---
title: "CFKS: Continuum-Feature Kernel PDE Solvers"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2610.38644v1"
permalink: "https://feed.astralor.com/items/itm_0812c481c73aa2002763957f"
published: "2026-10-28T10:54:34.000Z"
score: 57
channel: "tech"
tags: ["CFKS","Continuum-Feature Kernel PDE","Solvers Neural-network-based PDE","Fixing","Gaussian","RBF","Neural-network-based PDE"]
---

# CFKS: Continuum-Feature Kernel PDE Solvers

## At a glance

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 featur…

## Key points

- 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, ther…
- 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.

## Full text

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

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