EquiNO: A Physics-Informed Neural Operator for Multiscale Simulations

TL;DR

This work tackles the computational bottleneck of multiscale FE$^{2}$ simulations by introducing EquiNO, a physics-informed neural operator that predicts microscale responses while enforcing microscale equilibrium and periodic boundary conditions exactly. By projecting governing equations onto divergence-free POD modes, EquiNO delivers a reduced-order surrogate, enabling FE$^{2}$-level predictions with data-efficient, unsupervised training and large speedups (exceeding 8000×) relative to conventional methods. The paper also develops VPIONet as a variational-based competitor and demonstrates FE-OL integration for both microscale and multiscale runs across three RVEs and three macroscale tests, with EquiNO consistently achieving lower stress errors. Overall, EquiNO offers a principled balance between data-driven efficiency and physics-based fidelity, making it particularly suitable for many-query multiscale problems in solid mechanics and beyond.

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 $\textit{substituted}$ 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 $\textit{complementary}$ 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.

Figures (9)

• Figure 1: A schematic view of the EquiNO architecture.
• Figure 2: RVEs considered in this study. The blue and red regions indicate nonlinear and linear elastic materials, respectively.
• Figure 3: Comparison of the convergence behavior of EquiNO (magenta) and VPINN (blue) for three different RVEs. The plots on the left show the relative $L_2$ stress error percentages during training, averaged over ten independent samples and all stress components. The field visualizations on the right display the reference (top) and the predicted displacement and stress components for EquiNO (middle) and VPINN (bottom), with the relative $L_2$ error percentages indicated in parentheses.
• Figure 4: RVE1 field visualizations of the reference (first row) and the predicted displacement and stress components for EquiNO (second row) and VPIONet (fourth row). Results correspond to a sample that exhibits the median relative $L_2$ error on stress components among the test samples. The third and fifth rows display the absolute errors in the predictions obtained from EquiNO and VPIONet, respectively. The relative $L_2$ errors are reported in parentheses.
• Figure 5: Macroscale test cases employed for testing the performance of the FE-OL method against the reference FE$^{\,2}\,$ method.