Discovery of physically interpretable wave equations

TL;DR

This paper introduces a redesigned symbolic regression framework that simultaneously evolves functional forms and coefficients, ensuring physical interpretability, to discover wave equations from noisy, sparse data in various media.

Contribution

It presents a novel method combining genetic algorithms and physical constraints to discover interpretable wave equations with complete velocity terms from observational data.

Findings

Successfully discovers physically reasonable wave equations from noisy data.

Effective in both homogeneous and inhomogeneous media.

Handles realistic, sparse observational scenarios.

Abstract

Using symbolic regression to discover physical laws from observed data is an emerging field. In previous work, we combined genetic algorithm (GA) and machine learning to present a data-driven method for discovering a wave equation. Although it managed to utilize the data to discover the two-dimensional (x,z) acoustic constant-density wave equation u_tt=v^2(u_xx+u_zz) (subscripts of the wavefield, u, are second derivatives in time and space) in a homogeneous medium, it did not provide the complete equation form, where the velocity term is represented by a coefficient rather than directly given by v^2. In this work, we redesign the framework, encoding both velocity information and candidate functional terms simultaneously. Thus, we use GA to simultaneously evolve the candidate functional and coefficient terms in the library. Also, we consider here the physics rationality and interpretability in the randomly generated potential wave equations, by ensuring that both-hand sides of the equation maintain balance in their physical units. We demonstrate this redesigned framework using the acoustic wave equation as an example, showing its ability to produce physically reasonable expressions of wave equations from noisy and sparsely observed data in both homogeneous and inhomogeneous media. Also, we demonstrate that our method can effectively discover wave equations from a more realistic observation scenario.