数据驱动的局部径向基函数配点法形状参数统一优化策略求解偏微分方程

A Data-Driven Unified Optimization Strategy for Shape Parameters in Local Radial Basis Function Collocation for Solving PDEs

摘要: 在局部径向基函数配点法(LRBFCM)中，形状参数的选取对数值稳定性与精度具有决定性影响。传统方法通常依赖经验进行选取，且针对每个问题分别优化形状参数，难以适应不同方程源项与边界条件的变化。为此，本文提出一种基于数据驱动的局部径向基函数形状参数自适应优化方法，构建了一个具有算子级泛化能力的形状参数预测框架。该方法通过离线阶段的数据驱动训练，实现针对给定微分算子的一次性网络学习；在在线阶段，可针对任意源项与边界条件快速确定形状参数，从而避免通过试错法确定形状参数与针对同一微分算子的重复优化过程。通过几个数值实验结果表明，本文方法能够在保持高精度的同时显著降低优化开销，且在不同源项与边界条件下均表现出良好的稳定性与泛化性。

Abstract: In the Local Radial Basis Function Collocation Method (LRBFCM), the choice of the shape parameter plays a decisive role in determining numerical stability and accuracy. Traditional approaches usually rely on empirical selection or case-by-case optimization for each problem, making them difficult to adapt to variations in source terms and boundary conditions. To address this issue, this paper proposes a data-driven adaptive optimization method for the shape parameter in LRBFCM, establishing a shape-parameter prediction framework with operator-level generalization capability. Through an offline data-driven training phase, the proposed network learns the mapping between a given differential operator and its optimal shape parameter once and for all. In the online solving phase, the trained model can rapidly determine the appropriate shape parameter for arbitrary source terms and boundary conditions, thereby avoiding trial-and-error tuning and repeated optimization for the same differential operator. Numerical experiments demonstrate that the proposed method achieves high accuracy while significantly reducing optimization cost, and exhibits excellent stability and generalization performance across different source terms and boundary conditions.