cs.LGOct 28, 2026

CFKS: Continuum-Feature Kernel PDE Solvers

Authors: Vikas Dwivedi, Balaji Srinivasan, Monica Sigovan, Bruno Sixou

Organizations: CREATIS Biomedical Imaging Laboratory INSA, CNRS UMR 5220, Inserm, Université Lyon 1 Lyon 69621, France · Wadhwani School of Data Science and AI IIT Madras, India

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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

Figures & tables

ProblemGoverning equation and domainConditionsReference solution
First-order ODEu′=cos⁡(15x)u^{\prime}=\cos(15x) , x∈[−2π,2π]x\in[-2\pi,2\pi]u(0)=0u(0)=0uex=115sin⁡(15x)u_{\mathrm{ex}}=\frac{1}{15}\sin(15x)
Second-order ODEu′′=sin⁡(15x)u^{\prime\prime}=\sin(15x) , x∈[−2π,2π]x\in[-2\pi,2\pi]u(0)=0u(0)=0 , u′(0)=−1/15u^{\prime}(0)=-1/15uex=−1225sin⁡(15x)u_{\mathrm{ex}}=-\frac{1}{225}\sin(15x)
Multiscale ODEu′=cos⁡x+15cos⁡(15x)u^{\prime}=\cos x+15\cos(15x) , x∈[−2π,2π]x\in[-2\pi,2\pi]u(0)=0u(0)=0uex=sin⁡x+sin⁡(15x)u_{\mathrm{ex}}=\sin x+\sin(15x)
Poisson−Δu=f-\Delta u=f on the five-petal domainu=uexu=u_{\mathrm{ex}} on ∂Ω\partial\Omegauex=sin⁡(4πx)cos⁡(4πy)u_{\mathrm{ex}}=\sin(4\pi x)\cos(4\pi y) ; f=32π2uexf=32\pi^{2}u_{\mathrm{ex}}
BiharmonicΔ2u=f\Delta^{2}u=f on the five-petal domainu=uexu=u_{\mathrm{ex}} , ∂nu=∇uex ⁣⋅ ⁣n\partial_{n}u=\nabla u_{\mathrm{ex}}\!\cdot\!\mathbf{n} on ∂Ω\partial\Omegauex=sin⁡(5πx)cos⁡(5πy)u_{\mathrm{ex}}=\sin(5\pi x)\cos(5\pi y) ; f=2500π4uexf=2500\pi^{4}u_{\mathrm{ex}}
Singular perturbationu′−νu′′=0u^{\prime}-\nu u^{\prime\prime}=0 , x∈(0,1)x\in(0,1) , ν=10−3\nu=10^{-3}u(0)=0u(0)=0 , u(1)=1u(1)=1uex=e(x−1)/ν−e−1/ν1−e−1/ν\displaystyle u_{\mathrm{ex}}=\frac{e^{(x-1)/\nu}-e^{-1/\nu}}{1-e^{-1/\nu}}
Table 1: Summary of the numerical test problems.
Figure 1: ODE benchmarks. Top: first-order oscillatory problem u′=cos⁡(15x)u^{\prime}=\cos(15x) . Middle: second-order oscillatory problem u′′=sin⁡(15x)u^{\prime\prime}=\sin(15x) . Bottom: multiscale problem with [ω1,ω2]=[1,15][\omega_{1},\omega_{2}]=[1,15] . In each row, the left panel compares the continuum-feature kernel and exact solutions, while the right panel shows the pointwise absolute error.
CaseKernel assemblyNull-spaceReduced SVD/solveTotal solve
 (s)(s)(s)(s)
First-order ODE1.5180.0220.6032.155
Second-order ODE2.4270.0360.8233.296
Multiscale ODE1.7780.0040.7592.546
Table 2: Computational timing for the three ODE benchmarks.
MethodTraining / solution costArchitectural complexityAccuracy on high-frequency and multiscale tests
PINNApproximately 5×1045\times 10^{4} – 10510^{5} gradient steps; convergence deteriorates for high-frequency problemsSingle neural network; performance depends on depth, width, activation choice, and may benefit from Fourier featuresLarge errors for ω=15\omega=15 relative to FBPINNs; affected by spectral bias ( Moseley et al., 2023 )
FBPINNApproximately 5×1045\times 10^{4} – 5×1055\times 10^{5} gradient steps depending on the problemTypically 20–30 overlapping subdomains with multiple local networks; requires subdomain, overlap, and training-schedule choicesAccurately resolves the high-frequency and multiscale solutions; best reported first-order errors are of order 10−410^{-4} ( Moseley et al., 2023 )
CFKSNo backpropagation; deterministic kernel assembly and linear-algebra solve. Total solve times: 2.152.15 – 3.303.30 sNo neural architecture, finite hidden-layer width, or sampled center realization; prescribed kernel parameters and collocation constraintsSolution RMSEs of 1.70×10−61.70\times 10^{-6} , 9.96×10−59.96\times 10^{-5} , and 2.24×10−52.24\times 10^{-5} for the first-order, second-order, and multiscale problems, respectively
Table 3: Qualitative and quantitative comparison of PINN, FBPINN, and CFKS on the oscillatory and multiscale one-dimensional ODEs. PINN and FBPINN results are taken from Moseley et al. (2023) .
Figure 2: CFKS results on the five-petal domain. Top row: Poisson problem. Bottom row: biharmonic problem. In each row, the panels show the collocation geometry, CFKS solution, pointwise absolute error, and absolute PDE residual.
Figure 3: CFKS solution of the singularly perturbed problem. Top row: coarse predictor over the full domain, boundary-layer zoom, and independent residual indicator. Bottom row: adaptive corrector over the full domain, boundary-layer zoom, and residual-guided collocation enrichment. The adaptive refinement is concentrated near the boundary layer at x=1x=1 .
Figure 4: CFKS results for the linear advection problem. Top row: predictor solutions at representative times and the independently evaluated PDE residual. Bottom row: adaptive corrector solutions and the resulting space–time collocation enrichment. The enriched collocation region follows the trajectory of the advecting Gaussian pulse.
Figure 5: CFKS results for the advection–diffusion problem. Top row: predictor solutions at representative times and the independently evaluated PDE residual. Bottom row: adaptive corrector solutions and the corresponding space–time collocation enrichment. The enriched region follows the translating and diffusing Gaussian pulse.
Figure 6: Two-stage multiscale CFKS solution of viscous Burgers’ equation at ν=0.01\nu=0.01 . Top row: coarse predictor solutions at t=0.25t=0.25 , 0.50.5 , and 11 , together with the independently evaluated nonlinear PDE residual. Bottom row: adaptive corrector solutions at the corresponding times and the residual-guided collocation distribution. Dashed curves denote the Cole–Hopf reference solution. The corrector combines increased collocation resolution near the developing internal transition with narrower prescribed multiscale kernel widths.
 PredictorCorrector
ttRMSEL∞L^{\infty}RMSEL∞L^{\infty}
0.250.259.02×10−49.02\times 10^{-4}4.69×10−34.69\times 10^{-3}2.21×10−42.21\times 10^{-4}9.64×10−49.64\times 10^{-4}
0.500.504.65×10−24.65\times 10^{-2}3.32×10−13.32\times 10^{-1}3.44×10−43.44\times 10^{-4}2.32×10−32.32\times 10^{-3}
0.750.755.97×10−25.97\times 10^{-2}4.16×10−14.16\times 10^{-1}4.70×10−44.70\times 10^{-4}2.65×10−32.65\times 10^{-3}
1.001.004.83×10−24.83\times 10^{-2}3.01×10−13.01\times 10^{-1}4.55×10−44.55\times 10^{-4}1.81×10−31.81\times 10^{-3}
Table 4: Predictor and corrector errors for viscous Burgers’ equation at ν=0.01\nu=0.01 , evaluated against the Cole–Hopf reference solution.
Figure 7: Corrected multiscale CFKS solution of viscous Burgers’ equation at ν=0.01\nu=0.01 compared with the Cole–Hopf reference solution at t=0.25t=0.25 , 0.50.5 , 0.750.75 , and 11 . The corrected solution closely follows the reference while resolving the developing internal transition near x=0x=0 .
Figure 8: Finite-feature convergence toward CFKS. Left: relative errors of the empirical value and operator Gram matrices as the number of sampled Gaussian RBF features MM increases; shaded bands denote realization-to-realization variability and the dashed line indicates the M−1/2M^{-1/2} reference rate. Right: RMS difference between the finite-feature and CFKS solutions, together with the finite-feature error relative to the exact solution. The dashed horizontal line denotes the CFKS error relative to the exact solution.
Figure 9: Conditioning and regularization study for the first-order oscillatory ODE benchmark. Left: dependence of the CFKS solution RMSE and PDE-residual RMS on the relative Tikhonov parameter λrel\lambda_{\rm rel} at fixed σ=0.10\sigma=0.10 . Right: raw reduced-system condition number and regularized effective condition number as the Gaussian width σ\sigma is varied. Dashed vertical lines indicate the baseline parameters used in the numerical experiments.
Figure 10: Periodic advection of two separated Gaussian pulses. Top row: predictor solution at t=0t=0 , 0.50.5 , and 1.01.0 , followed by the independently evaluated residual over the space–time domain. Bottom row: adaptive-corrector solution at the same times and the resulting collocation-enrichment pattern. The residual traces both propagating pulses and directs increased collocation density along their corresponding space–time trajectories.
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