increasing the size of training samples can improve performance to some extent. When noise is added to functions, the irregular features are often obscured by the noise, making it challenging for both MLP and KAN to extract these features effectively. We hope these experiments provide valuable insights for future neural network research and encourage further investigations to overcome these challenges.

###### Index Terms:

## I Introduction

The Kolmogorov-Arnold networks (KAN) have garnered significant attention since their appearance on arXiv []. These networks utilize the Kolmogorov-Arnold representation theorem, which posits that any multivariate continuous function can be expressed as a combination of continuous single-variable functions and addition. Unlike conventional Multi-Layer Perceptron (MLP) networks, KANs incorporate learnable activation functions. According to [], this feature provides KANs with enhanced interpretability and accuracy over MLPs.

Numerous investigations into KAN applications have rapidly surfaced, covering areas such as smart energy grid optimization[][], chemistry data analysis[][], image classification[][][], deep function learning[], quantum architecture search[], medical image analysis and processing[][], disease risk predictions[], graph learning tasks[][][], asset pricing models[], 3D object detection (in autonomous driving)[], sentiment analysis[], and deep kernel learning[].

On the contrary, a growing body of research has highlighted the imperfections of KANs compared to MLPs. For example, [] and our work [] noted the vulnerability of KANs to noise, indicating that even minor noise can lead to a significant rise in test loss. Additionally, [] claimed that KANs do not outperform MLPs in highly complex datasets and require considerably more hardware resources. Furthermore, [] noted that MLPs generally have higher accuracy than KANs across various standard machine learning tasks, with the exception of tasks involving symbolic formula representation.

Moreover, it is widely recognized that the regularity of the functions being approximated significantly influences the efficacy of neural networks. Functions that are smooth and continuous, known as regular functions, are generally approximated more precisely by KANs. On the other hand, functions that exhibit discontinuities or abrupt variations, referred to as irregular functions, present a greater difficulty.

It is essential to emphasize that the research by [] focused on achieving a fair comparison between KANs and MLPs, yet it excluded the effect of irregularities or noise. In this investigation, we assess the performance of MLP and KAN in modeling irregular or noisy functions. To ensure fairness, we control the number of parameters and the amount of training data. Moreover, we investigate the influence of different optimizers on the accuracy of fitting specific functions. This research continues directly and naturally from our recent study on the efficacy of KANs in fitting noisy functions [].

The structure of this paper is organized as follows: Section II provides an introduction to the Kolmogorov-Arnold Theorem and KANs, discussing their benefits and limitations, and enumerates the six types of functions. Section III evaluates the performance of MLP and KAN in approximating regular and irregular functions. KAN performs better than MLP in the case of regular functions, whereas for certain irregular functions, MLP shows superior performance over KAN. Increasing the number of training samples can enhance performance to a certain degree. Moreover, the impact of different optimizers on specific functions is examined. Adam generally performs better than L-BFGS during extensive training on most functions. Section IV introduces noise to the previously utilized functions and continues the comparison between MLP and KAN. It is observed that noise can obscure the characteristics of irregular components of functions, making it difficult for both MLP and KAN to capture these features accurately. Finally, Section V summarizes the findings of our experiments.

## II Komogorov-Arnold Theorem and KANs

**TABLE I: Several Types of Functions and Their Examples**
| Regular | Smooth |
| --- | --- |
| $\qquad\qquad\ f_{1}(x)=x^{2}\qquad\qquad\$ | $f_{2}(x)=e^{x}$ |
|  |  |
| Irregular | Continuous everywhere except at points of non-differentiability |
| $f_{3}(x)=|x|$ | $f_{4}(x)=1-\sqrt{|x|}$ |
|  |  |
| Jump |  |
| $f_{5}(x)=\begin{cases}1,\;|x|<0.5\\<br>0,\;\text{other}\\<br>\end{cases}$ | $f_{6}(x)=\begin{cases}1-4x^{2},\;|x|<0.5\\<br>1,\;\text{other}\\<br>\end{cases}$ |
|  |  |
| Singular |  |
| $f_{7}(x)=\frac{1}{x}$ | $f_{8}(x)=\frac{1}{1-x^{2}}-1$ |
|  |  |
| Coherent oscillation |  |
| $f_{9}(x)=\cos(\frac{1}{x})$ | $f_{10}(x)=\cos(\frac{2\pi}{1-x^{2}})$ |
|  |  |
| Noisy | Noisy |
| $y=x+n(x)$, where $n(x)$ denotes additive noise. |  |
|  |  |

The Kolmogorov-Arnold theorem pertains to expressing multivariable continuous functions. According to the theorem, any continuous function involving multiple variables can be expressed as a combination of continuous single-variable functions and addition [] [] []. Formally, it can be stated as:

###### Theorem 1 .

[Kolmogorov-Arnold Theorem]
Let $f:\ [0,1]^{n}\to\mathbb{R}$ be any multivariate continuous function, there exist continuous univariate functions $\phi_{i}$ and $\psi_{ij}$ such that:

$$ $f(x_{1},x_{2},\ldots,x_{n})=\sum_{i=1}^{2n+1}\phi_{i}\left(\sum_{j=1}^{n}\psi_ {ij}(x_{j})\right).$ (1) $$

Leveraging the Kolmogorov-Arnold theorem, KANs introduce a novel neural network architecture. Unlike traditional Multi-Layer Perceptrons (MLPs) which use fixed activation functions, KANs employ flexible activation functions. Each weight in the network is replaced by a univariate function, often modeled as a spline function. This methodology is theoretically advantageous in enhancing the adaptability of KANs across different datasets and applications.

Unfortunately, Theorem 1 was proven using a non-constructive approach, without providing a constructive proof. In 2009, [] presented a constructive proof for this theorem. However, issues can arise when dealing with functions that show irregular behaviors. In mathematical analysis, it is common to classify at least five distinct types of irregularities. Details and examples of these types are presented in Table I.

###### Theorem 1 .

## III Comparison on irregular functions

Here, we offer some explanations for the initial five types mentioned in Table I. We compare these functions using multiple sets of KAN and MLP networks that have similar parameter counts. The parameter numbers for each network are presented in Table II. We implement the L-BFGS optimizer for these functions as it shows better performance in small-scale training. For functions with singularities or coherent oscillations, which might need more training samples and iterations, we also investigate the Adam optimizer’s capability.

**TABLE II: The number of parameters for each KAN and MLP network.**
| MLP | KAN |  |  |  |  |
| --- | --- | --- | --- | --- | --- |
| width | parameter | width | grid | k | parameter |
| [1,7,1] | 22 | [1,1,1] | 3 | 3 | 24 |
| [1,39,1] | 118 | [1,5,1] | 3 | 3 | 120 |
| [1,79,1] | 238 | [1,10,1] | 3 | 3 | 240 |

### III-A Regular functions

First, consider the functions exhibiting strong regularity. Such functions are continuous and differentiable at all points, exhibiting no discontinuities or non-differentiable regions, similar to $f_{1}$ and $f_{2}$. We reconstruct these two functions using two sets of MLP and KAN networks that have comparable parameters but are trained with different sample sizes. The outcomes are displayed in Fig 1. It can be observed that for this category of functions, KAN outperforms MLP.

### III-B Continuous functions with points where derivatives do not exist

The functions $f_{3}$ and $f_{4}$ serve as prime examples of this category. They maintain continuity at all points except at $x=0$, where they are non-differentiable. Likewise, these functions are retrieved individually using independent MLP and KAN network sets.

The outcomes are illustrated in Fig 2. For these particular functions, the KAN’s performance is worse than the MLP’s. Despite the MLP network’s slower convergence, it eventually reaches a lower test loss. Additionally, it can be noted that amplifying the training sample size marginally enhances the performance of both networks. However, in the vicinity of the non-differentiable point, the MLP shows more significant improvement than the KAN. More visually, it is evident that the fitting performance of MLP and KAN around $x=0$ (the non-differentiable point) is approximately the same. Yet, with a larger training sample size, the MLP demonstrates superior fitting performance near $x=0$ compared to the KAN.

### III-C Functions with jump

The examples of this category include $f_{5}$ and $f_{6}$. These functions have jump discontinuities at $x=\pm 0.5$, where the function values abruptly change between 0 0 and $1$. The experimental outcomes for these functions are depicted in Fig 3. Similar to the earlier type, the results show that the MLP outperforms the KAN. Moreover, expanding the training dataset size can enhance both networks’ performance to a certain extent. Nevertheless, KAN consistently fails to match the performance of MLP.

### III-D Functions with singularities

Functions possessing singularities display distinct behaviors, marked by a rapid change rate as they near these points, with the absolute value of their first derivative tending towards positive infinity at the singularity. Additionally, for any chosen continuous interval that omits these singularities, the functions remain continuous and differentiable across the interval.

To avoid division by zero and guarantee clear fitting results, the ranges of the functions $f_{7}$ and $f_{8}$ are limited to $[0.001,1]$ and $[-0.999,0.999]$, respectively.

Before performing specific fitting experiments, we examined the effects of the training sample size, the number of Epochs, and the selection of optimizer on the fitting performance. As illustrated in Fig 4, simply enlarging the sample size by itself does not substantially enhance the performance when recovering $f_{7}$ and $f_{8}$.

Pykan offers two optional optimizers: Adam and L-BFGS. As depicted in Fig 5, while the L-BFGS optimizer achieves faster convergence with fewer epochs, the Adam optimizer shows superior performance in test loss with an increased number of epochs, particularly within MLP networks. This suggests that using the Adam optimizer and increasing the number of epochs can be an effective method to improve fitting performance. However, it is crucial to recognize that with a fixed learning rate, the improvement from this strategy is naturally constrained.

An additional limitation of the L-BFGS algorithm is its substantial time complexity. As illustrated in Table IV, for an identical number of epochs, the training duration of the network with the L-BFGS optimizer is frequently several times greater than with the Adam optimizer.

Drawing from earlier experiments and findings, the fitting tests will utilize the Adam optimizer set with a learning rate of 0.01. As shown in Fig. 6, KAN outperformed MLP in terms of fitting functions with singularities at the same number of epochs. However, it is noteworthy that during our experiments, the computation time for KAN was frequently much longer compared to that of MLP (refer to the Appendix for more information).

Fig 7 sheds light on a potential reason for the below-par performance of the two networks in such function fitting tasks. Notably, within a given error tolerance limit, it has been noted that higher absolute values of the function’s first derivative are associated with a greater number of epochs needed for convergence. Additionally, because of restrictions posed by the learning rate, increasing the number of epochs offers only marginal benefits in fitting accuracy, thus limiting the networks’ overall performance.

**TABLE III: Time consumption of L-BFGS and Adam optimizers in fitting functions $f_{7}$ and $f_{8}$ using MLP and KAN**
| Function | Network | Optimizer | Time(s) |
| --- | --- | --- | --- |
| $f_{7}$ | MLP | L-BFGS | 16.0677 |
| $f_{7}$ | MLP | Adam | 8.6278 |
| $f_{7}$ | KAN | L-BFGS | 901.4664 |
| $f_{7}$ | KAN | Adam | 122.8072 |
| $f_{8}$ | MLP | L-BFGS | 15.7724 |
| $f_{8}$ | MLP | Adam | 8.4364 |
| $f_{8}$ | KAN | L-BFGS | 941.1880 |
| $f_{8}$ | KAN | Adam | 150.0152 |

### III-E Functions with coherent oscillations

A different kind of functional singularity, referred to as an ’coherent oscillatory singularity,’ is illustrated by functions $f_{9}$ and $f_{10}$. These functions exhibit ’unreachable points’ (for example, $x=0$ in the case of $f_{9}$) where, as the function gets closer to these points, its values oscillate more rapidly, crossing the x-axis an infinite number of times. At these ’unreachable points’, not only does the absolute value of the first derivative lack a generalized limit, but the function values themselves cannot be represented within the extended real number system.

In the experimental phase, taking a similar approach as described in section D, we initially explored the impacts of raising the sampling rate, the number of Epochs, and the selection of optimizer. As shown in Fig 8 and 9, an increase in the sampling rate did not markedly enhance fitting accuracy. Additionally, while increasing the number of Epochs was advantageous, its effectiveness diminished after surpassing a certain Epoch threshold, leading to a slower reduction in test loss.

In particular, within the KAN network framework, the optimizer L-BFGS outperformed Adam for function $f_{9}$, while for function $f_{10}$, Adam showed superior results. On the other hand, when fitting both functions with an MLP, Adam consistently performed better than L-BFGS.

In a similar manner, Table IV demonstrates that employing the L-BFGS optimizer during the fitting process usually resulted in an additional increase in computational time. Figure 10 demonstrates that KAN consistently surpasses MLP when comparing performance over the same number of epochs.

## IV Noisy functions

In the following, we discuss the roles of noise functions. These functions have a wide range of applications, as noise can be added to any function, including those previously discussed. Thus, we introduce noise to these functions and proceed to evaluate the performance of MLP and KAN. According to the conclusions drawn in the preceding section, we will classify the functions into three categories: normal functions, functions with localized irregularities, and functions with severe discontinuities.

### IV-A Regular functions

We introduce noise to functions exhibiting strong regularity, and subsequently fit these noisy data using KAN and MLP. The experimental outcomes are depicted in Fig 11. Our observations indicate that KAN achieves a lower test loss with low noise levels but performs worse under high noise conditions. When comparing the function fitting effect, the conclusion remains consistent: MLP shows better performance with minor noise interference, but KAN rapidly outperforms MLP as the training sample size increases.

### IV-B Adding noise to functions with irregularities

Noise is subsequently added to $f_{3}$, $f_{4}$, $f_{5}$, and $f_{6}$. The experimental findings are shown in Fig 13. It is clear that the noise nearly completely conceals the irregular characteristics of the functions. When compared to the original data, fitting noisy data is more difficult for neural networks. Although increasing the number of training samples greatly enhances the fitting performance, it still falls short of the noise-free level. For $f_{3}$ and $f_{4}$, the network can still capture some of the irregular features with a larger training sample. However, for $f_{5}$ and $f_{6}$, both KAN and MLP perform poorly. The networks still have difficulty identifying the jump discontinuities, even with an increased sample size.

### IV-C Irregular functions adding noise

In Fig 14, it becomes clear that the KAN fitting performance greatly surpasses that of MLP when noise is added to functions with singularities or coherent oscillation. Interestingly, in these extreme and unusual function scenarios, the impact of adding noise to the fitting process is minimal, to the point that fitting with noisy data produces results that are either on par with or better than those derived from the original, noise-free data. This minimal influence of noise highlights the ineffectiveness of strategies relying solely on increased sampling rates in such instances, as they fail to exploit the subtle benefits that noise brings.

## V Conclusion

In this study, we evaluate the effectiveness of KAN and MLP in approximating irregular or noisy functions. Our analysis concentrates on two main factors: the relative performance of KAN and MLP in fitting functions with different types according to regularity, and their ability to handle noise during the fitting process.

At first, in order to thoroughly evaluate the performance of the two networks, we designed three sets of comparative experiments. These focus on testing their effectiveness in fitting different functions under varied conditions: changing sampling rates, varying numbers of epochs, and, particularly for functions $f_{7}-f_{10}$, examining the effect of different optimizers (exclusively to identify the optimal optimizer).

Secondly, as identified in [] and additionally explored in this study, raising the sampling rate is a potent method to enhance the fitting performance of functions $f_{1}-f_{6}$. Particularly, this strategy shows greater advantages when handling noisy data versus clean data. Nevertheless, the improvement in the fitting accuracy for functions with low regularity ($f_{7}-f_{10}$) is minimal, irrespective of the presence of noise.

Thirdly, we also compared the fitting performance under varying Epochs from two distinct perspectives: convergence speed and stabilized test loss. KAN exhibits a faster convergence rate than MLP across all tested functions. However, MLP outperforms KAN on test functions $f_{3}-f_{6}$ on stabilized test loss.

Fourthly, via experimental analysis (fitting $f_{7}-f_{10}$), it was observed that Adam exceeded L-BFGS in performance for both networks in every instance, except for function $f_{9}$. Notably, when fitting function $f_{9}$ with the KAN, L-BFGS demonstrated better results than Adam.

At last, when dealing with noisy functions, KAN exhibits superior performance over MLP for regular functions or irregular functions. Conversely, for functions that with jump discontinuities or singularities, MLP outperforms KAN.

## Acknowledgment

The authors wish to extend their sincere appreciation to Dr. Aijun Zhang for his enlightening discussions about the negative effects of noise on the KANs network and for offering numerous invaluable suggestions for this paper.

## Appendix

Given the considerably extended computation times for KAN compared to MLP when fitting functions exhibiting singularities or coherent oscillations, as detailed in Section 3, we carried out further experiments to explore this matter. Nonetheless, owing to the persistent effects of coding-level optimizations, these findings do not serve as conclusive evidence. Therefore, they are included in the Appendix for documentation and further examination.

Table: TABLE V: Performance of MLP and KAN in recovering $f_{7}$, $f_{8}$, $f_{9}$ and $f_{10}$

| Function | Network | Epochs | Time(s) | Test Loss<br>(RMSE) |
| --- | --- | --- | --- | --- |
| $f_{7}$ | MLP | 2000 | 9.4114 | 22.17 |
| $f_{7}$ | KAN | 2000 | 124.5332 | 3.04 |
| $f_{8}$ | MLP | 2000 | 10.2214 | 11.20 |
| $f_{8}$ | KAN | 2000 | 128.6405 | 4.72 |
| $f_{7}$ | MLP | 25000 | 135.06336 | 0.95 |
| $f_{7}$ | KAN^* | 2000 | 149.6529 | 3.04 |
| $f_{8}$ | MLP | 25000 | 103.8949 | 2.71 |
| $f_{8}$ | KAN^* | 2000 | 121.0088 | 4.72 |
| $f_{9}$ | MLP | 2000 | 8.6413 | 0.32 |
| $f_{9}$ | KAN | 2000 | 993.7106 | 0.30 |
| $f_{10}$ | MLP | 2000 | 10.0180 | 0.42 |
| $f_{10}$ | KAN | 2000 | 134.2680 | 0.28 |
| $f_{9}$ | MLP | 220000 | 858.6642 | 0.23 |
| $f_{9}$ | KAN^* | 2000 | 1036.1291 | 0.30 |
| $f_{10}$ | MLP | 22000 | 112.6193 | 0.37 |
| $f_{10}$ | KAN^* | 2000 | 118.9079 | 0.28 |

- *
denotes replicated experiments. For the sake of ensuring full alignment of the data with original results, they are explicitly presented.

As shown in Table V, by extending the number of epochs for training the MLP until its computation time matches that of the KAN, the MLP demonstrates better performance than the KAN in fitting the functions $f_{7}$ and $f_{8}$. However, the KAN outperforms the MLP for functions $f_{9}$ and $f_{10}$.

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## Peer Reviews

confidence: 4
contribution: 1
initial_rating: 1
presentation: 1
questions: It is possible to include more challenging problems for comparison? It is well established that MLP can model fairly complicated functions.
soundness: 1
strengths: Experiment codes are provided for reproducibility.

Do provide some insight on what KAN may be good at modeling.
summary: This work empirically compares Kolmogorov-Arnold Networks with Multi-Layer Perceptron on
learning irregular or noisy functions. The experiment results show that KAN do not always perform
the best.
weaknesses: The finding is purely empirical.

The paper does not clearly state the experiment setting in the main text.

The experiment does not provide conclusive results.

The experiment only tries to fit relatively simple functions. The result may not be relevant to real-world problems.

---

confidence: 3
contribution: 1
initial_rating: 3
presentation: 2
questions: I do not have a complete understanding of KAN, but I think KAN appears to be a generalization of projection pursuit regression. Is this correct?
soundness: 1
strengths: - The author compared KAN and MLP on various irregular and noisy functions and experimentally demonstrated in which cases KAN is worse than MLP.
summary: Authors compare the performance of Kolmogorov-Arnold Networks (KAN) and Multi-Layer Perceptron (MLP) networks on irregular or noisy functions. The author experimentally demonstrated that KAN does not always outperform MLP.
weaknesses: - The author merely compared KAN and MLP experimentally but did not analyze why KAN or MLP performs poorly in certain situations.

- The author experimentally demonstrated that KAN is sometimes inferior to MLP. It would be better to propose a new, improved KAN model to address this.

- There are no experiments on high-dimensional functions. In one dimension, both KAN and MLP are likely to approximate well to some extent, but more experiments are needed to explore how they perform in high-dimensional spaces with irregular points.

- If the experiments are conducted only on univariate functions, many models besides MLP can be compared with KAN. It would be beneficial to include other models commonly used in machine learning in the experiments.

---

confidence: 4
contribution: 1
initial_rating: 3
presentation: 3
questions: Given that the structures of both MLP and KANs are well-known, a deeper analysis of their capabilities and limitations in the related work section would be beneficial. More thorough research could uncover more significant findings. For instance, some limitations of KANs identified in the paper are not due to the Kolmogorov-Arnold Theorem but rather due to B-spline, a critical component in KANs that is not discussed in the paper at all.
soundness: 3
strengths: The structure of the paper is clear and well-organized.
The experimental results are clearly presented.
The experiments validate that KANs are not consistently superior to MLP.
summary: The paper conducts a comparative analysis of experiments between MLP and KANs, discussing the outcomes. It challenges the assumption that KANs consistently outperform MLP in modeling mathematical equations, particularly with irregular functions. The experiments involve applying MLP and KAN to various functions—regular, non-differentiable, discontinuous, singular, and coherent oscillation, with and without noise. These functions are single input and single output. Variations include different training sample sizes, iteration counts, and optimizers. The findings demonstrate that KANs do not always surpass MLP.

While this paper serves as a great exploration to KANs and does establish that KANs are not invariably superior to MLP, it falls short by only providing experimental evidence without introducing new theoretical insights or network structures, thus lacking substantial academic contribution.
weaknesses: The experiments could be designed more targeted. For instance, in the experiments for non-differentiability, both functions feature only a single non-differentiable point. A comparison between functions with single versus multiple non-differentiable points would be more insightful, given the focus on the impact of these points.

The discussion lacks depth. Given the simplicity of both the functions and network structures used, there is potential for a more detailed examination of how parameters are trained and the reasons behind specific outcomes.

The discussion section does not yield any intriguing or unexpected conclusions, nor does it propose any novel theories or structures.

---

confidence: 4
contribution: 1
initial_rating: 3
presentation: 2
questions: If authors can provide some theoretical insights backing the observed empirical results in all or some of the function classes tested, it would make the work more promising and considerable for this conference.
soundness: 2
strengths: The work has considered different classes of functions with some common irregularities. The empirical comparison is sound, and extensive plots provided let's the reader to compare the performance of tested KAN and MLPs in each case.
summary: In this empirical study, the authors compare the performance of two ad-hoc versions of KAN and MLPs, both with identical parameter counts, in learning ten single-dimensional real functions, with and without added noise.
weaknesses: The comparison primarily centers on the resulting test accuracy curves; however, it lacks the necessary theoretical justification and fundamental analysis to substantiate the findings.

While the authors have structured the text well, the plots are somewhat cluttered and could be presented more effectively for better clarity. 

Overall, the work appears basic and does not demonstrate the level of novelty typically expected from submissions to ICLR.

Meta-review: The paper compares the performance of Kolmogorov-Arnold Networks and Multi-Layer Perceptrons in approximating various types of functions, including irregular and noisy ones. Experiments demonstrate that while KANs often outperform MLPs for certain regular or singular functions, MLPs show better performance for functions with localized irregularities or jumps. 

The paper is motivated by an interesting question but lacks sufficient evidence and deep discussion. It also offers no new interpretations, theoretical implications, or methodological suggestions for improving the problem. The reviewers agree that much can still be done with this paper.

Decision: Reject