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

2026-10-07

AI Summary

This study addresses the displacement oscillations, integration errors, and high computational costs encountered when using physics-informed neural networks to simulate heterogeneous composite materials. To overcome these limitations, this work proposes a mesh-based neural energy method that integrates finite element discretization with a radial basis function neural network architecture. Kinematic constraints are enforced through shape function interpolation to suppress spurious oscillations, while algebraic derivatives combined with high-order Gaussian quadrature replace automatic differentiation, significantly enhancing the resolution of interfacial gradients. Experimental results demonstrate that the proposed approach reduces stress errors by three orders of magnitude and improves computational efficiency by one to two orders of magnitude. Furthermore, the method achieves stable convergence even under extreme stiffness contrast scenarios.

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).

Problem

• heterogeneous composites
• neural energy method
• physics-informed neural networks
• displacement oscillations
• computational cost

Innovation

• Mesh-based neural energy method
• Heterogeneous composites
• Physics-informed neural networks
• Radial basis function neural networks
• Gaussian quadrature