Kinetic-based regularization: Learning spatial derivatives and PDE applications

2026-03-06

AI Summary

Accurately estimating spatial derivatives from discrete, noisy data remains a central challenge in scientific machine learning and numerical solutions of partial differential equations. This work introduces kinetic-based regularization (KBR) into derivative learning for the first time, proposing both explicit and implicit fully localized schemes that achieve noise-adaptive, high-accuracy derivative estimation without global solves or heuristic smoothing. The approach is based on local multidimensional kernel regression with a single trainable parameter, combining closed-form predictions with solutions to perturbed linear systems. In one dimension, it offers theoretical guarantees of second-order accuracy and quadratic convergence, while naturally preserving conservation laws. Preliminary applications to one-dimensional hyperbolic conservation laws successfully capture shocks in a stable manner, and the method extends naturally to higher dimensions.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• spatial derivatives
• noisy data
• PDEs
• irregular point clouds
• conservation laws

Innovation

• kinetic-based regularization
• spatial derivative learning
• localized kernel regression
• second-order accuracy
• conservative PDE solvers