Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis

TL;DR

KEqL delivers a data-efficient, kernel-based framework for learning differential equations and their solution maps from scarce data, unifying equation learning, operator learning, and PDE solving under a computational-graph view. It offers two main learning strategies: a 2-step approach that first recovers states $u^m$ with kernel interpolation and then learns the operator $P$, and a 1-step approach that jointly learns $u^m$ and $P$ with a PDE constraint, aided by a representer theorem and LM optimization; a reduced 1-step variant further improves efficiency. The framework is supported by quantitative worst-case error bounds and convergence theory in RKHS Sobolev spaces, with rates tied to fill-distances and smoothness. Empirically, KEqL achieves 1–2 orders of magnitude improvements in accuracy over state-of-the-art baselines on Duffing, Burgers, and Darcy problems, while displaying robustness to hyper-parameter choices and enabling novel capabilities such as one-shot operator learning for variable-coefficient PDEs in extremely data-scarce regimes.

Abstract

We introduce a novel kernel-based framework for learning differential equations and their solution maps that is efficient in data requirements, in terms of solution examples and amount of measurements from each example, and computational cost, in terms of training procedures. Our approach is mathematically interpretable and backed by rigorous theoretical guarantees in the form of quantitative worst-case error bounds for the learned equation. Numerical benchmarks demonstrate significant improvements in computational complexity and robustness while achieving one to two orders of magnitude improvements in terms of accuracy compared to state-of-the-art algorithms.

Figures (13)

• Figure 1: (A) Schematic depiction of the computational graph of \ref{['form-of-P']} in the context of equation learning for a single pair $(u,f)$. Red objects are unknown nonlinearities that need to be learned. Blue objects are data for the problem, while black objects (the map $\Phi$) are assumed to be known. The left and right panels show the solution and right-hand side of an example second order PDE depending on $y, u, \partial_y u$, and $\partial_{yy} u$ while the middle panel shows $\Phi(y, u)$; (B) The computational graph of 2-step KEqL. Red edges are unknown nonlinear maps to be learned. Blue boxes denote data that is known for various nodes with dashed lines denoting where the data is injected. Note that the graphs for $u^m$ and $P$ are disconnected, hence the learning of $u^m$ and $P$ is performed sequentially in two steps; (C) The computational graph for 1-step KEqL. Coloring conventions follow panel (B) with the main difference being that the $u^m$ and $P$ are now connected and have to be learned simultaneously.
• Figure 2: Representative numerical results for the Duffing oscillator \ref{['duffing_ODE']}: (A) Shows the training data and the ground truth state of the system $u$ in comparison to the filtered state $\widehat{u}$ using 1-step KEqL and 2-step methods; (B) Quantitative values of relative filtering and operator learning errors. The operator learning errors are reported for different time windows and essentially constitute extrapolation errors. These values were averaged over three novel initial conditions; (C) visualization of three extrapolated dynamics used to compute $\mathcal{R}_{\text{opl}}$.
• Figure 3: Representative numerical results for Burgers' PDE \ref{['burgers-PDE']}: (A) The filtering and equation learning errors computed for the training functions for 1-step KEqL, SINDy, and PINN-SR using $M=1$ training pairs with different number of interior observations $N_\mathcal{Y}$; (B) Similar experiment as panel (A) but with randomized initial conditions; (C) An example application for an initial condition that leads to multiple shocks with scarce observations depicting the quality of filtering obtained using 1-step and 2-step methods; (D) Similar setup to row (C) with a smooth solution that is only observed on the boundary; (E) Depicting the solution to the PDEs that were learned in row (D) for a new initial condition.
• Figure 4: Representative numerical results for Darcy's flow PDE \ref{['darcy_PDE']}: The first three figures from the left show the equation learning errors computed over training, ID test, and OOD test data while the last panel shows the ID operator learning errors. R1-step here denotes the reduced 1-step KEqL method and the labels on the graphs denote the number of interior observations points $N_\mathcal{Y}$.
• Figure 5: Convergence history of LM for 1-step KEqL for the Duffing ODE \ref{['duffing_ODE']}.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Proposition 1
• Lemma 1
• proof
• Lemma 2: Representer theorem for interpolation owhadi2019operator
• Proposition 2: Sobolev embedding theorem adams2003sobolev
• Proposition 3: Sobolev sampling inequality
• Remark 1
• Proposition 4