Equilibrium Conserving Neural Operators for Super-Resolution Learning

TL;DR

This paper introduces a framework for super-resolution learning in solid mechanics problems that eliminates the reliance on high-fidelity data and reduces the upfront cost of data collection by two orders of magnitude, offering a robust pathway for resource-efficient surrogate modeling in materials modeling.

Abstract

Neural surrogate solvers can estimate solutions to partial differential equations in physical problems more efficiently than standard numerical methods, but require extensive high-resolution training data. In this paper, we break this limitation; we introduce a framework for super-resolution learning in solid mechanics problems. Our approach allows one to train a high-resolution neural network using only low-resolution data. Our Equilibrium Conserving Operator (ECO) architecture embeds known physics directly into the network to make up for missing high-resolution information during training. We evaluate this ECO-based super-resolution framework that strongly enforces conservation-laws in the predicted solutions on two working examples: embedded pores in a homogenized matrix and randomly textured polycrystalline materials. ECO eliminates the reliance on high-fidelity data and reduces the upfront cost of data collection by two orders of magnitude, offering a robust pathway for resource-efficient surrogate modeling in materials modeling. ECO is readily generalizable to other physics-based problems.

Figures (13)

• Figure 1: Overview of the physics-based neural super-resolution framework.
• Figure 2: Architecture. Physics-based neural super-resolution is performed with a UNet backbone implemented with the proposed ECO formulation. The UNet takes the microstructure $\boldsymbol{m}(x)$ as its input. s-ECO strongly enforces the conservation laws through the strain compatibility and equilibrium blocks while weakly imposing the constitutive law. w-ECO, on the other hand, strongly enforces the constitutive law and weakly imposes the conservation law via its loss function. Data-driven supervision is performed only at low resolution (LR), aided by coarser and cheaper simulations (DNS:LR).
• Figure 3: Strains and stresses predictions. We compare the material response predicted by the physics-based neural super-resolution implementations -- w-ECO (center column) & s-ECO (right column) -- with the DNS (left column) at twice the resolution of the training data. Top two rows compares the stresses and elastic strains in the embedded pores case study. Bottom two rows compares the stresses and elastic deformation gradients in the polycrystalline microstructures case study.
• Figure 4: Stress divergence. (a) Spatial distribution of stress divergence for a sample randomly selected from the test dataset for low- (DNS:64) and high-resolution (DNS:128) direct numerical simulation (DNS) results and compared to predictions from the (weakly enforced) physics-informed equilibrium conserving operator (w-ECO) and the strong ECO (s-ECO). (b) Spatial average of stress divergence across all 250 samples in the test dataset.
• Figure 5: Importance of incorporating physics in physics-based neural super-resolution. We compare the high-resolution (HR) material response, for the embedded pore case study, predicted by data-driven UNet (middle column) and the present s-ECO (right column) with the HR reference simulated using FFT solver (left column, DNS:HR). Note that the neural surrogates were trained on cheaper DNS low resolution whose resolution was half of the DNS:HR test dataset. Embedding the physics enables s-ECO to avoid the non-physical artifacts present in the standard UNet prediction.