Operator-aware physics-informed neural networks for forward and inverse problems in solid mechanics

Amir Ghorbani Ghezeljehmeidan, Willem Dirk van Driel, Justin Dauwels

Abstract Physics-informed neural networks (PINNs) on complex domains are limited by input representations that encode geometry but do not reflect the physics of the governing PDE. We propose an operator-aware PINN for solid mechanics problems that embeds precomputed eigenmodes of the problem’s own discrete operator as geometry and physics-aware features within a weak-form variational formulation. The displacement field is represented by a neural mapping enriched by operator-aligned features rather than being constrained to a finite element trial space. A hybrid chain-rule formulation propagates spatial derivatives through both coordinate and eigenmode branches, yielding strain fields consistent with operator structure while preserving neural expressivity. The approach maintains the stability of variational formulations while mitigating the spectral bias and geometry-encoding limitations of classical PINNs. For inverse elasticity, the same eigenmodes act as virtual test functions in a weak-form residual, mitigating the stiffness-collapse pathology. The framework provides an instance-based alternative to differentiable finite element method without labeled training data. It achieves competitive accuracy and connects spectral operator theory with physics-informed neural computation.