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

References in corpus (13)

• Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems
• A Physics Informed Neural Network for Time-Dependent Nonlinear and Higher Order Partial Differential Equations
• Simulation of multi-species flow and heat transfer using physics-informed neural networks
• Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl
• When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?
• Sparse identification of nonlinear dynamics with low-dimensionalized flow representations
• Physics-Informed Neural Networks for Shell Structures
• Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensors
• Mixing ADAM and SGD: a Combined Optimization Method
• Machine learning based data-driven discovery of nonlinear phase-field dynamics
• Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Method
• Black and Gray Box Learning of Amplitude Equations: Application to Phase Field Systems
• Momentum Diminishes the Effect of Spectral Bias in Physics-Informed Neural Networks

Cited by in corpus (1)

• Turbulence Closure in RANS and Flow Inference around a Cylinder using PINNs and Sparse Experimental Data