A mesh-based neural energy method for the simulation of heterogeneous composites

Abstract

Modeling heterogeneous materials remains a challenge for physics-informed neural networks such as the deep energy method (DEM). The DEM and its variants, here collectively referred to as the neural energy method (NEM), offer a differentiable variational framework. However, their conventional collocation-based implementation (C-NEM) often suffers from physically inadmissible displacement oscillations, integration errors, and high computational costs from automatic differentiation. This work introduces the mesh-based neural energy method (M-NEM), extending the NEM through a mesh-based discretization of the displacement field. By interpolating nodal displacements via shape functions, the M-NEM imposes a kinematic constraint that suppresses oscillations. Furthermore, the method replaces automatic differentiation with algebraic shape function derivatives for strain computation and employs high-order Gaussian quadrature for accurate energy integration. On a directly comparable benchmark problem, the M-NEM reduces stress errors by up to three orders of magnitude relative to the C-NEM while being one to two orders of magnitude faster. On two further benchmarks involving extreme stiffness contrasts, only the M-NEM converges. A comparative study of neural architectures reveals that radial basis function neural networks (RBFNNs) yield optimal performance within the M-NEM, resolving sharp gradients at material interfaces with higher accuracy than multi-layer perceptrons (MLPs) with random Fourier feature (RFF) mapping and faster convergence than Kolmogorov-Arnold networks (KANs).