spatial domain decomposition

To address the low parallel efficiency and load imbalance in solving large-scale 3D magnetostatic problems—such as synchronous machine modeling—on complex multi-patch geometries, this paper proposes a multi-patch domain decomposition algorithm tailored for isogeometric analysis (IGA). The method partitions the geometric model into multiple parameterized patch subdomains and integrates Schwarz-type iterative solvers with METIS-based graph partitioning to achieve automatic load balancing and distributed task scheduling. This work represents the first extension of isogeometric domain decomposition to multi-patch configurations, significantly enhancing both strong and weak scalability on heterogeneous CPU clusters. Experiments on problems with tens of millions of degrees of freedom demonstrate a 2–5× speedup over conventional approaches, while achieving over 92% load balance efficiency. The method preserves high numerical accuracy, computational efficiency, and compatibility with industrial-grade CAD geometries.

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 large-scale or sparse-reward environments, reinforcement learning often suffers from slow value propagation due to the locality of temporal difference (TD) updates. This work proposes a finite element method–inspired spatial domain decomposition framework that partitions the state space into overlapping subdomains and enforces boundary-consistent TD updates, enabling efficient local learning while preserving global value consistency. By introducing, for the first time in reinforcement learning, the scientific computing principle of boundary-consistent domain decomposition, the method accelerates long-range credit assignment without modifying the reward function, Bellman operator, or incorporating explicit planning mechanisms. Evaluated on hazardous, dense grid worlds with diverse geometric structures, the framework significantly improves convergence speed, cumulative reward, and stability of algorithms such as Q-learning, SARSA, and Dyna-Q. Moreover, multi-resolution grids effectively mitigate premature convergence and enhance distant value propagation.

Efficiently solving partial differential equations (PDEs) on complex geometries remains challenging for existing neural network solvers, which suffer from poor generalization, inability to handle discontinuous media, and scalability limitations. Method: We propose the first learning-based PDE solver that integrates a pre-trained neural operator into a domain decomposition framework: a physics-pretrained neural operator (PPNO) is pre-trained on simple geometries and deployed as a surrogate model for each subdomain; theoretical analysis establishes its approximation existence within PDE domain decomposition. Contribution/Results: The method achieves strong generalization to unseen microstructures and resolution invariance without retraining, seamlessly adapting to complex geometries and multiscale discontinuous media. On elliptic PDE benchmarks, it significantly outperforms state-of-the-art methods in accuracy, computational efficiency, and cross-domain generalization.

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.

该研究通过结合强化学习与重叠Schwarz交替方法，实现全阶模型和降阶模型的在线自适应混合耦合，解决了瞬态问题中高保真区域随时间变化的问题。

This work addresses a key limitation of conventional neural PDE surrogates, which model field evolution on fixed grids and thereby overlook the critical role of mesh design in allocating spatial resolution and spectral bandwidth. The study introduces, for the first time, adaptive discretization as a physics-constrained conditional generation task, proposing a two-stage diffusion framework: it first generates an r-adaptive displacement grid conditioned on observed dynamics and then predicts solution evolution on this adaptive mesh. By incorporating physics-aware regularization, geometric validity constraints, and local spectral concentration, the method achieves learnable, interpretable, and numerically stable mesh adaptation. Extensive experiments across five classes of PDE problems demonstrate substantial improvements over traditional adaptive and reduced-order methods, with particularly notable gains in complex domains.

This work addresses the long-standing challenge of automatically generating high-quality all-quadrilateral meshes for arbitrary geometries by introducing the first unified multi-agent reinforcement learning framework that jointly optimizes geometric decomposition and mesh generation. The meshing process is formulated as a Markov decision process, wherein three cooperative agents—responsible for topological simplification, geometric regularization, and quadrilateral mesh generation—operate in an end-to-end pipeline. Leveraging a parameterized decoupled Soft Actor-Critic algorithm, a hybrid discrete-continuous action space, and a curriculum learning strategy, the method enables recursive and parallel processing of subregions without requiring post-processing. It produces globally consistent all-quad meshes and significantly outperforms existing approaches across diverse complex geometric benchmarks, achieving notable advances in automation, robustness, and mesh quality.

该研究通过消除几何学框架探讨局部最优对象能否由共享部署规则实现，分析信息、架构等因素对缺陷修复的影响。

This study addresses the challenge of approximating the theoretical lower bound of vertex irregularity, as defined by the discrete Gauss-Bonnet theorem, in planar quadrilateral mesh decomposition. To this end, reinforcement learning is introduced for optimal mesh decomposition for the first time. Methodologically, the Proximal Policy Optimization algorithm is combined with behavioral cloning to overcome exploration difficulties under sparse rewards, while a connectivity-based graph convolutional network is designed to enable cross-scale generalization. Experimental results demonstrate that the proposed approach achieves theoretical optimality in 90% of 96 test domains, significantly outperforming the conventional tool Gmsh. Furthermore, it maintains low excess error on large-scale domains, effectively overcoming the performance bottlenecks of existing methods.