Toward Efficient Kernel-Based Solvers for Nonlinear PDEs
This work addresses computational bottlenecks in solving nonlinear PDEs for researchers and practitioners in scientific computing, though it appears incremental as it builds on existing kernel methods.
The paper tackles the inefficiency of kernel-based solvers for nonlinear PDEs by eliminating differential operators from the kernel, using standard kernel interpolation and differentiating the interpolant to compute derivatives, resulting in scalable computation with reduced costs for large numbers of collocation points.
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.