用随机神经网络解曲面偏微分方程，免网格、省计算，动静皆宜。
Randomized Neural Networks for Partial Differential Equation on Static and Evolving Surfaces
- 隐藏层参数随机固定，输出层通过最小二乘快速求解。
- 静态与动态曲面均适用，精度高且避免反复重剖分。
- 适合需高效处理复杂曲面演化问题的研究者。
曲面偏微分方程在众多科学与工程领域中广泛应用。由于几何复杂性，其在静态和动态曲面上的数值求解仍具挑战性，尤其对于动态几何，需反复更新网格并转移几何或解信息。尽管基于神经网络的方法可实现无网格离散化，但非凸训练往往代价高昂，且实际应用中难以保证高精度。本文提出一种随机神经网络（RaNN）方法，用于求解静态与动态曲面上的偏微分方程：隐藏层参数随机生成并固定，输出层系数通过求解最小二乘问题高效确定。针对静态曲面，提出参数化曲面、隐式水平集曲面及点云几何的公式，并对参数化形式给出具有界面连续性的理论分析。对于拓扑保持不变的动态曲面，引入基于流映射表示的RaNN策略，在时空共点集上求解曲面PDE，避免重剖分。大量数值实验表明该方法在典型基准测试中具有广泛适用性和优异的精度-效率表现。
原文摘要 · Abstract (English)
Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the need for repeated mesh updates and geometry or solution transfer. While neural-network-based methods offer mesh-free discretizations, approaches based on nonconvex training can be costly and may fail to deliver high accuracy in practice. In this work, we develop a randomized neural network (RaNN) method for solving PDEs on both static and evolving surfaces: the hidden-layer parameters are randomly generated and kept fixed, and the output-layer coefficients are determined efficiently by solving a least-squares problem. For static surfaces, we present formulations for parametrized surfaces, implicit level-set surfaces, and point-cloud geometries, and provide a corresponding theoretical analysis for the parametrization-based formulation with interface compatibility. For evolving surfaces with topology preserved over time, we introduce a RaNN-based strategy that learns the surface evolution through a flow-map representation and then solves the surface PDE on a space--time collocation set, avoiding remeshing. Extensive numerical experiments demonstrate broad applicability and favorable accuracy--efficiency performance on representative benchmarks.
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