{"ID":2852022,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.18266","arxiv_id":"2510.18266","title":"SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Laws","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.","short_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...","url_abs":"https://arxiv.org/abs/2510.18266","url_pdf":"https://arxiv.org/pdf/2510.18266v1","authors":"[\"Hua Su\",\"Lei Zhang\",\"Jin Zhao\"]","published":"2025-10-21T03:34:49Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"cs.AI\",\"cs.LG\",\"math.AP\"]","methods":"[]","has_code":false}