# Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?

arXiv: https://arxiv.org/abs/2412.06158 · HTML: https://mlsanity.com/paper/2412.06158/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 29 references that are in the map.
- First posted on arXiv 9 Dec 2024.

## Builds on (most cited first, 14 of 14)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning](https://mlsanity.com/paper/2101.04371/) (2021)
- [Active Importance Sampling for Variational Objectives Dominated by Rare Events: Consequences for Optimization and Generalization](https://mlsanity.com/paper/2008.06334/) (2020)
- [Deep learning neural networks for the third-order nonlinear Schrodinger equation: Solitons, breathers, and rogue waves](https://mlsanity.com/paper/2104.14809/) (2021)
- [Deep learning soliton dynamics and complex potentials recognition for 1D and 2D PT-symmetric saturable nonlinear Schr"odinger equations](https://mlsanity.com/paper/2310.02276/) (2023)

## Built on by (most cited first, 5 of 5)

- [Convergence and Sketching-Based Efficient Computation of Neural Tangent Kernel Weights in Physics-Based Loss](https://mlsanity.com/paper/2511.15530/) (2025)
- [Unveiling the Mechanism of Continuous Representation Full-Waveform Inversion: A Wave Based Neural Tangent Kernel Framework](https://mlsanity.com/paper/2603.22362/) (2026)
- [Objective Value Change and Shape-Based Accelerated Optimization for the Neural Network Approximation](https://mlsanity.com/paper/2508.20290/) (2025)
- [Pseudo-differential-enhanced physics-informed neural networks](https://mlsanity.com/paper/2602.14663/) (2026)
- [Deciphering Neural Reparameterized Full-Waveform Inversion with Neural Sensitivity Kernel and Wave Tangent Kernel](https://mlsanity.com/paper/2605.14370/) (2026)

## Abstract (arXiv)

In this paper, we study the neural tangent kernel (NTK) for general partial differential equations (PDEs) based on physics-informed neural networks (PINNs). As we all know, the training of an artificial neural network can be converted to the evolution of NTK. We analyze the initialization of NTK and the convergence conditions of NTK during training for general PDEs. The theoretical results show that the homogeneity of differential operators plays a crucial role for the convergence of NTK. Moreover, based on the PINNs, we validate the convergence conditions of NTK using the initial value problems of the sine-Gordon equation and the initial-boundary value problem of the KdV equation.