schur complement reconstruction

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.

Balancing computational efficiency and accuracy remains challenging in solving multiscale, dynamic, multiphysics partial differential equations (PDEs). Method: This paper proposes an adaptive hybrid solver integrating the finite element method (FEM) with a physics-informed DeepONet. It introduces, for the first time, a dynamic subdomain decomposition mechanism based on the Schwarz alternating method, enabling automatic subdomain evolution to capture transient fine-scale features; within each Newmark time step, DeepONet is embedded to establish a tightly coupled FEM–neural operator architecture—eliminating the need for remeshing. Contribution/Results: The solver achieves 20% speedup on static and dynamic solid mechanics problems while maintaining global error below 1%. It rigorously enforces inter-subdomain solution continuity, removes dependence on fine meshes, and significantly suppresses long-term error accumulation. This work establishes a new paradigm for multiscale physical modeling—delivering both high fidelity and high efficiency.

Predicting structural responses under dynamic loading via finite element methods (FEM) incurs prohibitive computational cost, while conventional RNN-based DeepONets struggle to capture continuous-time dynamical behavior. Method: We propose the Multi-Input Operator Network (MIONet), featuring a dual-branch architecture that explicitly encodes spatially discretized meshes and continuous temporal evolution; incorporates mass, damping, and stiffness matrices to embed physical priors; enforces physics-informed constraints derived from dynamic equilibrium equations; and employs Schur complement-based dimensionality reduction for efficient training-domain compression. Contribution/Results: On benchmark cases—simply supported beams and the KW-51 bridge—MIONet achieves FEM-level accuracy with sub-second inference time per query, accelerating predictions by over two orders of magnitude relative to GRU-DeepONet. This enables real-time structural health monitoring and high-fidelity digital twin deployment.

This work addresses the challenge of scaling traditional finite element surrogate models, which rely on costly reference solutions for supervised training. The authors propose a novel unsupervised training approach that eliminates the need for reference solutions by rigorously establishing, for the first time, an exact equivalence between discrete potential energy and stiffness-norm error. Leveraging this relationship, they design a gradient-consistent training mechanism that integrates discrete energy functionals, stiffness-weighted error analysis, and a JEPA (Joint-Embedding Predictive Architecture) framework. Validation across synthetic benchmarks and 16 experimental cases demonstrates that the resulting energy gap effectively controls displacement error. Furthermore, the study reveals that Euclidean error is ill-suited as a primary evaluation metric and delineates the method’s applicability boundaries.

This work proposes implicit Finite Operator Learning (iFOL), a physics-informed operator learning framework for multiphysics problems governed by coupled partial differential equations on arbitrary domains, which operates without requiring ground-truth labeled data. Built upon the finite element weighted residual formulation, iFOL establishes a resolution-independent mapping from input parameters to the solution space and provides a unified treatment of complex geometries and multiphysics coupling. Implemented within the Folax system on the JAX platform, the approach integrates FNO, DeepONet, and iFOL, leveraging finite element residuals to construct physics-constrained loss functions. Experiments demonstrate that iFOL achieves high efficiency on complex geometries in two- and three-dimensional nonlinear thermo-mechanical coupling and industrial casting scenarios, while FNO excels in accuracy on regular domains; furthermore, a single-network end-to-end training strategy significantly outperforms baseline methods.

This work addresses the prohibitive computational cost of full-field loss evaluation in physics-informed operator learning for microstructure surrogate modeling. To overcome this challenge, the authors propose a novel framework that integrates Equivariant Neural Operators (EquiNO) with a QR-based Discrete Empirical Interpolation Method (Q-DEIM). By constructing reduced-order representations of displacement and stress fields using periodic, divergence-free basis functions and evaluating constitutive relations only at a small number of spatial points, the method—introducing Q-DEIM to this domain for the first time—dramatically reduces training expenses. It directly predicts homogenized stresses without reconstructing full fields and demonstrates strong interpolation and extrapolation capabilities even with very few training snapshots. Numerical experiments show a reduction of approximately three orders of magnitude in per-step training cost and acceleration of homogenization computations by factors of $10^3$–$10^4$ compared to full-field simulations, while accurately capturing both microscopic stress fields and macroscopic responses.

This work addresses the challenge of efficiently and accurately estimating quantities of interest (QoI) in multi-query linear problems, where conventional approaches suffer from high computational costs and strong dependence on load configurations. The authors propose a novel reduced-order modeling paradigm based on the adjoint problem, shifting the focus of model reduction from the primal to the adjoint equation for the first time. By introducing a parameterized kernel function to replace the full external load, the method constructs a load-independent surrogate model. Demonstrated on Poisson’s equation and plane-stress elasticity problems, the approach achieves rapid convergence and significantly outperforms traditional primal-based reduction strategies. It enables high-fidelity QoI estimation while supporting fast multi-scenario evaluation and virtual chart generation, thereby greatly enhancing the generality and efficiency of early-stage design optimization.