# opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation

arXiv: https://arxiv.org/abs/2207.01765 · HTML: https://mlsanity.com/paper/2207.01765/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 60 references that are in the map.
- First posted on arXiv 5 Jul 2022.

## Builds on (most cited first, 11 of 11)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach](https://mlsanity.com/paper/1911.09843/) (2019)
- [The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach](https://mlsanity.com/paper/2009.13280/) (2020)
- [A spectral collocation method for the Landau equation in plasma physics](https://mlsanity.com/paper/2006.15885/) (2020)

## Built on by (most cited first, 8 of 8)

- [Separable Physics-informed Neural Networks for Solving the BGK Model of the Boltzmann Equation](https://mlsanity.com/paper/2403.06342/) (2024)
- [Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEs](https://mlsanity.com/paper/2404.17868/) (2024)
- [FourierSpecNet: Neural Collision Operator Approximation Inspired by the Fourier Spectral Method for Solving the Boltzmann Equation](https://mlsanity.com/paper/2504.20408/) (2025)
- [Micro-Macro Tensor Neural Surrogates for Uncertainty Quantification in Collisional Plasma](https://mlsanity.com/paper/2512.24205/) (2025)
- [JKO for Landau: a variational particle method for homogeneous Landau equation](https://mlsanity.com/paper/2409.12296/) (2024)
- [Pseudo-Physics-Informed Neural Operators: Enhancing Operator Learning from Limited Data](https://mlsanity.com/paper/2502.02682/) (2025)
- [Learning collision operators from plasma phase space data using differentiable simulators](https://mlsanity.com/paper/2601.10885/) (2026)
- [A Theory-guided Weighted L2 Loss for solving the BGK model via Physics-informed neural networks](https://mlsanity.com/paper/2604.04971/) (2026)

## Abstract (arXiv)

We propose a hybrid framework opPINN: physics-informed neural network (PINN) with operator learning for approximating the solution to the Fokker-Planck-Landau (FPL) equation. The opPINN framework is divided into two steps: Step 1 and Step 2. After the operator surrogate models are trained during Step 1, PINN can effectively approximate the solution to the FPL equation during Step 2 by using the pre-trained surrogate models. The operator surrogate models greatly reduce the computational cost and boost PINN by approximating the complex Landau collision integral in the FPL equation. The operator surrogate models can also be combined with the traditional numerical schemes. It provides a high efficiency in computational time when the number of velocity modes becomes larger. Using the opPINN framework, we provide the neural network solutions for the FPL equation under the various types of initial conditions, and interaction models in two and three dimensions. Furthermore, based on the theoretical properties of the FPL equation, we show that the approximated neural network solution converges to the a priori classical solution of the FPL equation as the pre-defined loss function is reduced.