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

Technical Summary: Optimizing Parameterized Physics-Informed Neural Networks for Multilayered Static Linear Elastic PDEs

Problem Statement
The design of multilayered materials for controlled deformation is currently bottlenecked by the high computational costs and time requirements of traditional Finite Element Methods (FEM). These expenses severely restrict the exploration of design spaces and the application of gradient-based optimization. The specific challenge addressed in this work is the simulation of multilayer elastic plate-like structures under surface loading. The goal is to develop a parametric model capable of predicting mechanical responses across variations in Young's modulus and layer thickness while maintaining physical consistency and quantitative accuracy against numerical solutions.

Methodology
The authors propose a framework for Parameterized Physics-Informed Neural Networks (P2INNs) that integrates three core components: a trusted FEM baseline, a parametric PINN displacement field model, and an automated FEM-referenced evaluation system.

1. Problem Formulation: The study models a 3D rectangular plate ($L_x=L_y=1.0, H=0.1$) partitioned into $N_L$ layers (specifically $N_L=1$ and $N_L=3$). The governing physics is static linear elasticity defined by the Navier-Cauchy equations ($-\nabla \cdot \sigma = 0$). The model accepts spatial coordinates $(x, y, z)$ and parameter vectors representing layer thicknesses ($t_i$) and Young's moduli ($E_i$).
2. FEM Baseline: A standard FEM solver using 8-node hexahedral elements with trilinear basis functions and 2-point Gaussian quadrature serves as the ground truth. Material properties are assigned based on element centroid positions to capture discontinuities at layer interfaces.
3. P2INN Architecture:
   • Network Structure: A multilayer perceptron (MLP) with 4 hidden layers (64 neurons each) and hyperbolic tangent activation.
   • Input Handling: Spatial coordinates and material parameters are normalized. For multilayer cases, the network encodes layer boundaries via normalized signed distances and soft layer indicators to handle material discontinuities.
   • Compliance Scaling: A critical innovation involves scaling the network output by effective stiffness and thickness parameters ($E_{eff}, H_{eff}$) to reflect the mechanics of multilayer plate responses, preventing the network from struggling with varying magnitude scales.
   • Boundary Conditions: Side boundaries are clamped ($u=0$) using a hard constraint (multiplicative mask), while traction conditions on the top/bottom surfaces are enforced via soft penalties.
4. Training Strategy & Loss Functions: The total loss is a weighted sum of six terms: PDE residuals, boundary conditions, load patch traction, energy consistency, interface continuity, and optional sparse data supervision.
   • Physics-First Design: The framework employs layerwise PDE decomposition to balance residuals across stiff and soft layers, preventing stiff layers from dominating the loss landscape.
   • Interface Enforcement: A dedicated loss term enforces traction continuity across layer interfaces, with increased sampling density at these boundaries.
   • Sparse Supervision: For the three-layer model, lightweight FEM supervision is introduced at parameter extremes (corners of the parameter space) to stabilize training in high-compliance regimes without turning the model into a pure data-driven surrogate.
   • Optimization: Training utilizes a two-phase approach: SOAP optimizer for initial exploration followed by L-BFGS for convergence (in the three-layer model).

Key Contributions

• Unified Workflow: The paper presents an end-to-end workflow combining a qualified FEM baseline, a parametric PINN with specific physics constraints, and automated evaluation.
• Novel Training Mechanisms: The introduction of layerwise PDE decomposition, interface continuity penalties, and compliance-aware scaling specifically addresses the challenges of multilayer elasticity, where traditional PINNs often fail due to optimization difficulties and magnitude variations.
• Hybrid Training: The strategic use of sparse FEM supervision at parameter extremes allows the model to generalize across the full parameter space while maintaining a physics-driven core.

Results
The framework was evaluated against the FEM baseline using Volume Mean Absolute Error (VolMAE) across 100 random parameter configurations for both one-layer and three-layer setups.

• One-Layer Configuration (Pure Physics-Driven):
  • Mean VolMAE: 1.56%
  • Worst-case VolMAE: 2.87%
  • The model achieved these results without any FEM supervision data.
• Three-Layer Configuration (Physics + Sparse Supervision):
  • Mean VolMAE: 2.53%
  • Worst-case VolMAE: 4.66%
  • This meets the target of a near-5% worst-case error for preliminary design-space exploration.
• Computational Efficiency:
  • The trained P2INN model reduces inference time from ~0.47 seconds (FEM assembly + solve) to ~0.0025–0.0030 seconds (single forward pass).
  • For a fine sweep of $10^6$ configurations, the PINN approach requires approximately 0.8–0.9 hours compared to 122–132 hours for FEM, representing a speedup of roughly 150–188x.

Significance and Claims
The paper claims that the proposed P2INN framework provides a lightweight surrogate model capable of drastically reducing the need for expensive FEM evaluations. By achieving near-5% worst-case accuracy, the method is positioned as a viable tool for preliminary design-space exploration and inverse design of advanced layered architectures, particularly for protective structures in load-heavy or collision-prone fields.

The authors maintain a modest stance, noting that while the model is accurate within the tested parameter ranges, it is intended as a precursor to more in-depth workflows. They explicitly state that the current static formulation cannot capture transient stress waves, nonlinearities, or time-dependent systems (such as sudden collision spikes), and that broader robustness claims are tied strictly to the tested sweep and extreme-case settings. The work emphasizes reproducibility, with the code made publicly available.