Kinetic-based regularization: Learning spatial derivatives and PDE applications
This work provides a method for accurately estimating spatial derivatives from noisy data, which is crucial for scientific machine learning and numerical PDE solutions, particularly for researchers working with irregular point clouds.
This paper extends kinetic-based regularization (KBR) to learn spatial derivatives from discrete and noisy data, achieving provable second-order accuracy in 1D. The proposed explicit and implicit schemes demonstrate quadratic convergence, matching second-order finite difference for clean data, and enable stable shock capture in 1D hyperbolic PDEs when coupled with conservative solvers.
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.