Discovering Interpretable Physical Models using Symbolic Regression and Discrete Exterior Calculus

TL;DR

This paper introduces a novel framework combining Symbolic Regression and Discrete Exterior Calculus to automatically discover interpretable and mathematically consistent physical models from experimental data, enhancing generalization and understanding.

Contribution

The paper presents a new method that integrates DEC with SR to discover physical models, ensuring mathematical consistency and broad applicability beyond previous SR applications.

Findings

Successfully rediscovered Poisson, Euler's Elastica, and Linear Elasticity models from synthetic data.

DEC enables a strongly-typed SR that guarantees model consistency.

The approach generalizes well with limited data.

Abstract

Computational modeling is a key resource to gather insight into physical systems in modern scientific research and engineering. While access to large amount of data has fueled the use of Machine Learning (ML) to recover physical models from experiments and increase the accuracy of physical simulations, purely data-driven models have limited generalization and interpretability. To overcome these limitations, we propose a framework that combines Symbolic Regression (SR) and Discrete Exterior Calculus (DEC) for the automated discovery of physical models starting from experimental data. Since these models consist of mathematical expressions, they are interpretable and amenable to analysis, and the use of a natural, general-purpose discrete mathematical language for physics favors generalization with limited input data. Importantly, DEC provides building blocks for the discrete analogue of field theories, which are beyond the state-of-the-art applications of SR to physical problems. Further, we show that DEC allows to implement a strongly-typed SR procedure that guarantees the mathematical consistency of the recovered models and reduces the search space of symbolic expressions. Finally, we prove the effectiveness of our methodology by re-discovering three models of Continuum Physics from synthetic experimental data: Poisson equation, the Euler's Elastica and the equations of Linear Elasticity. Thanks to their general-purpose nature, the methods developed in this paper may be applied to diverse contexts of physical modeling.