# KP-PINNs: Kernel Packet Accelerated Physics Informed Neural Networks

arXiv: https://arxiv.org/abs/2506.08563 · HTML: https://mlsanity.com/paper/2506.08563/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 45 references that are in the map.
- First posted on arXiv 10 Jun 2025.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Improved Training of Physics-Informed Neural Networks with Model Ensembles](https://mlsanity.com/paper/2204.05108/) (2022)
- [BdryGP: a new Gaussian process model for incorporating boundary information](https://mlsanity.com/paper/1908.08868/) (2019)
- [Kernel Packet: An Exact and Scalable Algorithm for Gaussian Process Regression with Mat'ern Correlations](https://mlsanity.com/paper/2203.03116/) (2022)
- [High-Dimensional Non-Parametric Density Estimation in Mixed Smooth Sobolev Spaces](https://mlsanity.com/paper/2006.03696/) (2020)

## Abstract (arXiv)

Differential equations are involved in modeling many engineering problems. Many efforts have been devoted to solving differential equations. Due to the flexibility of neural networks, Physics Informed Neural Networks (PINNs) have recently been proposed to solve complex differential equations and have demonstrated superior performance in many applications. While the L2 loss function is usually a default choice in PINNs, it has been shown that the corresponding numerical solution is incorrect and unstable for some complex equations. In this work, we propose a new PINNs framework named Kernel Packet accelerated PINNs (KP-PINNs), which gives a new expression of the loss function using the reproducing kernel Hilbert space (RKHS) norm and uses the Kernel Packet (KP) method to accelerate the computation. Theoretical results show that KP-PINNs can be stable across various differential equations. Numerical experiments illustrate that KP-PINNs can solve differential equations effectively and efficiently. This framework provides a promising direction for improving the stability and accuracy of PINNs-based solvers in scientific computing.