A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data

TL;DR

This paper introduces a hybrid framework combining extended physics-informed neural networks and symbolic regression to accurately uncover unknown nonlinear equations of motion from noisy data, with minimal data requirements.

Contribution

It develops a novel gray-box learning framework that integrates X-PINNs with symbolic regression, enhancing equation discovery from data with noise and limited samples.

Findings

Accurately predicts equations of motion using at least 50% data.

Framework is robust against significant noise in data.

Successfully uncovers closed-form equations from simulated data.

Abstract

We propose a framework and an algorithm to uncover the unknown parts of nonlinear equations directly from data. The framework is based on eXtended Physics-Informed Neural Networks (X-PINNs), domain decomposition in space-time, but we augment the original X-PINN method by imposing flux continuity across the domain interfaces. The well-known Allen-Cahn equation is used to demonstrate the approach. The Frobenius matrix norm is used to evaluate the accuracy of the X-PINN predictions and the results show excellent performance. In addition, symbolic regression is employed to determine the closed form of the unknown part of the equation from the data, and the results confirm the accuracy of the X-PINNs based approach. To test the framework in a situation resembling real-world data, random noise is added to the datasets to mimic scenarios such as the presence of thermal noise or instrument errors. The results show that the framework is stable against significant amount of noise. As the final part, we determine the minimal amount of data required for training the neural network. The framework is able to predict the correct form and coefficients of the underlying dynamical equation when at least 50\% data is used for training.