A Multi-Frequency Deep Energy Submodeling Framework for Localized High-Gradient Responses in Solid Mechanics

Physics-informed and energy-based neural methods have shown strong potential for solving solid-mechanics boundary-value problems without labeled solution data. However, existing approaches still face difficulties in problems with localized high-gradient or singular responses. To address this issue, this work develops a multi-frequency deep energy submodeling framework (MFDES) for solid mechanics. The proposed framework combines mesh-based energy integration, a mask-DOF strategy for the strict enforcement of physical and artificial boundaries, and a two-stage global-local submodeling procedure with multi-frequency Fourier networks, so that the smooth global response and the localized high-gradient response can be resolved separately and coupled consistently within a unified deep energy setting. Four representative examples are considered, including boundary-enforcement verification, strongly singular problems with one and multiple localized critical regions, and a three-dimensional stress-concentration problem with weakly localized high-gradient response. The results show that MFDES provides stable and accurate treatment for both physical and artificial boundaries. For strongly singular problems, it achieves a favorable local accuracy-cost performance compared with representative baseline methods. The results also indicate that, once the local submodel becomes sufficiently refined, the final accuracy becomes sensitive to the quality of the transferred artificial-boundary data. For the three-dimensional stress-concentration problem, MFDES improves accuracy, robustness under refinement, and cost-growth behavior.