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Symbolic recovery of PDEs from measurement data

Erion Morina, Philipp Scholl, Martin Holler

TL;DR

This work studies identifiability and reconstructibility of symbolic physical laws governing PDEs from indirect, noisy data. It introduces symbolic networks built from rational-function transformations to express the right-hand side $f$ in a function-space PDE via $\partial_t u = f(t, \mathcal{J}_\kappa u)$ and analyzes an all-at-once learning formulation with regularization. The main theoretical contribution is an identifiability result showing that, when the true law lies in the chosen symbolic-architecture, the state and law can be recovered in the noiseless, complete-data limit, with the learned law minimizing a sparsity-promoting objective. Complementary regularity results for the networks in function space and a ParFam-based numerical validation corroborate the theory, demonstrating practical recoverability of concise, interpretable physical laws from indirect measurements. The findings advance symbolic PDE discovery by uniting continuous PDE structure, identifiability theory, and data-driven symbolic regression, with implications for interpretable scientific modeling and verification.

Abstract

Models based on partial differential equations (PDEs) are powerful for describing a wide range of complex relationships in the natural sciences. Accurately identifying the PDE model, which represents the underlying physical law, is essential for a proper understanding of the problem. This reconstruction typically relies on indirect and noisy measurements of the system's state and, without specifically tailored methods, rarely yields symbolic expressions, thereby hindering interpretability. In this work, we address this issue by considering existing neural network architectures based on rational functions for the symbolic representation of physical laws. These networks leverage the approximation power of rational functions while also benefiting from their flexibility in representing arithmetic operations. Our main contribution is an identifiability result, showing that, in the limit of noiseless, complete measurements, such symbolic networks can uniquely reconstruct the simplest physical law within the PDE model. Specifically, reconstructed laws remain expressible within the symbolic network architecture, with regularization-minimizing parameterizations promoting interpretability and sparsity in case of $L^1$-regularization. In addition, we provide regularity results for symbolic networks. Empirical validation using the ParFam architecture supports these theoretical findings, providing evidence for the practical reconstructibility of physical laws.

Symbolic recovery of PDEs from measurement data

TL;DR

This work studies identifiability and reconstructibility of symbolic physical laws governing PDEs from indirect, noisy data. It introduces symbolic networks built from rational-function transformations to express the right-hand side in a function-space PDE via and analyzes an all-at-once learning formulation with regularization. The main theoretical contribution is an identifiability result showing that, when the true law lies in the chosen symbolic-architecture, the state and law can be recovered in the noiseless, complete-data limit, with the learned law minimizing a sparsity-promoting objective. Complementary regularity results for the networks in function space and a ParFam-based numerical validation corroborate the theory, demonstrating practical recoverability of concise, interpretable physical laws from indirect measurements. The findings advance symbolic PDE discovery by uniting continuous PDE structure, identifiability theory, and data-driven symbolic regression, with implications for interpretable scientific modeling and verification.

Abstract

Models based on partial differential equations (PDEs) are powerful for describing a wide range of complex relationships in the natural sciences. Accurately identifying the PDE model, which represents the underlying physical law, is essential for a proper understanding of the problem. This reconstruction typically relies on indirect and noisy measurements of the system's state and, without specifically tailored methods, rarely yields symbolic expressions, thereby hindering interpretability. In this work, we address this issue by considering existing neural network architectures based on rational functions for the symbolic representation of physical laws. These networks leverage the approximation power of rational functions while also benefiting from their flexibility in representing arithmetic operations. Our main contribution is an identifiability result, showing that, in the limit of noiseless, complete measurements, such symbolic networks can uniquely reconstruct the simplest physical law within the PDE model. Specifically, reconstructed laws remain expressible within the symbolic network architecture, with regularization-minimizing parameterizations promoting interpretability and sparsity in case of -regularization. In addition, we provide regularity results for symbolic networks. Empirical validation using the ParFam architecture supports these theoretical findings, providing evidence for the practical reconstructibility of physical laws.
Paper Structure (31 sections, 9 theorems, 99 equations, 2 figures)

This paper contains 31 sections, 9 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Inverse problems perspective on model learning.
  3. Learning interpretable models from data.
  4. Identifiability of physical laws.
  5. Scope and contributions of this work.
  6. Structure of the paper.
  7. Symbolic networks
  8. Rational functions
  9. Network design
  10. Architecture.
  11. Parameterization.
  12. Universal approximation
  13. Extension to function space
  14. Framework
  15. Regularity
  16. ...and 16 more sections

Key Result

Proposition 3

Consider symbolic networks in symbolic_network with $d=n_i^r =n_i^\sigma$ and Lipschitz continuous activations $\sigma_i:\mathbb{R}^d\to \mathbb{R}^d$, such that $\sigma_i^{-1}\in\mathcal{C}^3([0,1]^d)$ for $1\leq i\leq L$. Then, for every $f\in\mathcal{C}^3([0,1]^d)$ there exists a sequence of netw If $\nabla\sigma_i$ is Lipschitz continuous for $1\leq i\leq L$, then $(\mathfrak{S}_\sigma^m)_m$ c

Figures (2)

  • Figure 1: Scheme of symbolic network $\mathfrak{S}_\sigma$
  • Figure 2: Numerical performance.

Theorems & Definitions (29)

  • Definition 1: Rational and base function
  • Definition 2: Positive polynomials
  • Proposition 3
  • proof
  • Remark 5: State space regularity
  • Proposition 6
  • proof
  • Remark 8: Approximation of $K^\dagger$
  • Remark 9: Choice of $\mathcal{R}_0$
  • Remark 10: Uniqueness of admissible solution
  • ...and 19 more