math.NAMar 6, 2026

Kinetic-based regularization: Learning spatial derivatives and PDE applications

Authors: Abhisek Ganguly, Santosh Ansumali, Sauro Succi

Organizations: Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bengaluru-560064, Karnataka, India · Center for Life Nano-Neuroscience at La Sapienza, Fondazione Istituto Italiano di Tecnologia, Viale Regina Elena 291, 00161 Roma, Italy

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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.

Figures & tables

Figure 1: We show convergence of implicit and explicit derivative prediction schemes ( ϕ^′\hat{\phi}^{\prime} and ϕ^′′\hat{\phi}^{\prime\prime} ) for two 1D functions with random sampling. RMSE is computed on a random test set of 5,0005,000 points, benchmarked against second-order non-uniform FD.
ϕ(x)\phi(x)DNNKBR-ImplicitKBR-Explicit
 ϕ′\phi^{\prime}ϕ′\phi^{\prime}ϕ′′\phi^{\prime\prime}ϕ′\phi^{\prime}ϕ′′\phi^{\prime\prime}
sin⁡(x)\sin(x)7.5×10−67.5\times 10^{-6}6.60×10−126.60\times 10^{-12}2.22×10−42.22\times 10^{-4}3.12×10−123.12\times 10^{-12}6.50×10−56.50\times 10^{-5}
x2x^{2}1.1×10−31.1\times 10^{-3}8.14×10−138.14\times 10^{-13}3.27×10−43.27\times 10^{-4}3.24×10−193.24\times 10^{-19}3.07×10−143.07\times 10^{-14}
ln⁡(x)\ln(x)3.6×10−63.6\times 10^{-6}8.92×10−128.92\times 10^{-12}2.52×10−42.52\times 10^{-4}3.88×10−123.88\times 10^{-12}1.34×10−41.34\times 10^{-4}
Table 1: Comparison of non-normalized MSE (DNN vs KBR) on deployment/test data
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=0.3t=0.3 , and t=0.15t=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.
RegionMetricsKBR-RoeRoeMUSCL-TVD
Shock 1 (Region 1)L1 Error (Global)9.409×10−39.409\times 10^{-3}1.014×10−21.014\times 10^{-2}5.311×10−35.311\times 10^{-3}
L ∞\infty Error (Global)9.305×10−29.305\times 10^{-2}9.021×10−29.021\times 10^{-2}8.614×10−28.614\times 10^{-2}
Shock Thickness2.874×10−22.874\times 10^{-2}3.221×10−23.221\times 10^{-2}2.450×10−22.450\times 10^{-2}
Post-shock Osc.4.919×10−24.919\times 10^{-2}5.162×10−25.162\times 10^{-2}4.392×10−24.392\times 10^{-2}
Total Variation1.010×10−11.010\times 10^{-1}9.646×10−29.646\times 10^{-2}1.265×10−11.265\times 10^{-1}
Shock 2 (Region 2)L1 Error9.409×10−39.409\times 10^{-3}1.014×10−21.014\times 10^{-2}5.311×10−35.311\times 10^{-3}
Table 2: Shock-capturing metrics for density ρ\rho in the Sod shock tube problem on a non-uniform grid (KBR-Roe, Roe first-order, MUSCL-TVD). Region 1: shock [0.62,0.67][0.62,0.67] , post-shock [0.64,0.72][0.64,0.72] ; Region 2: shock [0.73,0.77][0.73,0.77] , post-shock [0.755,0.80][0.755,0.80] .
Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

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 θ\theta and validation error in KBR. Training data consists of 501501 randomly sampled points, with an 80:20 split between train and validation.
 A=[∑iPi∑ixiPi∑ixi2Pi∑iPi−∑ixiPi−∑ixi2Pi−∑iPi+∑ixiPi+∑ixi2Pi+]A=\begin{bmatrix}\sum_{i}P_{i}&\sum_{i}x_{i}P_{i}&\sum_{i}x_{i}^{2}P_{i}\\ \sum_{i}P_{i}^{-}&\sum_{i}x_{i}P_{i}^{-}&\sum_{i}x_{i}^{2}P_{i}^{-}\\ \sum_{i}P_{i}^{+}&\sum_{i}x_{i}P_{i}^{+}&\sum_{i}x_{i}^{2}P_{i}^{+}\end{bmatrix} 
Appendix
Algorithm 3 Implicit scheme for KBR derivatives
Figure 5: The actual and predicted fields, gradients, Laplacian and Hessian is shown for the 2D Camel function on a uniform 101×101101\times 101 grid using KBR Implicit spatial derivative scheme.
MetricDescription
L1 errorMean absolute deviation from exact solution
L ∞\infty errorMaximum absolute deviation
Shock thickness10%–90% transition width of the discontinuity (measures smearing)
Post-shock oscillationMaximum deviation in the post-shock region
Total variationSum of absolute differences in the shock window
Appendix
Table 3: Metrics used for shock-capturing evaluation in the Sod shock tube problem.
Figure 6: Schematic of (left) a one-dimensional non-uniform grid with interface flux locations, and (right) a 2D Voronoi diagram representation constructed from scattered points, representing an unstructured control volume discretization.
PDEConservative formIC ( t=0t=0 )BC
Burgers’∂u∂t+∂∂x(u22)=0\dfrac{\partial u}{\partial t}+\dfrac{\partial}{\partial x}\left(\dfrac{u^{2}}{2}\right)=0u={1x<0.50x≥0.5u=\begin{cases}1&x<0.5\\ 0&x\geq 0.5\end{cases}u(0,t)=1u(0,t)=1
   u(1,t)=0u(1,t)=0
Euler∂∂t(ρρuρE)+∂∂x(ρuρu2+pu(ρE+p))=0\dfrac{\partial}{\partial t}\begin{pmatrix}\rho\\ \rho u\\ \rho E\end{pmatrix}+\dfrac{\partial}{\partial x}\begin{pmatrix}\rho u\\ \rho u^{2}+p\\ u(\rho E+p)\end{pmatrix}=\mathbf{0}(ρup)={(101)x<0.5(0.12500.1)x≥0.5\begin{pmatrix}\rho\\ u\\ p\end{pmatrix}=\begin{cases}\begin{pmatrix}1\\ 0\\ 1\end{pmatrix}&x<0.5\\ \begin{pmatrix}0.125\\ 0\\ 0.1\end{pmatrix}&x\geq 0.5\end{cases}outflow
Appendix
Table 4: Conservative form, initial conditions and boundary conditions for the numerical solution of the tested PDEs.
ComponentValue / DescriptionNotes
Input dimensions2(x,t)(x,t)
Output dimensions1uu
Hidden layers4 
Neurons per hidden layer64 
Activation function (hidden)tanh⁡\tanh 
Output activationsoftplus + 10−610^{-6}Ensures positivity
Appendix
Table 5: PINN architecture and training hyperparameters used for both Euler equation (Sod shock tube test) and inviscid Burgers’equation with Riemann type initial conditions.

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