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.

翻译：本文提出了稳定物理信息核演化（SPIKE）方法，用于无粘双曲守恒律的数值计算。SPIKE方法解决了一个根本性悖论：强形式残差最小化如何能够捕捉包含间断的弱解。SPIKE采用具有正则化参数演化的再生核表示，其中Tikhonov正则化提供了穿越激波形成的平滑过渡机制，使动力学能够跨越激波奇点。该方法在无需显式激波探测或人工粘性的统一框架内，自动保持守恒性、追踪特征线并捕捉满足Rankine-Hugoniot条件的激波。在标量和向量值守恒律上的数值验证证实了该方法的有效性。