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Kinetic-based regularization: Learning spatial derivatives and PDE applications

Abhisek Ganguly, Santosh Ansumali, Sauro Succi

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.

Kinetic-based regularization: Learning spatial derivatives and PDE applications

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.
Paper Structure (9 sections, 1 theorem, 26 equations, 6 figures, 5 tables, 3 algorithms)

This paper contains 9 sections, 1 theorem, 26 equations, 6 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Results
  4. Conclusion and Outlook
  5. Appendix
  6. Exact Second-Order Correction
  7. Explicit scheme for derivative calculation
  8. Implicit scheme for derivative calculation
  9. Numerical experiment details

Key Result

Lemma A.1.1

Let $\{x_i,\phi(x_i)\}_{i=1}^N$ be a set of training points, For fully converged Lagrange multipliers $\tilde{x}$, define $\tilde{\xi}_i = x_i - \tilde{x}$. The corrected second-order prediction, satisfies $\phi^{(2)}(x) = \phi(x)$ exactly for quadratic $\phi(x) = a+bx+cx^2$ at test points $x\notin\{x_i\}$.

Figures (6)

  • Figure 1: We show convergence of implicit and explicit derivative prediction schemes ($\hat{\phi}'$ and $\hat{\phi}"$) for two 1D functions with random sampling. RMSE is computed on a random test set of $5,000$ points, benchmarked against second-order non-uniform FD.
  • Figure 2: Convergence plot (top panel) comparing errors from KBR-predicted spatial derivatives with second-order FD for the 1D Camel function on a non-uniform grid. Bottom panel shows the errors in extracted derivative errors when the training data is corrupted with varying degrees of gaussian noise. Cubic spline-smoothened results are given as a reference.
  • Figure 3: The figure shows the numerical solution at time $t=0.3$, and $t = 0.15$ for 1D inviscid Burgers' and 1D Euler equations respectively, using strict FD and KBR-integrated schemes on a grid size of 251 points. Trained PINN predictions and exact solutions are given as a reference.
  • Figure 4: Comparison between the loss function landscapes for the case of self-correction and the proposed exact second-order correction, with respect to the single parameter $\theta$ and validation error in KBR. Training data consists of $501$ randomly sampled points, with an 80:20 split between train and validation.
  • Figure 5: The actual and predicted fields, gradients, Laplacian and Hessian is shown for the 2D Camel function on a uniform $101\times 101$ grid using KBR Implicit spatial derivative scheme.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma A.1.1: Exact Second-Order KBR
  • proof