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Toward Efficient Kernel-Based Solvers for Nonlinear PDEs
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Proposes SKS, a kernel-interpolation PDE solver that removes differential operators from the kernel and uses grid collocation with product kernels to get Kronecker-structured computation that scales to over 100K collocation points.
Problem
Solving nonlinear PDEs with kernel methods is expensive because the Gram matrix over solution and derivative evaluations grows beyond the number of collocation points, limiting scalability to the dense collocation sets needed for rough or high-frequency solutions.
Approach
Model the solution as a standard kernel interpolant over collocation points with free nodal values, differentiate the interpolant directly rather than embedding operators in the kernel, and place points on a grid with a product kernel so the kernel matrix is a Kronecker product that avoids forming the full Gram matrix.
Key results
- On Allen-Cahn (a=15), SKS reaches L2 error 4.44E-06 on a 200x200 grid (40000 points), while PINN stays around 10^0 and DAKS around 10^-1 at 4800 points.
- On Burgers' (nu=0.001), SKS with 120000 collocation points achieves L2 error 2.28E-03, versus PINN 4.13E-03 at the same count.
- SKS per-iteration runtime is about 100x faster than the naive full-matrix variant, which runs out of memory at 22,500 Burgers' points.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use SKS instead of DAKS | Kernel PDE solvers · Eikonal, 300 collocation points | L² error: 1.01E-01 -> 6.23E-04 | Helped |
| use SKS instead of DAKS | Kernel PDE solvers · Nonlinear elliptic, 2400 collocation points | L² error: 1.68E-07 -> 1.83E-06 | Hurt |
| use SKS instead of PINN | PDE solvers · Allen-Cahn a=15, 40000 collocation points | L² error: 5.86E0 -> 4.44E-06 | Helped |
| increase collocation points using the SKS grid method | SKS · Burgers ν=0.001, 4800 to 120000 collocation points | L² error: 1.89E-02 -> 2.28E-03 | Helped |
| use a 96 × 50 grid rather than a 70 × 70 grid (ablation) | SKS · Burgers ν=0.02, 4800 collocation points | L² error: 3.21E-04 -> 7.54E-05 | Helped |
| increase spatial resolution from 600 × 200 to 1200 × 100 (ablation) | SKS · Burgers ν=0.001, 120000 collocation points | L² error: 2.28E-03 -> 1.63E-03 | Helped |
| optimize basis coefficients α instead of solution values η (ablation) | SKS · Burgers ν=0.02, 1200 collocation points | L² error: 5.40E-03 -> 2.80E-02 | Hurt |
| set the length-scale to 0.1 instead of 0.05 (ablation) | SKS · Nonlinear elliptic, 35 × 35 grid | L² error: 1.19E-02 -> 6.80E-06 | Helped |
| use grid rather than random collocation points (ablation) | DAKS · Burgers ν=0.02, 600 collocation points | L² error: 1.75E-02 -> 4.09E-01 | Hurt |
| use Kronecker structure rather than naive full-matrix computation (ablation) | SKS · Burgers ν=0.001, 4800 collocation points | runtime per iteration: 5.4E-02 -> 9.8E-4 | Helped |
| use SKS instead of finite differences | PDE solvers · Nonlinear elliptic, 18 × 18 grid | L² error: 3.36E-02 -> 1.26E-02 | Helped |
Surprising findings
- On simpler benchmarks with few hundred to few thousand points, DAKS and PINN outperform SKS in some cases, notably Burgers' at 600-2400 points for PINN and nonlinear elliptic at 2400 points for DAKS.
- Optimizing kernel coefficients alpha instead of nodal values eta worsened error by an order of magnitude in tested settings.
Limitations
- Relies on ADAM with many epochs; second-order methods such as L-BFGS and relaxed Gauss-Newton fail to improve or diverge on most benchmarks.
- Kronecker structure requires collocation points on a tensor grid; irregular domains use a virtual grid.
- Performance is sensitive to kernel length-scale and grid shape, and it is weaker than DAKS on some low-point settings.
Experimental setup
Datasets: Burgers' equation (Cole-Hopf reference solution) nonlinear elliptic PDE (manufactured solution) regularized Eikonal PDE (finite difference reference) 2D Allen-Cahn equation (manufactured solution)
Benchmarks: Burgers' equation nonlinear elliptic PDE Eikonal PDE 2D Allen-Cahn equation
Metrics: L2 relative error runtime per iteration runtime total
Baselines: DAKS PINN finite difference Newton-Krylov RBF methods
Builds on: Derivative-augmented kernel PDE solver (DAKS) physics-informed neural networks Gaussian process regression
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