用局部核回归学习空间导数，实现高精度无网格微分求解。
Kinetic-based regularization: Learning spatial derivatives and PDE applications
- 基于动能正则化，通过局部核回归直接学习导数。
- 在1维下达到二阶精度，噪声数据中表现稳定。
- 适合处理不规则点云上的守恒型偏微分方程求解。
从离散且含噪数据中准确估计空间导数是科学机器学习与偏微分方程数值求解的核心问题。本文将动能正则化（KBR）——一种具有单个可训练参数的局部多维核回归方法——扩展至1维空间导数学习，并证明其具备二阶收敛性。提出两种导数学习方案：基于显式闭式预测表达式的显式方法，以及在目标点求解扰动线性系统的隐式方法。完全局部化的形式无需全局求解或启发式平滑，能自适应噪声。两种方法均呈现二次收敛，对清洁数据性能等同于二阶有限差分，且可拓展至高维。初步结果表明，将KBR与保守型求解器结合，可在1维双曲型PDE中实现稳定激波捕捉，为在更高维度不规则点云上求解守恒律方程提供可能。
原文摘要 · Abstract (English)
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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