New kernel method improves solving complex partial differential equations
CFKS: Continuum-Feature Kernel PDE Solvers
Machine Learning
Summary
Solving partial differential equations (PDEs) with neural networks often requires guessing how many features or layers to use, which can be tricky. The authors developed a method called CFKS that removes this guesswork by using an infinite, continuous set of simple building blocks called Gaussian features. This approach turns the problem into working with smooth kernels, making it easier to enforce physical rules and solve both linear and nonlinear PDEs. Their tests show CFKS can handle a range of challenging problems with oscillations, sharp changes, and unusual shapes.
What this means in practice
- •For computational engineers: Solve complex engineering PDE models on irregular domains with improved control over feature representation and physical constraints.
- •For simulation software developers: Enhance PDE solver modules by integrating continuous-feature kernel methods for better accuracy in nonlinear and multiscale simulations.
Authors
Vikas Dwivedi, Balaji Srinivasan, Monica Sigovan, Bruno Sixou
Abstract
Neural-network-based PDE solvers require architectural choices, particularly the number of layers and neurons per layer. Fixing a single Gaussian radial basis function (RBF) feature layer and taking the number of features to the continuum limit removes the need to prescribe a finite feature count or select individual feature centers, thereby reducing representation arbitrariness. Motivated by this principle, we introduce Continuum-Feature Kernel Solver (CFKS), a deterministic framework for PDEs obtained by analytically marginalizing a bounded continuum of Gaussian RBF features. CFKS works directly with the resulting continuum-feature mean kernel and its operator-applied counterparts, parameterized by prescribed Gaussian widths and bounded center distributions. For linear PDEs, CFKS leads to symmetric operator-kernel collocation systems with null-space enforcement of physical constraints. Nonlinear PDEs are addressed through linearized initialization and nonlinear least-squares residual minimization using analytical kernel derivatives and Jacobians. Residual-guided collocation enrichment and sequential space--time-slab marching provide additional resolution strategies, while prescribed multiscale kernel refinement helps resolve localized nonlinear structures. Finite-feature experiments confirm convergence of empirical value and operator Gram matrices toward their continuum counterparts. Numerical studies include oscillatory and multiscale ODEs, irregular-domain Poisson and biharmonic problems, singular perturbations, transient transport equations, and viscous Burgers' equation. The results establish CFKS as a direct computational connection between finite Gaussian-RBF representations and continuum-feature kernel PDE solvers. Codes: https://github.com/vikas-dwivedi-2022/CFKS_Codes