SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Step 1: Choose the form and incorporate prior knowledge. A human expert should encode their prior knowledge of the problem into a dictionary of non-degenerate extended derivatives  $\mathcal{Q} = \{\hat{\mathcal{E}}_p\}_{p\in [P]}$ . We use this dictionary to search over a finite-dimensional subspace of closed-form operators spanned by this set. In other words, we assume that the VR-PDE is of the form:
$$
\sum_ {p = 1} ^ {P} \beta_ {p} \hat {\mathcal {E}} _ {p} [ \boldsymbol {u} ] (\boldsymbol {x}) - g (\boldsymbol {x}, \boldsymbol {u} (\boldsymbol {x})) = 0 \forall \boldsymbol {x} \in \Omega \tag {13}
$$
where  $\beta \in \mathbb{R}^P$ ,  $g$  is any closed-form function of  $M + N$  variables, and  $\hat{\mathcal{E}}_p = (\alpha_p, a_p, h_p)$ .
For instance, a dictionary might include only the partial derivatives up to a certain order. For a  $1 + 1$  second-order equation that means  $\mathcal{Q} = \{\partial_t,\partial_x,\partial_{tx},\partial_t^2,\partial_x^2\}$ . That is already enough to discover heat and wave equations with any closed-form source. If, for instance, the user suspects the presence of the advection term  $uu_{x}$  (as in the Burgers' equation), the term  $\partial_x(u^2)$  can be included in the library.
It's important to note that we do not assume any particular form of  $g$  apart from being closed-form.
Step 2: Estimate the fields. As the dataset  $\mathcal{D}$  consists of noisy and infrequently sampled fields, we first need to estimate the "true" fields  $\hat{\pmb{u}}^{(d)}$  from  $\pmb{v}^{(d)}$ . Any choice of reconstruction algorithm can be used and the user should choose it according to the problem setting and their domain knowledge.
Step 3: Optimize. We minimize the loss function in Equation 11 for the estimated fields  $\{\hat{\pmb{u}}^{(d)}\}_{d = 1}^{D}$  among all PDEs of the form in Equation 13. We solve the following optimization problem:
$$
\min  _ {g} \min  _ {| | \boldsymbol {\beta} | | _ {1} = 1} \sum_ {d = 1} ^ {D} \sum_ {s = 1} ^ {S} \left(\sum_ {p = 1} ^ {P} \mathcal {F} \left(\beta_ {p} \hat {\mathcal {E}} _ {p}, \hat {\boldsymbol {u}} ^ {(d)}, \phi_ {s}\right) - \int_ {\Omega} g (\boldsymbol {x}, \hat {\boldsymbol {u}} ^ {(d)} (\boldsymbol {x})) \phi_ {s} (\boldsymbol {x}) d \boldsymbol {x}\right) ^ {2} \tag {14}
$$
As we want to discover both  $g$  and  $\beta$  we cannot use the standard penalties on  $\beta$  such as the  $\lambda ||\beta||_2$  or  $\lambda ||\beta||_1$ , as the loss would be minimized by  $g = 0$  and  $\beta = 0$ . Therefore we put the constraint  $||\beta||_1 = 1$ . We choose the L1 norm to encourage sparsity in the coordinates of the vector  $\beta$ .
The inner minimization in Equation 14 can be rewritten as a constrained least-squares problem.
$$
\min  _ {\left| \left| \boldsymbol {\beta} \right| \right| _ {1} = 1} \sum_ {(d, s) \in [ D ] \times [ S ]} \left(\boldsymbol {\beta} \cdot \boldsymbol {z} ^ {(d, s)} - w ^ {(d, s)}\right) ^ {2} \tag {15}
$$
where  $\hat{\mathcal{E}}_p = (\alpha_p, a_p, h_p)$  and  $\boldsymbol{z}^{(d,s)} \in \mathbb{R}^P$ ,  $w^{(d,s)} \in \mathbb{R}$  are defined as
$$
z _ {p} ^ {(d, s)} = \int_ {\Omega} h _ {p} (\boldsymbol {x}, \hat {\boldsymbol {u}} ^ {(d)} (\boldsymbol {x})) (- 1) ^ {| \boldsymbol {\alpha} _ {p} |} \partial^ {\boldsymbol {\alpha} _ {p}} \left(a _ {p} (\boldsymbol {x}) \phi_ {s} (\boldsymbol {x})\right) d \boldsymbol {x} \tag {16}
$$
$$
w ^ {(d, s)} = \int_ {\Omega} g (\pmb {x}, \hat {\pmb {u}} ^ {(d)} (\pmb {x})) \phi_ {s} (\pmb {x}) d \pmb {x}
$$
We show the full derivation in Appendix C.  $\pmb{z}^{(d,s)}$  can be precomputed at the beginning of the algorithm without estimating the derivatives of the reconstructed fields. They can be easily calculated if the derivatives of the testing functions  $\phi_s$  and the derivatives of  $a_p$  can be analytically computed.
As the optimization problem in Equation 15 has to be solved many times for different closed-form expressions  $g$ , it poses some unique challenges. As standard approaches are not sufficiently fast, we design a new heuristic algorithm to solve this problem, CoLLie, and describe it in the next section.
Figure 1: This diagram describes how the algorithm works. After the optimization procedure is finished, we get the best found closed-form function  $g$  and use CoLLie to find the best vector  $\beta$ . The found equation has the form  $\sum_{p=1}^{P} \beta_p \hat{\mathcal{E}}_p[\boldsymbol{u}](\boldsymbol{x}) - g(\boldsymbol{x}, \boldsymbol{u}(\boldsymbol{x})) = 0$
7 CoLLie
The problem in Equation 15 from the previous section can be formulated as follows. Given matrix  $\mathbf{A} \in \mathbb{R}^{m \times n}$  and vector  $\mathbf{b} \in \mathbb{R}^m$ , find a vector  $\mathbf{z} \in \mathbb{R}^n$  that minimizes  $||\mathbf{A}\mathbf{z} - \mathbf{b}||_2^2$  such that  $||\mathbf{z}||_1 = 1$ . The task is challenging as the unit L1 sphere is not convex. A method that guarantees an optimal solution is based on an observation that the  $(n - 1)$ -dimensional L1 sphere consists of  $2^n$ $(n - 1)$ -simplices (which are convex). Minimizing  $||\mathbf{A}\mathbf{z} - \mathbf{b}||_2^2$  on a simplex is a quadratic program [8] with many available solvers [2, 51, 3]. However, that means that the computation time scales exponentially with the number of dimensions. This is prohibitively long for the inner optimization of our algorithm. Therefore, we design a heuristic algorithm CoLLie (Constrained L1 Norm Least Squares) that finds an approximate solution but is significantly faster (Figure 2). We observe that this optimization problem is related to the one encountered in LASSO. Denote  $z_0$  the solution that minimizes  $||\mathbf{A}\mathbf{z} - \mathbf{b}||_2^2$  (no constraints). If  $||\mathbf{z}_0|| \geq 1$ , the problem is equivalent to finding  $\lambda$  (in the Lagrangian form of LASSO, Equation 47) such that the LASSO solution has the norm 1. Least Angle Regression (LARS) [17] is a popular algorithm used to minimize the LASSO objective that computes complete solution paths. These paths show how the coefficients of the solution change as  $\lambda$  moves from 0 to  $\lambda_{max}$  (from no constraints to effectively imposing the solution to be 0). See Figure 6 in Appendix D.2. CoLLie uses these solution paths to calculate the exact solution to the optimization problem. The case  $0 < ||\mathbf{z}_0|| < 1$  is harder as it corresponds to  $\lambda < 0$ . CoLLie addresses this challenge by extending the solution paths generated by LARS beyond  $\lambda = 0$  for  $\lambda < 0$ . We assume that the paths continue to be piecewise linear and keep their slope (Figure 6 in Appendix
D.2). CoLLie then uses these assumptions to efficiently find an approximate solution. We provide a detailed description of CoLLie in Appendix D.
Figure 2: We compare CoLLie with an algorithm that uses CVXOPT [2] to solve each of the convex subproblems. We report the relative error between the loss obtained by CoLLie and the minimum loss achieved by CVXOPT. Panels B and C show the averages and the distributions of relative errors. The average relative error is below 0.005 and the bulk of the distribution is below  $10^{-7}$ . At the same time CoLLie is orders of magnitude faster (Panel A).
8 Experiments
We perform a series of synthetic experiments to show how well D-CIPHER is able to discover some well-known differential equations $^4$  (Table 2). First, we demonstrate that D-CIPHER performs better than current methods when discovering PDEs in a linear combination form (Section 8.1). Then we demonstrate it can discover PDEs with a closed-form  $\partial$ -free part that cannot be expressed as a linear combination and thus are beyond the capabilities of current methods (Section 8.2). We contrast D-CIPHER with its ablated version where the derivatives are estimated and the standard MSE loss is used instead of the variational loss (details in Appendix E.1). For additional information about the experiments (e.g., implementation details, data generation, experimental settings) see Appendix E.
Table 2: Equations used in the experiments. "LC" column specifies if the equation can be represented as a linear combination (Equation 5). "VR" column specifies if the PDE is Variational-Ready
Name	Equation	LC	VR
Homogeneous heat equation	∂t u - θ1∂x2u = 0	✓	✓
Burger's equation	∂t u + u∂x u - θ1∂x2u = 0	✓	✓
Kuramoto-Sivashinsky equation	∂t u + ∂x2u + ∂x4u + u∂x u = 0	✓	✓
Forced and damped harmonic oscillator	∂t2 + 2θ1θ2∂t u + θ22 u = θ3sin(θ4t)	X	✓
SLM model (Appendix F.1)	∂t u + ∂x u = -2eθ1ux	X	✓
Inhomogeneous heat equation	∂t u - θ1∂x2u = θ2eθ3t	X	✓
Inhomogeneous wave equation	∂t2 u - θ1∂x2u = θ2et sin(θ3t)	X	✓
Evaluation metrics. To establish how well a discovered PDE matches the ground truth, we evaluate its  $\partial$ -free and  $\partial$ -bound parts separately. For the  $\partial$ -free part, we assign a binary variable indicating whether the correct functional form of the equation was recovered (please check Appendix E.8 for details). For the  $\partial$ -bound part, we measure the RMSE between the found coefficients of  $\beta$  and the target ones. We report the averages and standard deviations for both parts. We call the averages respectively Success Probability and Average RMSE.
Implementation. We use B-Splines [15] as the testing functions and we estimate the fields in Step 2 of D-CIPHER with a Gaussian Process [60]. The outer optimization in Step 3 is performed using a modified genetic programming algorithm [26] and the inner optimization by CoLLie (Section 7). We also show additional experiments with different estimation algorithms in Appendix F.
8.1 Discovering Linear Combinations: comparison with other methods
We compare D-CIPHER against two variants of PDE-FIND [47] and WSINDy [46] (as implemented in PySINDy library [24, 16]) with optimization performed by Stepwise Sparse Regression [7] or
4All experiment code can be found at https://github.com/krzysztof-kacprzyk/D-CIPHER
Forward Regression Orthogonal Least-Squares [5]. We note that D-CIPHER is specifically designed to discover PDEs that are beyond the capabilities of current methods, i.e., where the derivative-free part can be any closed-form expression. Current methods are usually tested on equations where the derivative-free part is trivial (identically equal to 0). Even though these algorithms are specialized to discover these simpler kinds of equations, D-CIPHER performs better than (or equally well as) PDE-FIND and WSINDy, regardless of the optimization algorithm, when tested on Burgers' equation the homogeneous heat equation, and Kuramoto-Sivashinsky equation (Figure 3). This demonstrates gain from both the variational loss and the new optimization routine.
Figure 3: Simulation results for the Burgers' equation, homogeneous heat equation, and Kuramoto-Sivashinsky equation. We report the Average RMSE of the  $\partial$ -bound part of the equation. Note that some of the benchmarks overlap
8.2 Discovering equations beyond current methods
Forced and damped harmonic oscillator. As the oscillator is described by a second-order ODE, it cannot be discovered by D-CODE [42]. D-CIPHER discovers the correct functional form of the  $\partial$ -free part and achieves a low RMSE for the coefficients of  $\beta$  in most of the experimental settings. The performance is higher than or comparable to the ablated version of D-CIPHER, thus demonstrating gain from using the variational approach. We present the results in Figure 4.
Figure 4: Success probability of discovering the correct  $\partial$ -free part of the equation and the average RMSE between the recovered  $\partial$ -bound part and the target one across different experimental settings. We compare D-CIPHER against its ablated version (Abl. D-CIPHER).
Inhomogeneous heat equation. D-CIPHER is able to discover the correct equation even in settings with very high noise. It performs better than the ablated version, thus showing the importance of the variational objective. The result are presented in Table 3.
Table 3: We report the success probability of discovering the  $\partial$ -free part and the Average RMSE of the  $\partial$ -bound part for the inhomogeneous heat equation. Standard deviations shown in brackets.
Method	Success probability	Average RMSE
σR=0.05	0.1	0.2	σR=0.05	0.1	0.2
D-CIPHER	0.64 (.07)	0.42 (.07)	0.12 (.05)	0.15 (.009)	0.21 (.007)	0.24 (.005)
Ablated D-CIPHER	0.46 (.07)	0.20 (.06)	0.04 (.03)	0.18 (.009)	0.24 (.008)	0.27 (.007)
Inhomogeneous wave equation. This equation does not have the standard evolution form, as it does not involve the  $\partial_t$  term. Thus, even without the source term, most of the current methods cannot be applied directly to discover this equation. In Figure 5 we show the absolute difference between the true field and the fields computed from the sources discovered by D-CIPHER and its ablated version across different measurement settings. D-CIPHER finds the correct functional form with coefficients not far from the ground truth. The ablated version fails to discover the correct functional form and the found  $\partial$ -free part does not reproduce the correct behavior of the equation.
Figure 5: We solve the inhomogeneous wave equation for the  $\partial$ -free parts found by the D-CIPHER and its ablated version Abl. D-CIPHER. We show the absolute difference between the computed fields and the true field generated by  $\partial$ -free part  $2 \times e^{t} \sin(3t)$ .
9 Discussion
Applications. As D-CIPHER can potentially discover any closed-form  $\partial$ -free part, it is especially useful when this part of the PDE captures an essential component of the phenomenon. We demonstrate it by finding the heat and vibration sources as well as the driving force of an oscillator. Beyond the spatio-temporal physical equations, D-CIPHER might prove useful in discovering population models structured by age, size, and spatial position [59, 58], age-dependent epidemiological models [21], and predator-prey models with age-structure [41]. All these equations are VR-PDEs where the  $\partial$ -free parts are crucial elements of the equations signifying the rates of mortality, infection, recovery, or growth. We believe that discovering closed-form equations for these systems would prove invaluable in understanding their behavior.
Limitations and open challenges. D-CIPHER may fail in some scenarios, either due to challenging experimental settings or a challenging underlying PDE. Challenging experimental settings might include unobserved variables, high measurement noise, infrequent sampling, and inadequate domain (e.g., small time horizon). Challenging PDE forms might include a PDE outside of the VR-PDE class or a  $\partial$ -free part with a complex expression that is difficult to find. We note that we address some of these challenges by utilizing a variational approach, defining VR-PDEs to be a very general class of equations, and designing CoLLie, enabling a thorough search across closed-form expressions.
Ethics Statement. We want to emphasize that D-CIPHER was designed to facilitate the process of scientific discovery by extracting closed-form PDEs from data. It is not intended to or capable of replacing human experts in the modeling process. No human-derived data was used.
This work is supported by Cancer Research UK and Roche. We want to thank Samuel Holt, Jonathan Crabbé, and anonymous reviewers for their useful comments and feedback on earlier versions of this work. We are grateful to Professor Yuanzhang Xiao for insightful discussions about the optimization algorithms.
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Table of supplementary materials
1. Appendix A: notation and definitions
2. Appendix B variational formulation for linear PDEs and the proof of Theorem 1
3. Appendix C: details of the D-CIPHER framework, including pseudocode
4. Appendix D: details of the CoLLie algorithm
5. Appendix E: details of experiments and the implementation
6. Appendix F: additional experiments and discussion
A Notation and definitions
Table 4: Symbols used in this work
Symbol	Meaning
[n]	a set of numbers {1, ..., n}
N	a set of natural numbers, i.e., {1, 2, 3, ...,}
N0	a set of non-negative integers, i.e., {0, 1, 2, 3, ...}
M	the dimension of the domain of a vector field
N	the dimension of the codomain of a vector field
K	denotes the smoothness of functions or the maximum order of derivatives
D	the size of the dataset of observed fields
S	the number of testing functions
Ω	an open set in R^M
\(\dot{u}(t)\)	the derivative of u at t
α	a multi-index, an element of N^M
|α|	the order of α, |α| = Σi^M αi
\(\partial_i^{\alpha_i}\)	\(\alpha_i^{\text{th}}\)-order partial derivative with respect to theith variable
\(\partial^\alpha\)	\(\partial_1^{\alpha_1} \partial_2^{\alpha_2} \ldots \partial_M^{\alpha_M}\)
C^K	a set of functions with continuous partial derivatives \(\partial^\alpha\) for all \(|\alpha| \leq K\)
\(\mathcal{E}\)	an extended derivative, Definition 1
\(\mathcal{E}_{[P]}\)	an extended differential operator, Definition 1
\(\mathcal{F}\)	the functional used in the variational loss, Definition 3
\(\mathcal{L}(\mathcal{E}_{[P]}, g)\)	the variational loss, Equation 11
\(\mathcal{G}\)	a sampling grid, Definition 10
u	a true field, Definition 9
v	an observed field, Definition 10
\(\mathcal{D}\)	a dataset of observed trajectories
ε	the noise
\(\mathcal{Q}\)	a dictionary of non-degenerate extended derivatives
β	a vector describing the \(\partial\)-bound part of the VR-PDE
\(\sigma_R\)	a noise ratio
\(\Delta t\)	a sampling interval
A.2 Definitions
In this section, we collect the definitions of some of the important terms used in the paper for easy reference.
Definition 4 (Closed-form expressions and functions). A closed-form expression is a mathematical expression that consists of a finite number of variables, constants, arithmetic operations, and certain well-known functions (e.g., logarithm, trigonometric functions). A function  $f$  is called closed-form if it can be represented by a closed-form expression. E.g.,  $f(x,y) = x^{2}\log (y) + \sin (3z)$ .
Remark. In practice, we do not want to consider any finite expression. Any symbolic regression algorithm penalizes expressions that are too long putting a soft constraint on the number of elements used. That is why deep neural networks are not considered closed-form even if they satisfy the conditions in Definition 4.
Definition 5 (Multi-index). An  $n$ -dimensional multi-index  $\alpha$  is an  $n$ -tuple
$$
\boldsymbol {\alpha} = \left(\alpha_ {1}, \alpha_ {2}, \dots , \alpha_ {n}\right)
$$
where  $\forall i\in [n]\alpha_{i}\in \mathbb{N}_{0}$ . Thus  $\pmb {\alpha}\in \mathbb{N}_0^n$ . We define the order of  $\pmb{\alpha}$  as  $|\pmb {\alpha}| = \sum_{i = 1}^{n}\alpha_{i}$
Definition 6. For any  $n$ -dimensional multi-index  $\alpha$  we define a mixed derivative
$$
\partial^ {\alpha} = \partial_ {1} ^ {\alpha_ {1}} \partial_ {2} ^ {\alpha_ {2}} \dots \partial_ {n} ^ {\alpha_ {n}}
$$
where  $\partial_i^{\alpha_i} = \partial^{\alpha_i} / \partial x_i^{\alpha_i}$  is a  $\alpha_{i}^{\mathrm{th}}$ -order partial derivative with respect to  $x_{i}$  (the  $i^{\mathrm{th}}$  independent variable). We call  $\partial^{\alpha}$  non-trivial if  $|\pmb {\alpha}| > 0$ . We denote the list of all non-trivial partial derivatives of  $u$  up to order  $K$  as  $\partial^{[K]}u$ .
Definition 7 (Closed-form Partial Differential Equation). Let  $f$  be a closed-form real smooth function. We say that a vector field  $\mathbf{u} : \Omega \to \mathbb{R}^N$  is governed by a  $K^{\mathrm{th}}$ -order closed-form PDE described by  $f$  if
$$
f (\boldsymbol {x}, \boldsymbol {u} (\boldsymbol {x}), \partial^ {[ K ]} \boldsymbol {u} (\boldsymbol {x})) = 0 \forall \boldsymbol {x} \in \Omega \tag {17}
$$
where  $\partial^{[K]}\pmb{u}$  are all non-trivial mixed derivatives of all  $u_{j}$  ( $j\in [N]$ ) up to the  $K^{\mathrm{th}}$  order.
Definition 8 (Testing function). Support of a function  $\phi : \Omega \to \mathbb{R}$  is defined as
$$
s u p p \phi = \overline {{\{\boldsymbol {x} \in \Omega : \phi (\boldsymbol {x}) \neq 0 \}}}
$$
where  $\overline{\mathcal{B}}$  is the topological closure of  $\mathcal{B}$  in  $\Omega$ .
$\phi$  is called a testing function if it is a  $\mathcal{C}^K$  function with compact support.
Definition 9 (True Field). We define a true field on  $\Omega$  as a vector valued function  $\mathbf{u}:\Omega \to \mathbb{R}^N$  where each  $u_{j}:\Omega \to \mathbb{R}$  is a  $\mathcal{C}^K$  function.
Definition 10 (Observed field and sampling grid). We define a sampling grid  $\mathcal{G}$  to be a finite subset of  $\Omega$ . Let  $\pmb{u}:\Omega \to \mathbb{R}^N$  be a true field on  $\Omega$ . An observed field sampled from  $\pmb{u}$  on a grid  $\mathcal{G}$  is a function  $\pmb{v}:\mathcal{G}\rightarrow \mathbb{R}^{N}$  of the form
$$
v _ {j} (\boldsymbol {x}) = u _ {j} (\boldsymbol {x}) + \epsilon_ {j} (\boldsymbol {x}) \forall \boldsymbol {x} \in \mathcal {G} \forall j \in [ N ]
$$
where  $\epsilon_{j}(\pmb {x})$  corresponds to noise, a realisation of a zero-mean random variable.
Definition 11 (L1 sphere). Let  $n \in \mathbb{N}$ . We define  $n$ -dimensional L1 sphere to be a subset of  $\mathbb{R}^{n+1}$  defined as:
$$
\left\{\boldsymbol {x} \in \mathbb {R} ^ {n + 1} \mid | | x | | _ {1} = 1 \right\} \subset \mathbb {R} ^ {n + 1} \tag {18}
$$
Definition 12 (Standard simplex). Let  $n \in \mathbb{N}$ . We define standard  $n$ -simplex to be a subset of  $\mathbb{R}^{n+1}$  defined as:
$$
\left\{\boldsymbol {x} \in \mathbb {R} ^ {n + 1} \mid \sum_ {i = 1} ^ {n + 1} x _ {i} = 1 \wedge x _ {i} \geq 0 \forall i \in [ n + 1 ] \right\} \subset \mathbb {R} ^ {n + 1} \tag {19}
$$
B Variational-Ready PDEs
B.1 Variational Formulation of PDEs
In this section, we provide the standard variational formulation of PDEs for linear PDEs [19].
Definition 13 (Linear differential operator). Let  $\mathcal{A}$  be a finite set of multi-indices. A linear differential operator  $L$  is defined as
$$
L = \sum_ {\boldsymbol {\alpha} \in \mathcal {A}} a _ {\boldsymbol {\alpha}} \partial^ {\boldsymbol {\alpha}}
$$