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Robust identifiability for symbolic recovery of differential equations

Hillary Hauger, Philipp Scholl, Gitta Kutyniok

TL;DR

This paper develops a comprehensive mathematical framework to analyze the uniqueness of PDEs in the presence of noise and introduces new algorithms that account for noise, providing thresholds to assess uniqueness and identifying situations where excessive noise hinders reliable conclusions.

Abstract

Recent advancements in machine learning have transformed the discovery of physical laws, moving from manual derivation to data-driven methods that simultaneously learn both the structure and parameters of governing equations. This shift introduces new challenges regarding the validity of the discovered equations, particularly concerning their uniqueness and, hence, identifiability. While the issue of non-uniqueness has been well-studied in the context of parameter estimation, it remains underexplored for algorithms that recover both structure and parameters simultaneously. Early studies have primarily focused on idealized scenarios with perfect, noise-free data. In contrast, this paper investigates how noise influences the uniqueness and identifiability of physical laws governed by partial differential equations (PDEs). We develop a comprehensive mathematical framework to analyze the uniqueness of PDEs in the presence of noise and introduce new algorithms that account for noise, providing thresholds to assess uniqueness and identifying situations where excessive noise hinders reliable conclusions. Numerical experiments demonstrate the effectiveness of these algorithms in detecting uniqueness despite the presence of noise.

Robust identifiability for symbolic recovery of differential equations

TL;DR

This paper develops a comprehensive mathematical framework to analyze the uniqueness of PDEs in the presence of noise and introduces new algorithms that account for noise, providing thresholds to assess uniqueness and identifying situations where excessive noise hinders reliable conclusions.

Abstract

Recent advancements in machine learning have transformed the discovery of physical laws, moving from manual derivation to data-driven methods that simultaneously learn both the structure and parameters of governing equations. This shift introduces new challenges regarding the validity of the discovered equations, particularly concerning their uniqueness and, hence, identifiability. While the issue of non-uniqueness has been well-studied in the context of parameter estimation, it remains underexplored for algorithms that recover both structure and parameters simultaneously. Early studies have primarily focused on idealized scenarios with perfect, noise-free data. In contrast, this paper investigates how noise influences the uniqueness and identifiability of physical laws governed by partial differential equations (PDEs). We develop a comprehensive mathematical framework to analyze the uniqueness of PDEs in the presence of noise and introduce new algorithms that account for noise, providing thresholds to assess uniqueness and identifying situations where excessive noise hinders reliable conclusions. Numerical experiments demonstrate the effectiveness of these algorithms in detecting uniqueness despite the presence of noise.

Paper Structure

This paper contains 9 sections, 4 theorems, 11 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Methods
  4. Singular Values and Finite Differences
  5. Linear PDEs
  6. Algebraic and analytic PDEs
  7. Experiments
  8. Discussion
  9. Conclusion

Key Result

Theorem 1

If $A,E \in \mathbb{R}^{m \times n}$ are two arbitrary matrices, then $|\sigma_k(A+E) - \sigma_k(A)| \leq \|E\|_F$ for all $k=1,...,\min\{m, n\}$.

Figures (3)

  • Figure 1: Figure \ref{['fig:nr_franco_results_one']} shows NR-FRanCo results for a non-unique PDE and three noise levels. The blue line is $\rho(\tilde{G})$ for FD orders 2, 4, 6, and 8. Red and green lines are bounds from Theorem \ref{['theorem:sfranco_unique_class']}. $\rho(\tilde{G})$ in the red segment indicates uniqueness, in the green segment non-uniqueness, and in both segments means no definitive conclusion. Figure \ref{['fig:nr_franco_results_all']} shows how many PDEs were correctly classified by NR-FRanCo for different orders and noise levels $0,10^{-10}, ..., 10^{-1}$ for the PDEs in Table \ref{['tab:linear_pdes']}.
  • Figure 2: JRC results for the PDE from Table \ref{['tab:jrc_pdes']}a).
  • Figure 3: JRC results averaged over all PDEs from Table \ref{['tab:jrc_pdes']}.

Theorems & Definitions (8)

  • Definition 1: Uniqueness scholl2023icasspscholl2023welldefinedness
  • Theorem 1: Weyl_1912Horn_Johnson_2012
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof