GradNorm Physics-Informed Neural Networks for Linear Elasticity: Adaptive Loss Balancing and Comparative Finite Element Analysis
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Abstract
This study presents a comparative analysis between the classical Finite Element Method (FEM) and Physics-Informed Neural Networks (PINNs) for linear elasticity. A well-known challenge in PINN formulations stems from imbalances among loss components associated with governing equations and boundary conditions, which often induce training instabilities and ill-conditioned optimization dynamics. To address this issue, we employ GradNorm, a gradient-based adaptive loss-balancing strategy, to dynamically adjust the weighting parameters during training, thereby mitigating optimization stiffness and improving convergence. The proposed PINN approach is systematically evaluated against high-fidelity FEM benchmarks across various geometries and loading conditions. Numerical results demonstrate that, while FEM remains a highly computationally efficient method for linear elastic problems, PINNs equipped with GradNorm-based adaptive weighting constitute a robust, mesh-free alternative with comparable accuracy. These findings underscore the efficacy of adaptive loss-balancing strategies for enhancing the reliability of PINNs in computational mechanics.
Keywords
Finite element method, linear elasticity, forward problem, partial differential equations, physics-informed neural networks.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
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