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Symbolic Recovery of Differential Equations: The Identifiability Problem

Philipp Scholl, Aras Bacho, Holger Boche, Gitta Kutyniok

TL;DR

This work analyzes when a function $u$ solving a differential equation of the form $ rac{ ^n u}{ t^n}=F(g_1,...,g_k)$ uniquely determines the governing PDE within various function classes. It develops necessary and sufficient conditions for PDE uniqueness across linear, polynomial/algebraic, analytic, and continuous/smooth categories, and then specializes to autonomous and non-autonomous ODEs as concrete cases. The authors propose numerical algorithms—FRanCo, S-FRanCo, and Jacobian Rank Computation—to certify uniqueness from data and validate them with extensive experiments (including KdV, algebraic, and analytic examples) demonstrating both unique and non-unique learned equations. The findings highlight that non-uniqueness is common, stressing the need for explicit identifiability analyses in symbolic recovery, and they offer practical tools to assess reliability before drawing physical conclusions. The work also outlines limitations (notably sensitivity to noise) and outlines future directions toward noise-robust, symmetry-aware, and physics-informed identifiability criteria.

Abstract

Symbolic recovery of differential equations is the ambitious attempt at automating the derivation of governing equations with the use of machine learning techniques. In contrast to classical methods which assume the structure of the equation to be known and focus on the estimation of specific parameters, these algorithms aim to learn the structure and the parameters simultaneously. While the uniqueness and, therefore, the identifiability of parameters of governing equations are a well-addressed problem in the field of parameter estimation, it has not been investigated for symbolic recovery. However, this problem should be even more present in this field since the algorithms aim to cover larger spaces of governing equations. In this paper, we investigate under which conditions a solution of a differential equation does not uniquely determine the equation itself. For various classes of differential equations, we provide both necessary and sufficient conditions for a function to uniquely determine the corresponding differential equation. We then use our results to devise numerical algorithms aiming to determine whether a function solves a differential equation uniquely. Finally, we provide extensive numerical experiments showing that our algorithms can indeed guarantee the uniqueness of the learned governing differential equation, without assuming any knowledge about the analytic form of function, thereby ensuring the reliability of the learned equation.

Symbolic Recovery of Differential Equations: The Identifiability Problem

TL;DR

This work analyzes when a function solving a differential equation of the form uniquely determines the governing PDE within various function classes. It develops necessary and sufficient conditions for PDE uniqueness across linear, polynomial/algebraic, analytic, and continuous/smooth categories, and then specializes to autonomous and non-autonomous ODEs as concrete cases. The authors propose numerical algorithms—FRanCo, S-FRanCo, and Jacobian Rank Computation—to certify uniqueness from data and validate them with extensive experiments (including KdV, algebraic, and analytic examples) demonstrating both unique and non-unique learned equations. The findings highlight that non-uniqueness is common, stressing the need for explicit identifiability analyses in symbolic recovery, and they offer practical tools to assess reliability before drawing physical conclusions. The work also outlines limitations (notably sensitivity to noise) and outlines future directions toward noise-robust, symmetry-aware, and physics-informed identifiability criteria.

Abstract

Symbolic recovery of differential equations is the ambitious attempt at automating the derivation of governing equations with the use of machine learning techniques. In contrast to classical methods which assume the structure of the equation to be known and focus on the estimation of specific parameters, these algorithms aim to learn the structure and the parameters simultaneously. While the uniqueness and, therefore, the identifiability of parameters of governing equations are a well-addressed problem in the field of parameter estimation, it has not been investigated for symbolic recovery. However, this problem should be even more present in this field since the algorithms aim to cover larger spaces of governing equations. In this paper, we investigate under which conditions a solution of a differential equation does not uniquely determine the equation itself. For various classes of differential equations, we provide both necessary and sufficient conditions for a function to uniquely determine the corresponding differential equation. We then use our results to devise numerical algorithms aiming to determine whether a function solves a differential equation uniquely. Finally, we provide extensive numerical experiments showing that our algorithms can indeed guarantee the uniqueness of the learned governing differential equation, without assuming any knowledge about the analytic form of function, thereby ensuring the reliability of the learned equation.
Paper Structure (38 sections, 18 theorems, 21 equations, 9 figures, 1 table, 3 algorithms)

This paper contains 38 sections, 18 theorems, 21 equations, 9 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Symbolic regression.
  4. Symbolic recovery of differential equations.
  5. Parameter estimation and identifiability in differential equations.
  6. Our Contributions
  7. Outline
  8. Uniqueness of PDEs
  9. Linear PDEs
  10. Polynomial and Algebraic PDEs
  11. Analytic PDEs
  12. Continuous and Smooth PDEs
  13. Uniqueness of ODEs
  14. Autonomous ODEs
  15. First Order ODEs $u_t=F(u)$
  16. ...and 23 more sections

Key Result

Proposition 1

Define the mapping $g=(g_1,...,g_k):U\rightarrow\mathbb{R}^k$, with $U\subset\mathbb{R}^{m+1}$ open and $g_1,...,g_k$ as in Definition def:unique. Let $V$ be any class of functions mapping from $\mathbb{R}^k$ to $\mathbb{R}$ which is closed under addition and subtraction. Assume there exists $F\in V

Figures (9)

  • Figure 1: Plot of the lowest singular value of $(u(t_i,x_j),u_x(t_i,x_j))_{i,j}\in\mathbb{R}^{60,000\times2}$, where $u_x$ was computed using finite differences of different orders for $u(t,x)=\exp(x-at)$.
  • Figure 2: Plot of the lowest singular value of $(u(t_i,x_j),u_x(t_i,x_j))_{i,j}\in\mathbb{R}^{60,000\times2}$, where $u_x$ was computed using finite differences of different orders for $u(t,x)=(x+bt)\exp(at)$. (adapted from scholl2023icassp)
  • Figure 3: Plot of the lowest singular value of $(u(t_i,x_j),u_x(t_i,x_j),u_{xx}(t_i,x_j),u_{xxx}(t_i,x_j))_{i,j}\in\mathbb{R}^{60,000\times4}$, where the derivatives were computed using finite differences of different orders for $u(t,x)=\frac{c}{2}sech^2(\frac{\sqrt{c}}{2}(x-ct-a))$.
  • Figure 4: Plot of the lowest singular value of the feature matrix consisting out of all monomials up to degree 2 of $u$, $u_x$, $u_{xx}$ and $u_{xxx}$, where the derivatives were computed using finite differences of different orders for $u(t,x)=\frac{c}{2}sech^2(\frac{\sqrt{c}}{2}(x-ct-a))$. (adapted from scholl2023icassp)
  • Figure 5: Smallest singular value of the Jacobian at different points $(t_i,x_j)$ of $g=(u,u_x)$ for $u(t,x)=1/(x+t)$. For the upper image, the derivatives were computed using 2nd-order finite differences and, for the lower image, 7th-order finite differences were used. (adapted from scholl2023icassp)
  • ...and 4 more figures

Theorems & Definitions (44)

  • Example 1
  • Definition 1: Uniqueness
  • Definition 2
  • Proposition 1
  • proof
  • Corollary 1: Uniqueness for linear PDEs
  • proof
  • Definition 3
  • Definition 4
  • Definition 5
  • ...and 34 more