domain decomposition

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.

This work addresses the limited accuracy and high computational cost of large neural network surrogate models in capturing local nonlinear features. The authors propose a domain-decomposition-based parallel neural network architecture that partitions the input space into multiple subdomains, each modeled independently by a lightweight subnet. Interface continuity across subdomains is enforced through Lagrange multipliers and an augmented Lagrangian formulation. This approach significantly improves modeling accuracy in locally nonlinear regions while enhancing training efficiency. Experimental results demonstrate that both constraint strategies outperform unconstrained global training, with the augmented Lagrangian method exhibiting faster convergence and superior scalability for large-scale problems, achieving better overall performance at only a marginal trade-off in accuracy.

In multi-scale, multi-physics simulations, high-fidelity models suffer from prohibitive computational cost and complex mesh generation. To address this, we propose a subdomain-local coupling framework that synergistically integrates the overlapping Schwarz alternating method (O-SAM) with non-intrusive operator inference (OpInf)-based reduced-order models (ROMs). The framework requires no modification to existing high-fidelity solvers, enables seamless integration of heterogeneous models, non-conforming meshes, and disparate time steps, and is inherently parallelizable. Its key innovation lies in the first-ever embedding of OpInf within an overlapping domain decomposition architecture, enabling efficient and stable coupling between ROMs and full-order models at the subdomain level. Numerical experiments on a 3D solid dynamics benchmark demonstrate up to 106× speedup over conventional full-order coupling while maintaining high accuracy, thereby validating the method’s efficiency, fidelity, and scalability.

To address the limited geometric generalization and strong data dependency of neural operators in solving partial differential equations (PDEs), this paper proposes a domain-decomposition-based operator learning framework. Its core method, Schwarz Neural Inference (SNI), partitions the global domain into overlapping subdomains, trains local neural operators on each subdomain, and achieves decoupled–fused solutions via iterative exchange of boundary information. We establish, for the first time, convergence analysis and rigorous error bounds for neural operators under domain decomposition. The framework enables zero-shot generalization across unseen geometries. Experiments demonstrate significant improvements in generalization performance across diverse PDEs and complex boundary conditions, substantially reducing data requirements while achieving state-of-the-art accuracy and stability on previously unseen geometries.

This work addresses the lack of a unified understanding of the design principles, applicability, and performance differences between Physics-Informed Neural Networks (PINNs) and Neural Operators (NOs), which hinders the development of reliable data-driven PDE solvers. It proposes the first unified analytical framework that systematically characterizes the design space of both approaches along three dimensions: learning objectives, mechanisms for embedding physical structure, and strategies for computational load distribution. By elucidating the intrinsic connections and fundamental distinctions between these methods, the study not only clarifies the positioning and performance origins of existing techniques but also provides theoretical guidance and novel pathways for designing efficient and robust PDE solvers that effectively integrate physical priors with data-driven learning.

This work addresses the limited reusability and heavy reliance on problem-specific data in conventional global surrogate-based neural PDE solvers. The authors propose NEST, a novel framework that integrates local neural operators with classical domain decomposition techniques. By training reusable local solvers on 3×3×3 voxel patches and coupling them via overlapping subdomains, Schwarz iterations, and a partition of unity, NEST constructs globally consistent solutions without dependence on a fixed family of problems. The method is geometry-agnostic, scalable, and demonstrates strong generalization to complex three-dimensional nonlinear elasticity problems far exceeding the scale and configuration of its training data, enabling efficient solution transfer across scales, geometries, and boundary conditions.

This work addresses the poor computational scalability of large-scale physics-constrained optimization by proposing a novel framework that integrates physics-informed machine learning with polyhedral reformulation. Unlike conventional approaches that treat physical and geometric priors merely as regularization terms, this method uniquely embeds such priors directly into the reconstruction of the optimization problem itself, thereby decoupling problem complexity from solution difficulty. The framework reformulates intricate constraints into an efficient polyhedral representation, enabling off-the-shelf solvers to achieve rapid convergence while preserving high solution quality. Evaluated on three canonical problem classes, the approach demonstrates up to a 6400× speedup and a 99.87% reduction in memory usage, with solution accuracy matching or surpassing state-of-the-art methods.

Existing neural operators struggle to accurately capture the strong sensitivity of partial differential equation (PDE) solutions to complex, non-homogeneous boundary conditions. This work proposes a general framework that embeds arbitrary boundary conditions—including mixed, component-wise, and piecewise types—into domain-to-domain neural operators by learning a mapping from boundary data to a global pseudo-extension function over the entire domain. This approach enables joint modeling of input functions and boundary conditions without requiring modifications to existing architectures or dataset-specific hyperparameter tuning. Evaluated across 18 challenging datasets encompassing Poisson equations, linear elasticity, and hyperelasticity problems, the method consistently outperforms current baselines, achieving state-of-the-art accuracy.

This study addresses the challenge of constructing surrogate solvers that can be trained on small subdomains and seamlessly extended to large spatial domains without requiring access to the underlying partial differential equation (PDE) operators. The authors propose a novel approach based on assembling local latent-space elements: LaSDI latent-space ODE models are trained on subdomains and coupled through learned directional interaction terms between neighboring elements, while a windowed weighting strategy ensures smooth global field reconstruction. This work introduces, for the first time, a finite element–inspired assembly paradigm into latent-space modeling, thereby eliminating the need for iterative solvers or interface residual computations. The resulting framework is modular, reusable, and interpretable. Experiments on the 1D Burgers and Korteweg–de Vries equations demonstrate that the model maintains high accuracy and strong scalability even when extrapolating to spatial domains significantly larger than those seen during training.

This work addresses the challenge that existing physics-informed machine learning methods struggle to enforce Dirichlet, Neumann, and Robin boundary conditions exactly on arbitrary curved quadrilateral domains, particularly due to compatibility constraints at corners where Neumann and Robin boundaries intersect. To overcome this limitation, the authors propose a systematic framework that integrates exact geometric mapping, the Theory of Functional Connections (TFC), and transfinite interpolation to construct trial functions that rigorously satisfy all boundary conditions and vertex compatibility requirements. These trial functions are embedded within an Extreme Learning Machine (ELM) to solve partial differential equations. The method achieves machine-precision enforcement of boundary conditions on complex curved quadrilateral domains—surpassing conventional approaches that only approximate such constraints—and demonstrates high accuracy and broad applicability across a range of linear/nonlinear and steady/unsteady problems.