Failure mechanisms and resolution in deep energy method

Abstract

The deep energy/Ritz method (DEM/DRM) offers advantages over physics-informed neural networks (PINNs), including reduced derivative orders and accelerated training. However, DEM encounters critical failure modes in both forward and inverse analyses, with underlying mechanisms and robust remedies remaining underexplored. To our knowledge, this work presents the first formal analysis that systematically identifies two distinct DEM failure modes, forward divergence and inverse collapse, and establishes their root causes along with sound countermeasures. In forward analysis, DEM training may diverge due to artificial energy minimization, where abrupt loss reductions below the physically admissible minimum occur with catastrophic errors, which are thermodynamically infeasible but remain unclarified. We prove that this stems from numerical integration inaccuracies in neural network representations, inducing pathological overfitting with escalating complexity. In inverse problems involving unknown material parameters or Neumann boundary conditions, we reveal that DEM fails because its variational formulation with respect to such unknown parameters is not well defined. To overcome these limitations, we propose a novel Energy-Informed Neural Operator Network (EINO), integrating a new regularization technique. Our framework incorporates: (1) a finite-element-informed regularization that lower-bounds the loss by the ground-truth FEM energy to ensure stability, and (2) a deep operator architecture with two-stage training that reconstructs unknown parameters/boundary conditions by embedding inverse constraints. Comprehensive benchmarks on 2D/3D linear/nonlinear solid mechanics and diffusion problems confirm EINO’s superiority over DEM. EINO resolves forward divergence even on very coarse meshes and achieves substantially lower parameter errors in inverse discovery (e.g., <2% relative error under 200% Gaussian noise). The elucidated failure mechanisms and the EINO framework collectively promote physics-constrained learning for surrogate modeling and inverse uncertainty quantification, minimizing the reliance on labeled data. Graphical abstract: [Figure not available: see fulltext.]