Neural Approaches to Computational Solid Mechanics: A Critical Review

Abstract

The rapid evolution of physics-informed and data-driven neural networks has transformed scientific computing, offering new possibilities for solving and inferring complex physical systems. Yet, within the broad realm of computational solid mechanics (CSM), these advances have produced relatively modest outcomes. Despite their theoretical appeal and other advantages such as smooth field representation and the potential to overcome numerical pathologies such as locking, physics-informed neural networks (PINNs) implemented in the strong form have not demonstrated clear advantages over established finite element methods (FEM). This limited success arises from the distinctive challenges of solid mechanics: the multi-field and differential-algebraic nature of its governing equations, the non-smooth and history-dependent character of inelastic responses, and the severe conditioning issues that accompany residual-based training.

In contrast, data-driven approaches, where neural networks augment or replace constitutive laws, have seen rapid progress and widespread adoption. Graph neural networks (GNNs) and operator-learning architectures have also emerged as promising frameworks that encode mesh topology and local physical interactions directly within message-passing rules, thereby bridging the gap between classical discretizations and modern learning paradigms. Nevertheless, these graph-based methods face their own bottlenecks--most notably in efficient backpropagation, physical interpretability, and thermodynamic consistency.

This article presents a critical and integrative review of neural approaches to computational solid mechanics. We analyze why physics-informed strategies have underperformed relative to their success in other fields. We also summarize key developments in data-driven and graph-based mechanics, and discuss how emerging probabilistic and variational formulations based on optimal transport of probabilities, weak forms, and variance-controlled loss functionals, may help overcome the current limitations. The review concludes with a perspective on future directions for building neural solvers that are physically consistent, computationally scalable, and thermodynamically grounded. This should help laying the conceptual foundations for a new generation of mechanics-informed deep learning frameworks.