Optimizing Parameterized Physics-Informed Neural Networks to Solve Multilayered Static Linear Elastic PDEs

Abstract

Designing multilayered materials for controlled deformation is heavily bottlenecked by the immense computational costs and time of traditional, industrial-grade finite element methods (FEM). This excessive expense severely limits design space exploration and gradient-based optimization. To streamline the workflow, we propose a framework for parameterized physics-informed neural networks (P2INNs). The framework encompasses three core components: I. a FEM baseline with a Hex8 element and trilinear basis functions, II. a P2INN displacement field model across variable material stiffness and layer thicknesses, and III. FEM-referenced evaluation. The PINN enforces static linear elasticity via Navier-Cauchy residuals with training strategies that prioritize physics, including layerwise PDE decomposition, interface continuity penalty, and compliance aware scaling. For one-layer, pure physics-driven configurations, the PINN achieves a mean volume MAE of 1.56% and worst-case volume MAE of 2.87% against the FEM benchmark. For three-layer configurations with controlled supervised training, the PINN achieves a mean volume MAE of 2.53% and worst-case volume MAE of 4.66%, meeting the near-5% worst-case volume-MAE target for preliminary design-space exploration. The trained P2INN model remains a lightweight model with a simple forward pass for future calculations, creating an alternative to traditional FEM. By drastically reducing the need for expensive FEM evaluations, this approach could accelerate the inverse design and optimization of advanced layered architectures for protective structures in various load-heavy or potentially collision-heavy fields.