通过区域分解和非侵入式神经模型降阶实现多尺度混合维模拟
Multiscale Mixed-Dimensional Simulation via Domain Decomposition and Non-Intrusive Neural Model Order Reduction
AI总结：
针对科学工程中多尺度计算模型求解难题，提出基于区域分解和非侵入式神经模型降阶的方法，将全局问题转化为局部可解问题，经训练神经代理嵌入迭代求解器，实现稳定精确求解且具良好扩展性。
AI中文摘要：
许多科学和工程中的计算模型具有多尺度结构，直接求解全局问题计算量过大。区域分解（DD）方法通过将全局问题替换为一系列耦合的局部问题来克服这一限制。本文介绍了一种区域分解降阶模型（DD-ROMs）方法，基于DD不仅能使解算子局部化，还能使其几何和参数依赖性局部化的观察。核心思想是DD将全局难处理的解映射转化为一系列可从局部数据学习的局部算子。通过非侵入式神经代理近似细尺度局部操作并嵌入迭代求解器。训练算法基于级联策略。将所得DD方法解释为扰动不动点迭代并证明全局误差受代理近似误差限制。以混合维椭圆问题为例进行实例化，数值实验表明所得DD-ROM稳定，能在未见微观几何上实现精确近似，具有良好的可扩展性。
英文摘要：
Many computational models arising in science and engineering exhibit a multiscale structure that makes the assembly or direct solution of the global problem computationally prohibitive. Domain Decomposition (DD) methods overcome this limitation by replacing the global problem with a sequence of coupled local problems, whose iterative solution reconstructs the global response. This work introduces a method in the family of Domain Decomposition Reduced Order Models (DD-ROMs), based on the observation that DD naturally localizes not only the solution operator but also its geometric and parametric dependence. The central idea is that DD transforms a globally intractable solution map into a family of locally representable operators learnable from affordable local data after identification with a common reference configuration, a concept that we formalize through the notion of local representability. Non-intrusive neural surrogates are then trained to approximate the fine-scale local operations and embedded into the iterative solver. The training algorithm is based on a cascaded strategy designed to match the distributions encountered by the deployed surrogate iteration. We interpret the resulting DD method as a perturbed fixed-point iteration and establish that the global error remains bounded by the surrogate approximation error. The framework is instantiated for mixed-dimensional elliptic problems coupling three-dimensional bulk domains with embedded one-dimensional inclusions, using a two-level non-overlapping Robin-Robin method. Numerical experiments show that the resulting DD-ROM is stable, achieves accurate approximation on unseen microscale geometries and features good scalability properties with respect to the number of subdomains, scaling to large size global problems while avoiding fine-scale operator assembly and local high-fidelity solvers in the online stage.
