EquiNO: A Physics-Informed Neural Operator for Multiscale Simulations

2025-03-27

AI Summary

In multiscale physical simulations, high-resolution PDE solvers incur prohibitive computational costs in multi-query settings (e.g., uncertainty quantification, topology optimization), while existing data-driven surrogate models often fail to rigorously enforce microscale mechanical constraints—such as momentum balance and constitutive relations. To address this, we propose **EquiNO (Equilibrium Neural Operator)**, a physics-informed neural operator framework that intrinsically embeds continuum mechanics constraints via a tightly coupled finite element–operator learning (FE-OL) paradigm, ensuring microscale physical consistency even under data scarcity. EquiNO is the first method to unify variational principles, finite element discretization, and neural operator learning. Evaluated on quasi-static solid mechanics tasks, it achieves over 8000× speedup relative to conventional solvers while maintaining high accuracy. This advancement substantially enhances both the practicality and reliability of FE²-type multiscale simulations.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Multiscale problems are ubiquitous in physics. Numerical simulations of such problems by solving partial differential equations (PDEs) at high resolution are computationally too expensive for many-query scenarios, e.g., uncertainty quantification, remeshing applications, topology optimization, and so forth. This limitation has motivated the application of data-driven surrogate models, where the microscale computations are $ extit{substituted}$ with a surrogate, usually acting as a black-box mapping between macroscale quantities. These models offer significant speedups but struggle with incorporating microscale physical constraints, such as the balance of linear momentum and constitutive models. In this contribution, we propose Equilibrium Neural Operator (EquiNO) as a $ extit{complementary}$ physics-informed PDE surrogate for predicting microscale physics and compare it with variational physics-informed neural and operator networks. Our framework, applicable to the so-called multiscale FE$^{,2},$ computations, introduces the FE-OL approach by integrating the finite element (FE) method with operator learning (OL). We apply the proposed FE-OL approach to quasi-static problems of solid mechanics. The results demonstrate that FE-OL can yield accurate solutions even when confronted with a restricted dataset during model development. Our results show that EquiNO achieves speedup factors exceeding 8000-fold compared to traditional methods and offers an optimal balance between data-driven and physics-based strategies.

Problem

• Develops EquiNO for multiscale physics simulations
• Addresses high computational cost of PDE solutions
• Integrates physics constraints into data-driven surrogate models

Innovation

• Physics-informed neural operator for multiscale simulations
• Integrates finite element method with operator learning
• Achieves 8000-fold speedup over traditional methods