Symbolic recovery of PDEs from measurement data

2026-02-17

AI Summary

Accurately identifying interpretable partial differential equation (PDE) models from noisy, indirect measurements remains a significant challenge that impedes the understanding of underlying physical laws. This work proposes a rational-function-based neural network architecture, such as ParFam, which enables symbolic modeling to directly reconstruct PDEs from system state data. The core contributions include establishing a theoretical framework for the unique identifiability of PDEs, proving that the simplest form can be uniquely recovered under noise-free and complete data conditions; introducing L¹ regularization to enhance sparsity and interpretability of the solution; and developing a regularity theory for symbolic networks. Experimental results demonstrate that the proposed method efficiently recovers concise, accurate, and physically interpretable PDE models.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• symbolic recovery
• partial differential equations
• measurement data
• physical laws
• interpretability

Innovation

• symbolic recovery
• partial differential equations
• rational neural networks
• identifiability
• L1 regularization