Robust PDE discovery under sparse and highly noisy conditions via attention neural networks

TL;DR

This paper presents ANN-PYSR, an attention neural network-based framework that robustly discovers PDEs from sparse, noisy data, outperforming existing methods in efficiency and accuracy across various benchmarks.

Contribution

The paper introduces ANN-PYSR, a novel integration of attention neural networks with symbolic regression for PDE discovery under challenging data conditions.

Findings

Successfully identifies PDEs in six benchmark examples.

Outperforms DLGA in robustness and efficiency with noisy, sparse data.

Effective in high noise levels up to 200% with limited sampling points.

Abstract

The discovery of partial differential equations (PDEs) from experimental data holds great promise for uncovering predictive models of complex physical systems. In this study, we introduce an efficient automatic model discovery framework, ANN-PYSR, which integrates attention neural networks with the state-of-the-art PySR symbolic regression library. Our approach successfully identifies the governing PDE in six benchmark examples. Compared to the DLGA framework, numerical experiments demonstrate ANN-PYSR can extract the underlying dynamic model more efficiently and robustly from sparse, highly noisy data (noise level up to 200%, 5000 sampling points). It indicates an extensive variety of practical applications of ANN-PYSR, particularly in conditions with sparse sensor networks and high noise levels, where traditional methods frequently fail.