Kinetic-based regularization: Learning spatial derivatives and PDE applications

• URL: http://arxiv.org/abs/2603.06380v1
• Date: Fri, 06 Mar 2026 15:32:27 GMT
• Title: Kinetic-based regularization: Learning spatial derivatives and PDE applications
• Abstract summary: We extend kinetic-based regularization to learn spatial derivatives with provable second-order accuracy in 1D.<n>The fully localized formulation enables efficient, noise-adaptive derivative estimation without requiring global system solving or smoothing.<n>Preliminary results show that coupling KBR with conservative solvers enables stable shock capture in 1D hyperbolic PDEs.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Accurate estimation of spatial derivatives from discrete and noisy data is central to scientific machine learning and numerical solutions of PDEs. We extend kinetic-based regularization (KBR), a localized multidimensional kernel regression method with a single trainable parameter, to learn spatial derivatives with provable second-order accuracy in 1D. Two derivative-learning schemes are proposed: an explicit scheme based on the closed-form prediction expressions, and an implicit scheme that solves a perturbed linear system at the points of interest. The fully localized formulation enables efficient, noise-adaptive derivative estimation without requiring global system solving or heuristic smoothing. Both approaches exhibit quadratic convergence, matching second-order finite difference for clean data, along with a possible high-dimensional formulation. Preliminary results show that coupling KBR with conservative solvers enables stable shock capture in 1D hyperbolic PDEs, acting as a step towards solving PDEs on irregular point clouds in higher dimensions while preserving conservation laws.

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