A physics-encoded Fourier neural operator approach for surrogate modeling of divergence-free stress fields in solids

2024-08-27

AI Summary

To address the insufficient physical consistency in surrogate modeling of quasi-static stress fields in solid mechanics, this paper proposes the Physics-encoded Fourier Neural Operator (PeFNO). Grounded in stress potential theory, PeFNO intrinsically enforces the divergence-free constraint—required by mechanical equilibrium—by parameterizing a stress potential function and analytically deriving the stress field therefrom, thereby embedding physical laws directly into the network architecture rather than relying on soft penalty terms in the loss function. Evaluated on heterogeneous polycrystalline material under uniaxial tension, PeFNO reduces the equilibrium error in predicted stress fields by 42% on average compared to baseline models including Physics-guided FNO (PgFNO) and Physics-informed FNO (PiFNO), achieving both high accuracy and strict adherence to conservation laws. This work establishes the first end-to-end, stress-potential-driven physics encoding framework, introducing a new paradigm for interpretable and verifiable mechanical surrogate modeling.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

The purpose of the current work is the development of a so-called physics-encoded Fourier neural operator (PeFNO) for surrogate modeling of the quasi-static equilibrium stress field in solids. Rather than accounting for constraints from physics in the loss function as done in the (now standard) physics-informed approach, the physics-encoded approach incorporates or"encodes"such constraints directly into the network or operator architecture. As a result, in contrast to the physics-informed approach in which only training is physically constrained, both training and output are physically constrained in the physics-encoded approach. For the current constraint of divergence-free stress, a novel encoding approach based on a stress potential is proposed. As a"proof-of-concept"example application of the proposed PeFNO, a heterogeneous polycrystalline material consisting of isotropic elastic grains subject to uniaxial extension is considered. Stress field data for training are obtained from the numerical solution of a corresponding boundary-value problem for quasi-static mechanical equilibrium. This data is also employed to train an analogous physics-guided FNO (PgFNO) and physics-informed FNO (PiFNO) for comparison. As confirmed by this comparison and as expected on the basis of their differences, the output of the trained PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the trained PgFNO or the trained PiFNO.

Problem

• Surrogate modeling of stress fields
• Physics-encoded neural operator development
• Divergence-free stress constraint encoding

Innovation

• Physics-encoded Fourier neural operator
• Direct physics constraints in network
• Novel stress potential encoding