Equilibrium Conserving Neural Operators for Super-Resolution Learning

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.

@article{oommen2025_2504.13422,
  title={ Equilibrium Conserving Neural Operators for Super-Resolution Learning },
  author={ Vivek Oommen and Andreas E. Robertson and Daniel Diaz and Coleman Alleman and Zhen Zhang and Anthony D. Rollett and George E. Karniadakis and Rémi Dingreville },
  journal={arXiv preprint arXiv:2504.13422},
  year={ 2025 }
}