domain-decomposed randomized networks

This work addresses the challenges of solving partial differential equations on unbounded domains, where conventional domain truncation incurs large errors and global spectral methods struggle with local features and multiscale behavior. The authors propose a domain-decomposition-based stochastic neural network framework that models near- and far-field regions with separate subnetworks, coupled through interface conditions. By optimizing only the output-layer coefficients, the method yields a linear least-squares system. The approach innovatively integrates domain decomposition with stochastic neural networks and, for the first time, establishes a provably bounded parametric approximation theory under broken Sobolev norms, along with a complete error decomposition. Combined with Petrov–Galerkin formulations for semi-infinite elliptic problems and collocation strategies for fully unbounded, multiply connected, or time-dependent settings, the method demonstrates high accuracy, strong adaptivity, and effective handling of complex geometries in numerical experiments on Poisson and time-dependent Schrödinger equations.

This paper addresses the modeling challenge of deterministic inputs yielding strongly non-Gaussian stochastic outputs in neural systems. Methodologically, it introduces the first probabilistic neural architecture generation framework based on manifold-valued hidden random fields: neural topology and synaptic weights are jointly modeled as anisotropic Gaussian random fields on compact, boundaryless, multiply connected manifolds, with connectivity defined via geodesic distance and field affinity; geometrically aware sparsification is achieved through percentile-based diffusion masking; and scalable stochastic inference is enabled via inhomogeneous Poisson sampling coupled with Monte Carlo likelihood estimation. Theoretical contributions include establishing expressivity and well-posedness foundations for stochastic mappings, proving measurability, expressive variability, and feasibility of single-sample supervised learning. The framework requires no predefined network architecture and inherently integrates geometric awareness with statistical interpretability.

To address the lack of native geometric processing capabilities in neural surface representations, this paper introduces spherical neural surface representation—a framework enabling seamless, mesh-free estimation of normals, first and second fundamental forms, gradients, divergence, and the Laplace–Beltrami operator on genus-0 neural surfaces. Our method leverages spherical parameterization with implicit neural representation, employs automatic differentiation to derive differential geometric operators, and establishes a numerical verification framework alongside neural spectral analysis tools. Key contributions include: (1) breaking the conventional “mesh-then-process” paradigm by establishing a systematic theoretical bridge between neural representations and classical differential geometry; (2) enabling geometric processing tasks—including neural heat flow and mean curvature flow—with robustness under isometric deformations; and (3) achieving numerical accuracy comparable to analytical solutions and mesh-based baselines, significantly outperforming existing neural surrogates.

Neural fields exhibit slow convergence under Adam-style stochastic optimization, hindering practical deployment. To address this, we propose the first curvature-aware diagonal preconditioning framework tailored for stochastic training of neural fields—overcoming the fundamental incompatibility of second-order methods (e.g., L-BFGS) with stochastic neural field optimization. Our method constructs a diagonal Hessian approximation from stochastic gradients, integrates adaptive learning-rate preconditioning with neural field parameterization, and supports end-to-end joint optimization for NeRF. Evaluated on image reconstruction, shape modeling, and NeRF tasks, it achieves an average 2.1× training speedup while maintaining or improving reconstruction accuracy and enhancing convergence stability. Our core contribution is the first theoretical foundation for second-order, curvature-aware diagonal preconditioning in the stochastic setting—realized as an efficient, scalable, and plug-and-play optimization accelerator for neural fields.

Kolmogorov–Arnold Networks (KANs) suffer from high training costs and limited efficacy in multiscale modeling. Method: We propose Finite-Basis KANs (FBKANs), the first KAN variant integrating domain decomposition and finite-basis representation: it partitions the global domain into subdomains, trains local KANs in parallel, and employs learnable basis functions to enable parameter sharing and embed physical constraints. FBKAN synergistically combines Kolmogorov–Arnold-theorem-inspired learnable activations, adaptive domain partitioning, finite-basis expansions, and physics-informed loss optimization. Contribution/Results: Experiments demonstrate that FBKAN significantly outperforms standard KANs and MLPs in noisy function approximation and partial differential equation solving—achieving higher accuracy, accelerating training by multiple-fold, and exhibiting superior generalization, robustness to noise, and scalability across problem sizes and domains.

This work proposes a novel approach to spatially localizing functional specialization in neural networks through structured noise, enabling a single network to efficiently store and distinguish multiple functions. By introducing a virtual noise field that generates spatially structured noise in a continuous auxiliary space, the method activates partially overlapping subnetworks and leverages cross-activation functions to achieve multi-level parameter sharing at the sample, statistical, and analytical levels. The key innovation lies in repurposing noise from a source of interference into an active regulatory mechanism that defines the topological structure of functional subnetworks. Experiments on one-dimensional function approximation demonstrate that memory capacity significantly increases when the spatial configuration of the noise field aligns with the similarity structure of target functions, while misalignment leads to degraded performance, revealing a critical relationship between noise structure and function representation.

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.

This work addresses the dual challenges of the curse of dimensionality and the lack of models that simultaneously ensure mathematical rigor and efficient inference in solving high-dimensional spatiotemporal-parametric partial differential equations (PDEs). The authors propose a Separable Neural Architecture (SNA) that decomposes function representations into local coordinate atoms and a global sparse low-rank interaction structure, and further develop a Variational SNA (VSNA) framework. VSNA serves as a Galerkin trial space satisfying the Lax–Milgram conditions, uniquely integrating tensor decomposition with neural approximation to yield a compact function class endowed with rigorous variational guarantees. This approach enables “solve-once, query-anywhere” continuous parameter manifold modeling, facilitating efficient inversion and uncertainty propagation. In seven-dimensional manufacturing simulations and thermal-property inversion for Inconel 718, it achieves one million Monte Carlo queries in just 102 seconds on a CPU—150,000× faster than finite-element baselines on an A100 GPU—and supports real-time generative inversion within <100 ms.

Existing neural operators struggle to efficiently model parametric and coupled partial differential equations (PDEs). This work addresses this limitation by extending the Fourier Neural Operator (FNO) with minimal architectural modifications: it introduces a hypernetwork-driven, parameter-aware modulation mechanism to condition the operator on physical parameters, and systematically designs an operator structure for coupled PDEs that balances shared representations with cross-variable interactions. The resulting approach significantly improves modeling accuracy while preserving computational efficiency. On benchmark problems including capacitively coupled plasma and the Gray–Scott system, the method reduces prediction errors by 55%–72% compared to strong baselines.

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.