Kinetic-based regularization: Learning spatial derivatives and PDE applications

TL;DR

This paper introduces an extension of kinetic-based regularization (KBR) for accurate, noise-robust estimation of spatial derivatives, enabling improved PDE solutions and shock capturing in irregular data settings.

Contribution

The paper develops a second-order accurate, localized KBR method for learning spatial derivatives, with explicit and implicit schemes, applicable to high-dimensional and irregular data.

Findings

Quadratic convergence in derivative estimation

Stable shock capture in 1D hyperbolic PDEs

Efficient noise-adaptive derivative estimation

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.