Thermodynamically-Informed Iterative Neural Operators for Heterogeneous Elastic Localization
This addresses a computational bottleneck in engineering mechanics for predicting material behavior, but it is incremental as it builds on existing neural operator methods with thermodynamic encodings.
The paper tackled the problem of predicting elastic deformation fields in heterogeneous materials with discontinuous coefficients, which is computationally expensive with traditional solvers and challenging for neural operators due to sharp transitions and data scarcity. The result was a Thermodynamically-informed Iterative Neural Operator (TherINO) that improved efficiency, accuracy, and flexibility, demonstrating an enhanced speed-accuracy tradeoff for elastic predictions.
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.