Crack Propagation Prediction Using Partial Differential Equations-Based Neural Network With Discovered Governing Equations

Abstract

This paper presents a regularization technique using discovered partial differential equations (PDEs) with an example of a surrogate model of crack propagation. Regularization techniques are so essential in machine learning to improve generalization performance. Recently, physics-informed neural networks (PINN) have been proposed, which can be trained to solve supervised learning tasks while respecting any given PDE(s). However, the observational data are often measured under complex physical phenomena with fluctuations. PINN is, therefore, difficult to apply to multiphysics problems. Thus, PDE(s) is discovered, and the loss is defined by the weighted sum of the prediction loss and the PDE(s) loss as regularization, which is expected to reduce the validation loss and improve the generalization performance. PDE(s) is discovered using data that have been partially differentiated using Pytorch's automatic differentiation package. Automatic differentiation allows us to discover PDE(s) consisting of input and output parameters. In the surrogate model of crack propagation, the interpretable PDEs were discovered using AI Feynman, which consists of the derivatives of crack propagation vectors and their rate with respect to the coordinates of the analysis model. The validation loss for training constrained by PDEs was reduced by about 93% compared to the unconstrained validation loss. The error in crack length reduced from 2.23% to 0.12%, and the error in crack propagation rate reduced from 19.26% to 1.60%. The effectiveness of the discovered PDE regularization is discussed based on training results.