# Fast Learning Rates for Physics-Informed Kernel Methods

arXiv: https://arxiv.org/abs/2609.18901 · HTML: https://mlsanity.com/paper/2609.18901/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 59 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 26 of 26)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [Machine learning for molecular simulation](https://mlsanity.com/paper/1911.02792/) (2019)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [Overcoming the curse of dimensionality with Laplacian regularization in semi-supervised learning](https://mlsanity.com/paper/2009.04324/) (2020)
- [Physics-informed machine learning as a kernel method](https://mlsanity.com/paper/2402.07514/) (2024)
- [Physics-informed kernel learning](https://mlsanity.com/paper/2409.13786/) (2024)
- [Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning](https://mlsanity.com/paper/2501.18708/) (2025)
- [Normal-guided Detail-Preserving Neural Implicit Function for High-Fidelity 3D Surface Reconstruction](https://mlsanity.com/paper/2406.04861/) (2024)
- [On the Sample Complexity of Subspace Learning](https://mlsanity.com/paper/1408.5032/) (2014)
- [Fast kernel methods: Sobolev, physics-informed, and additive models](https://mlsanity.com/paper/2509.02649/) (2025)
- [Learning Controlled Stochastic Differential Equations](https://mlsanity.com/paper/2411.01982/) (2024)
- [Precise asymptotic analysis of Sobolev training for random feature models](https://mlsanity.com/paper/2511.03050/) (2025)
- [PIKS: Universal Physics-Informed Kernel Methods](https://mlsanity.com/paper/2607.27062/) (2026)

## Abstract (arXiv)

In physics-informed machine learning, a target function u^* is learned from noisy value observations yi=u^*(xi)+ varepsiloni, together with differential information, given either by noisy observations dj=(Du^*)(zj)+xij or by a known physical constraint Du^*=v. We consider the setting where D is a linear differential operator and analyze a physics-informed kernel estimator hat u combining n value observations and m differential observations. In this context, we ask how much can differential information improve predictions, and how does this improvement depend quantitatively on n, m, and D. We prove finite-sample bounds, supported by numerical simulations, revealing a two-regime structure for the prediction error. When m is limited, the rate depends jointly on n and m; when m exceeds a problem-dependent threshold, the rate saturates and matches the oracle rate obtained when the perfect constraint D hat u = Du^* is imposed. Examples are discussed for Sobolev spaces which are reproducing kernel Hilbert spaces and include partial Laplacian constraints on the torus and gradient observations on bounded domains. These examples illustrate the range of possible learning rate improvements --- from the standard nonparametric n-1/4 to the parametric rate n-1/2. Finally, we derive physically consistent rates in a stronger norm that jointly controls the errors in hat u and Dhat u.