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

Bibliographic Details
Title: A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data
Authors: Kiyani, Elham, Shukla, Khemraj, Karniadakis, George Em, Karttunen, Mikko
Publication Year: 2023
Collection: Computer Science
Condensed Matter
Subject Terms: Condensed Matter - Disordered Systems and Neural Networks, Computer Science - Machine Learning
More Details: 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.
Document Type: Working Paper
DOI: 10.1016/j.cma.2023.116258
Access URL: http://arxiv.org/abs/2305.10706
Accession Number: edsarx.2305.10706
Database: arXiv
More Details
DOI:10.1016/j.cma.2023.116258