Equilibrium Conserving Neural Operators for Super-Resolution Learning

2025-04-18

AI Summary

Super-resolution modeling in solid mechanics heavily relies on high-fidelity, high-resolution training data—costly and often infeasible to acquire. Method: We propose a novel paradigm that trains high-resolution neural operators exclusively from low-resolution data. Our approach introduces the Equilibrium Conserving Operator (ECO), a physics-embedded architecture that rigorously enforces static equilibrium and mass continuity as strong constraints within a neural operator framework, augmented by multi-scale feature extraction and a conservation-aware loss function. Contribution/Results: To our knowledge, this is the first method enabling strict satisfaction of fundamental mechanical conservation laws in super-resolution learning—without requiring high-resolution ground-truth labels. Evaluated on porous media and polycrystalline material tasks, it reduces data acquisition cost by two orders of magnitude while ensuring mechanically consistent, equilibrium-satisfying predictions—effectively alleviating the high-resolution data bottleneck.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Develops neural operator for super-resolution learning in mechanics
• Trains high-resolution models using only low-resolution data
• Enforces physics-based constraints to reduce data needs

Innovation

• Trains high-resolution networks with low-resolution data
• Embeds physics into network via ECO architecture
• Enforces conservation laws in solution predictions