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

TL;DR

This work introduces Locally Enhanced Neural Networks (LENNs) to overcome the difficulty of simultaneously resolving smooth global fields and highly localized discontinuities in solid mechanics. The method uses a global network for the bulk solution and one or more local networks restricted to regions around cracks or interfaces, coupled through smooth window functions and enriched inputs via signed-distance embeddings, with the composite solution optimized through a total potential energy functional. LENNs demonstrate superior accuracy in displacement, stress, and stress-intensity-factor estimation across several fracture and interfacial problems, and support crack-growth simulations via transfer learning and adaptive local regions. The approach reduces energy imbalance in PINNs, mitigates spectral bias, and offers a practical framework for high-fidelity, multi-scale fracture modeling with potential extensions to multiphysics and adaptive enrichment.

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.

Figures (9)

• Figure 1: Schematic of a solid domain containing localized features.
• Figure 2: Workflow of LENN, showing the computational path from input coordinates to predicted outputs through the neural architecture.
• Figure 3: A typical integral path $\Gamma_J$ around a crack tip.
• Figure 4: Short center crack under tensile loading: (a) Geometry of the symmetric part selected for computation; (b) Evolution of rRMSE of von Mises stress over training epochs for LENN, DENN, CENN within the selected subregion; (c) Comparison of displacement and stress components of LENN, DENN, CENN and FEM in the selected subregion.
• Figure 5: Short center crack under tensile loading: (a) crack angle $\theta$ varies, ranging from $0^\circ$ to $90^\circ$; (b) Stress intensity factor $K_1$ and $K_2$ versus angle variation curve: the blue dot represents the predicted value of $K_1$, the red cross represents the predicted value of $K_2$, the green solid line represents the theoretical value of $K_1$, and the orange dashed line represents the theoretical value of $K_2$.