Automating the Discovery of Partial Differential Equations in Dynamical Systems

TL;DR

This paper introduces ARGOS-RAL, an automated method that uses sparse regression and adaptive lasso to identify partial differential equations from data, even with noise and limited prior knowledge.

Contribution

The paper extends the ARGOS framework with a new approach that automates PDE discovery using sparse regression and adaptive lasso, improving robustness and accuracy.

Findings

ARGOS-RAL outperforms existing methods in noisy data scenarios.

The method reliably identifies PDEs from limited and non-uniform data.

It effectively distinguishes signal from noise, even with severely compromised data.

Abstract

Identifying partial differential equations (PDEs) from data is crucial for understanding the governing mechanisms of natural phenomena, yet it remains a challenging task. We present an extension to the ARGOS framework, ARGOS-RAL, which leverages sparse regression with the recurrent adaptive lasso to identify PDEs from limited prior knowledge automatically. Our method automates calculating partial derivatives, constructing a candidate library, and estimating a sparse model. We rigorously evaluate the performance of ARGOS-RAL in identifying canonical PDEs under various noise levels and sample sizes, demonstrating its robustness in handling noisy and non-uniformly distributed data. We also test the algorithm's performance on datasets consisting solely of random noise to simulate scenarios with severely compromised data quality. Our results show that ARGOS-RAL effectively and reliably identifies the underlying PDEs from data, outperforming the sequential threshold ridge regression method in most cases. We highlight the potential of combining statistical methods, machine learning, and dynamical systems theory to automatically discover governing equations from collected data, streamlining the scientific modeling process.