新方法让求解非线性偏微分方程更高效，无需复杂矩阵计算。
Toward Efficient Kernel-Based Solvers for Nonlinear PDEs
- 用标准核插值建模解，通过求导得到导数，避免在核中嵌入微分算子。
- 网格点+乘积核构造克罗内克结构，省去完整格曼矩阵计算，可扩展至大量点。
- 理论证明收敛性与误差率，实测在多个基准问题上优于现有方法。
我们提出一种新型核学习框架，用于高效求解非线性偏微分方程。不同于将微分算子嵌入核中的主流方法（在大量采样点下面临挑战），我们的方法将微分算子从核中移除，采用标准核插值形式表示解，并通过对插值函数求导获得导数。该框架避免了解与其导数间复杂的格曼矩阵构建，实现简单实现与可扩展计算。以网格上均匀分布的采样点结合乘积核为例，插值结构呈现克罗内克积形式，从而无需计算全量格曼矩阵，显著降低计算成本并支持大规模点集。我们在合理正则性假设下提供了收敛性及误差率分析。数值实验表明，该方法在求解多个基准非线性PDE时具有明显优势。
原文摘要 · Abstract (English)
We introduce a novel kernel learning framework toward efficiently solving nonlinear partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that embeds differential operators within kernels, posing challenges with a large number of collocation points, our approach eliminates these operators from the kernel. We model the solution using a standard kernel interpolation form and differentiate the interpolant to compute the derivatives. Our framework obviates the need for complex Gram matrix construction between solutions and their derivatives, allowing for a straightforward implementation and scalable computation. As an instance, we allocate the collocation points on a grid and adopt a product kernel, which yields a Kronecker product structure in the interpolation. This structure enables us to avoid computing the full Gram matrix, reducing costs and scaling efficiently to a large number of collocation points. We provide a proof of the convergence and rate analysis of our method under appropriate regularity assumptions. In numerical experiments, we demonstrate the advantages of our method in solving several benchmark PDEs.
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