Embedding thermodynamics into neural operators for variable-coefficient PDEs

We propose a new class of data-driven models which predict local deformations over heterogeneous material microstructures. This problem -- termed the localization problem -- is a core component of numerous open challenges in multiscale materials design. The inherent stochasticity of manufacturing means that conducting forward (process -> property) or inverse (property -> process) uncertainty quantification requires solving a variable-coefficient PDE repeatedly for many microstructure instantiations. Moreover, the discontinuity and disorder of the PDE coefficients leads to a high-dimensional, poorly-conditioned system of equations. Our work combines traditional numerical solvers and data-driven methods to construct a hybrid approximation for the coefficient-to-solution map. In particular, we utilize the Lippmann-Schwinger formulation of localization to guide the design of a thermodynamically-informed implicit (Deep Equilibrium) neural operator. Applied to both two-phase composites and polycrystalline materials, our methodology shows improved accuracy and stability compared to existing machine learning methods. Finally, we find that embedding thermodynamic encodings into the architecture provides improved data efficiency and generalizability.