SPIKE: Stable Physics-Informed Kernel Evolution

• SPIKE is a method for solving hyperbolic conservation laws by evolving physics-informed, reproducing kernel representations that capture both smooth and shock-containing solutions.
• It utilizes strong-form residual minimization, RKHS theory, and Tikhonov regularization to evolve kernel amplitudes and locations, ensuring exact conservation and accurate shock tracking.
• The approach avoids ad hoc shock capturing and artificial viscosity by integrating regularization directly into the kernel evolution, enabling stable and interpretable numerical simulations.

Stable Physics-Informed Kernel Evolution (SPIKE) is a methodology for solving hyperbolic conservation laws and related PDEs via dynamic, reproducing kernel representations with regularized parameter evolution. It is specifically designed to address the challenge of accurately modeling both smooth and discontinuous (shock-containing) solutions within a unified, strongly physics-informed framework, without resorting to ad hoc shock capturing, artificial viscosity, or explicit weak-form formulations. The approach relies on minimization of strong-form residuals, RKHS theory, and Tikhonov regularization for stable numerical evolution across both scalar and vector-valued conservation laws (Su et al., 21 Oct 2025).

1. Motivation and Paradox of Strong-Form Residual Minimization

SPIKE was introduced to address a historic paradox: classical strong-form residual minimization typically fails when solutions contain discontinuities, due to ill-posedness at points of non-differentiability. Hyperbolic conservation laws (e.g., Burgers’, Buckley–Leverett, Euler equations) commonly evolve shocks from smooth initial data, where traditional PDE solvers must explicitly switch to weak formulations or build in numerical smoothing. SPIKE overcomes this by utilizing a physics-informed kernel representation designed to maintain stability and accurately transition through singularity formation.

2. Kernel-Based Representation and Parameter Evolution

The core solution ansatz is a superposition of translating, amplitude-modulated kernels:

$$q(x, t) \approx \sum_{i=1}^N a_i(t)\, \varphi(x - x_i(t)) + b(t),$$

where $\varphi$ is a piecewise quadratic function in $H^1(T)/\mathbb{R}$, constructed to have zero mean and satisfy RKHS reproducing properties. The kernel parameters (locations $x_i(t)$ and amplitudes $a_i(t)$) are evolved according to a regularized ODE system obtained by minimizing the strong-form residual

$$\mathcal{R}(q; \dot{\theta}) = \left| \frac{\partial q}{\partial \theta} \cdot \dot{\theta} + \partial_x f(q) \right|^2,$$

subject to Tikhonov regularization terms (e.g., $\lambda_a \sum_i |\dot{a}_i|^2$, $\lambda_x \sum_i |\dot{x}_i|^2$) to ensure convexity, bound parameter drift, and facilitate traversing non-smooth regions. The superposition structure and RKHS constraint allow for direct encoding of discontinuities via kernel amplitude differences, while regularization moderates infinite gradients at shocks.

3. Shock Capturing via Regularized Kernel Evolution

SPIKE’s formulation inherently transitions through shock formation by leveraging the regularization:

• Near an emergent shock, adjacent kernel amplitudes ($a_i$, $a_{i+1}$) acquire large opposite values, corresponding to a sharp localized gradient.
• The spline property of $\varphi$ and the constraint $\sum_i a_i = 0$ encode the jump where the second derivative is concentrated.
• Tikhonov regularization terms ensure that parameter evolution remains bounded and that finite-time blowup (possible in unregularized strong-form minimization) is averted, enabling a smooth passage through shock singularities.
• In the limit $\lambda \to 0$, the model converges to weak solutions satisfying the Rankine–Hugoniot conditions.

This mechanism obviates explicit shock detection and avoids artificial dissipation.

4. Conservation Laws and Characteristic Tracking

SPIKE is constructed to guarantee exact conservation and correctly follow characteristics:

• By using $\varphi$ of zero mean and incorporating a bias term $b(t)$ equal to $\int q(x,t)\,dx$, conservation of the integral is automatic.
• The ODE system for kernel locations $x_i$ yields, in smooth regions,

$$\dot{x}_i = f'(q(x_i)) + \text{higher-order corrections},$$

matching characteristic speeds dictated by the conservation law.

• In the presence of a shock, clustered kernel centers propagate together at the correct Rankine–Hugoniot velocity.

Consequently, SPIKE accurately tracks both propagation and interaction of discontinuities without additional physics or numerical artifacts.

5. Numerical Performance and Validation

SPIKE demonstrates robust performance on canonical and complex test problems:

• Scalar laws: Burgers’, Buckley–Leverett problems show SPIKE reconstructs shocks matching high-resolution finite-volume baselines, with errors localized to the discontinuity vicinity only.
• Systems: For the compressible Euler equations, SPIKE enables stable, long-term integration with accurate resolution of interacting shock waves and rarefaction fans.
• Compared methods: Physics-Informed Neural Networks (PINNs) tend to oversmooth shocks; Evolutional Deep Neural Networks (EDNNs) are prone to oscillatory artifacts. SPIKE’s kernel evolution and regularization result in superior accuracy and stability throughout discontinuity formation and propagation.
• Adaptive knot clustering and periodic redistribution strategies maintain numerical stability over extended simulation time frames.

6. Comparison with Classical and Machine Learning Solvers

SPIKE departs from classical finite-volume/high-order shock-capturing techniques and standard machine-learning solvers:

Method	Shock Handling	Conservation	Mesh Adaptivity	Regularization	Weak Form Needed
SPIKE	Implicit (kernel amplitudes)	Exact	Kernel clustering	Tikhonov (O(N))	No
PINN	Weak (loss residual)	Approximate	None	Data/physics loss	Yes
Finite-Volume	Explicit (flux limiters)	Controlled	Sometimes	Upwind/diffusion	Yes

SPIKE integrates physics directly into kernel evolution, dispensing with artificial viscosity, mesh refinement, or explicit weak-form integration.

7. Broader Implications and Future Directions

SPIKE establishes new theoretical and computational links between reproducing kernel theory, variational physics-informed modeling, and classical conservation law analysis. Key implications and plausible future extensions include:

• Applicability to large-scale, real-time simulations in computational fluid dynamics, gas dynamics, and wave propagation, where shocks and conservation properties are essential.
• Linear complexity (O(N)) algorithms via spline kernel structures suggest scalability to higher-dimensional and long-time problems.
• Potential for integration with adaptive knot management, more general domain geometries, and further theoretical exploration of regularization–viscosity connections.
• Suggests a unifying pathway beyond explicit weak-form formulations for stable, conservative, and accurate hyperbolic PDE solvers.

SPIKE’s foundational philosophy—evolving a physics-informed kernel representation via strong-form regularized optimization—offers an interpretable, stable, and effective route for future research in numerical methods for conservation laws and physics-informed machine learning (Su et al., 21 Oct 2025).