Constitutive parameterized deep energy method for solid mechanics problems with random material parameters

TL;DR

The paper introduces CPDEM, a physics-driven neural network framework that efficiently predicts solid mechanics responses under varying random material parameters without retraining.

Contribution

CPDEM reformulates the energy functional by embedding stochastic parameters into the neural network, enabling real-time, zero-shot predictions for unknown material properties.

Findings

Validated on linear elasticity, hyperelasticity, and contact mechanics benchmarks.

Achieves continuous, real-time displacement predictions without dataset generation.

First physics-driven deep learning method for multi-parameter solid mechanics problems.

Abstract

In practical structural design and solid mechanics simulations, material properties inherently exhibit random variations within bounded intervals. However, evaluating mechanical responses under continuous material uncertainty remains a persistent challenge. Traditional numerical approaches, such as the Finite Element Method (FEM), incur prohibitive computational costs as they require repeated mesh discretization and equation solving for every parametric realization. Similarly, data-driven surrogate models depend heavily on massive, high-fidelity datasets, while standard physics-informed frameworks (e.g., the Deep Energy Method) strictly demand complete retraining from scratch whenever material parameters change. To bridge this critical gap, we propose the Constitutive Parameterized Deep Energy Method (CPDEM). In this purely physics-driven framework, the strain energy density functional is reformulated by encoding a latent representation of stochastic constitutive parameters. By embedding material parameters directly into the neural network alongside spatial coordinates, CPDEM transforms conventional spatial collocation points into parameter-aware material points. Trained in an unsupervised manner via expected energy minimization over the parameter domain, the pre-trained model continuously learns the solution manifold. Consequently, it enables zero-shot, real-time inference of displacement fields for unknown material parameters without requiring any dataset generation or model retraining. The proposed method is rigorously validated across diverse benchmarks, including linear elasticity, finite-strain hyperelasticity, and complex highly nonlinear contact mechanics. To the best of our knowledge, CPDEM represents the first purely physics-driven deep learning paradigm capable of simultaneously and efficiently handling continuous multi-parameter variations in solid mechanics.