specPINN: Physics-Informed Neural Networks with Fast Spectral Collision Evaluation for the Fokker–Planck–Landau Equation

This repository presents specPINN, a physics-informed neural network (PINN) framework for solving the Fokker–Planck–Landau (FPL) equation by directly evaluating the collision operator using the fast spectral method, without introducing any surrogate model for the collision term.

Recent studies have proposed deep-learning-based solvers for the FPL equation. In particular, opPINN [1] introduces a two-stage strategy in which two neural networks (or neural operators) are trained to approximate the linear operators composing the FPL collision operator (referred to as Step 1 in the original paper).

In contrast, specPINN bypasses this surrogate-modeling step entirely. Instead of learning an approximation of the collision operator, we employ the fast spectral method [2], a well-established quasi-linear numerical approach, to compute the collision term of the distribution function on-the-fly during PINN training.

Advantages

Replacing Step 1 in opPINN with a numerical evaluation of the collision operator provides the following advantages:

1. Reduced training cost
   Since no surrogate model for the collision operator is trained, the overall training time of the PINN can be significantly reduced.

2. Improved generalization with respect to initial conditions
   The PINN can be trained for initial distributions that do not belong to, or are far from, the training set used for a surrogate collision model.

3. Potential computational efficiency
   The fast spectral method computes the collision term with $O(N^d \log N)$ computational complexity and $O(d^2 N^d)$ memory usage. Depending on the architecture and cost of the surrogate model, this direct evaluation may lead to faster or more stable training.

Experimental results

The following figure shows the relative $L^2_v$ error of the predicted solution over time. For the case of soft potential, the neural networks trained with the specPINN framework outperform the neural networks trained with the opPINN framework.

References

[1] Jae Yong Lee, Juhi Jang, Hyung Ju Hwang,
opPINN: Physics-informed neural network with operator learning to approximate solutions to the Fokker–Planck–Landau equation, Journal of Computational Physics, Volume 480, 2023, 112031.

[2] L. Pareschi, G. Russo, G. Toscani,
Fast Spectral Methods for the Fokker–Planck–Landau Collision Operator, Journal of Computational Physics, Volume 165, Issue 1, 2000, Pages 216–236.