Thermodynamically-Informed Iterative Neural Operators for Heterogeneous Elastic Localization

TL;DR

This paper introduces Thermodynamically-informed Iterative Neural Operators (TherINO), a novel machine learning approach that efficiently predicts elastic deformation in heterogeneous materials by leveraging thermodynamic encodings, improving accuracy and stability over traditional methods.

Contribution

The paper presents a new neural operator architecture that uses thermodynamic encodings and iterative solution space updates, enhancing prediction accuracy and computational efficiency for heterogeneous elasticity problems.

Findings

TherINO outperforms traditional neural operators in accuracy and speed.

TherINO demonstrates stability and good extrapolation on out-of-distribution data.

The approach offers a better speed-accuracy tradeoff for elastic quantity predictions.

Abstract

Engineering problems frequently require solution of governing equations with spatially-varying discontinuous coefficients. Even for linear elliptic problems, mapping large ensembles of coefficient fields to solutions can become a major computational bottleneck using traditional numerical solvers. Furthermore, machine learning methods such as neural operators struggle to fit these maps due to sharp transitions and high contrast in the coefficient fields and a scarcity of informative training data. In this work, we focus on a canonical problem in computational mechanics: prediction of local elastic deformation fields over heterogeneous material structures subjected to periodic boundary conditions. We construct a hybrid approximation for the coefficient-to-solution map using a Thermodynamically-informed Iterative Neural Operator (TherINO). Rather than using coefficient fields as direct inputs and iterating over a learned latent space, we employ thermodynamic encodings -- drawn from the constitutive equations -- and iterate over the solution space itself. Through an extensive series of case studies, we elucidate the advantages of these design choices in terms of efficiency, accuracy, and flexibility. We also analyze the model's stability and extrapolation properties on out-of-distribution coefficient fields and demonstrate an improved speed-accuracy tradeoff for predicting elastic quantities of interest.