Physics-informed sampled neural networks (PISaNN) for modelling nonlinear waves

Abstract

Physics-informed neural networks (PINNs) have emerged as a promising alternative to traditional numerical solvers for partial differential equations. This work presents a physics-informed sampled neural network (PISaNN) framework that combines causal sequential training with physics-informed adaptive sampling for nonlinear wave propagation problems in soft solids governed by cubic nonlinearity. The PISaNN consists of a fully connected dense multilayer perceptron trained without labelled data. We employ a physics-informed sampling strategy specifically designed to resolve steep gradients and discontinuities characteristic of shock-dominated shear wave propagation. This strategy is integrated into a causality-preserving algorithm that sequentially enforces the governing partial differential equation, ensuring stable learning of strongly nonlinear wave propagation. The resulting physics-informed sampled neural network represents the first PINN-based framework capable of capturing cubic nonlinear shear-wave behaviour. We evaluate the method using benchmark problems and compare its performance with that of the high-order Piecewise Parabolic Method solver for nonlinear shear waves. To the authors’ knowledge, this study provides the first application of a PINN-based framework to cubic nonlinear shear-wave propagation in soft solids. Across both linear and nonlinear regimes, PISaNN out performs the baseline PINNs while requiring no simulation or experimental data for training.