SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Laws

TL;DR

This work introduces SPIKE (Stable Physics-Informed Kernel Evolution), a kernel-based framework for solving 1D inviscid hyperbolic conservation laws by minimizing the strong-form residual with Tikhonov regularization. The solution is represented as a dynamic sum of shifted kernels $q(x,t)=\sum_{i=1}^N a_i(t)\varphi(x-x_i(t))+b(t)$, with evolving knot positions $x_i(t)$ and amplitudes $a_i(t)$; conservation is automatic thanks to a zero-mean kernel and a constant bias, and the evolution is made efficient via a cubic Hermite spline structure yielding a block tridiagonal system solvable in $O(N)$. Regularization smooths the parameter dynamics through shock formation, enabling a vanishing-regularization limit that recovers correct shock behavior while maintaining sharp discontinuities in the solution representation. Numerical experiments on Burgers’, Buckley–Leverett, and Euler equations show SPIKE achieving sharp shocks, robustness to long-time integration, and competitive computational efficiency compared with existing physics-informed approaches. These results reveal a deep link between reproducing kernel theory and hyperbolic PDEs, suggesting kernel-based physics-informed methods as a powerful tool for discontinuous problems.

Abstract

We introduce the Stable Physics-Informed Kernel Evolution (SPIKE) method for numerical computation of inviscid hyperbolic conservation laws. SPIKE resolves a fundamental paradox: how strong-form residual minimization can capture weak solutions containing discontinuities. SPIKE employs reproducing kernel representations with regularized parameter evolution, where Tikhonov regularization provides a smooth transition mechanism through shock formation, allowing the dynamics to traverse shock singularities. This approach automatically maintains conservation, tracks characteristics, and captures shocks satisfying Rankine-Hugoniot conditions within a unified framework requiring no explicit shock detection or artificial viscosity. Numerical validation across scalar and vector-valued conservation laws confirms the method's effectiveness.

Figures (8)

• Figure 1: Linear combination of two kernels to approximate a jump discontinuity, with $a_1=-a_2=10$ and $x_1=-x_2=-0.02$. The resulting piecewise linear approximation closely captures the sharp transition.
• Figure 1: Evolution of the exact zigzag solution (right) and SPIKE approximations with different regularization strengths: $\lambda_a=\lambda_x=10^{-3}$ (left), $10^{-6}$ (middle).
• Figure 1: (a) Solution contours of Burgers' equation using SPIKE, PINN, and EDNN, compared to a high-resolution reference. (b) Pointwise error relative to the reference solution.
• Figure 2: Convergence of SPIKE shape parameters $H(t)$ and $\delta(t)$ to the exact solution (dashed lines) as the regularization strength ($\lambda_a=\lambda_x=\lambda$) vanishes.
• Figure 2: (a) Solution contours of the Buckley-Leverett equation using SPIKE, PINN, and EDNN, compared to a high-resolution reference. (b) Pointwise error relative to the reference solution.

Theorems & Definitions (8)

• Remark 2.1: The Dual Role of Regularization
• Proposition 2.2
• Proof 1
• Proposition 2.3
• Proof 2
• Remark 2.4
• Remark 4.1
• Remark 4.2