LENNs: Locally Enhanced Neural Networks for High-Fidelity Modeling in Solid Mechanics

Abstract

Despite prior advances in PINNs, significant challenges remain in localized solid mechanics problems because of the limitations of single network formulations in simultaneous resolution of smooth global responses and near-tip singularities, and inadequacy in discontinuity representation, leading to unstable training and limited accuracy. To address the challenges, we propose Locally Enhanced Neural Networks (LENNs) that characterize localized discontinuities in solid mechanics via multilevel modeling. In particular, this novel framework employs a global network for the bulk solution and activates a local network in localized area for non-smooth response, coupled through a smooth window function that enables weighted superposition of local and global solutions. Moreover, the local network embeds additional functions that encode the discontinuous information into the input to capture localized non-smooth mechanical behaviors. Finally, the composite solution is substituted into the total potential energy functional for unified optimization. With this structure, the method resolves the conflict of single network in representing both smooth global and singular local fields without additional interface-loss terms and amplifies the contribution of localized critical features in energy optimization. We focus on a series of numerical experiments in solid mechanics to demonstrate the performance of the method. Results show that LENNs perform well in addressing localized discontinuous problems and provide accurate predictions for both displacement and stress fields.