Neural-Network PDE Solvers: Architectural Challenges
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.
Continuum-Feature Kernel Solver (CFKS) Framework
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.
Linear and Nonlinear PDE Handling
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.
Numerical Studies and Validation
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: CFKS_Codes