Evaluation of data driven low-rank matrix factorization for accelerated solutions of the Vlasov equation

TL;DR

This paper introduces a neural network-based low-rank matrix factorization method to accelerate Vlasov equation simulations, demonstrating faster inference and effective interpolation of plasma distribution data, with limitations in extrapolation.

Contribution

The paper presents a novel data-driven neural network approach for low-rank matrix factorization that improves computational efficiency in plasma simulations.

Findings

Faster inference than traditional linear algebra methods.

Effective interpolation of unseen time-series data.

Limited ability to extrapolate beyond trained data.

Abstract

Low-rank methods have shown success in accelerating simulations of a collisionless plasma described by the Vlasov equation, but still rely on computationally costly linear algebra every time step. We propose a data-driven factorization method using artificial neural networks, specifically with convolutional layer architecture, that trains on existing simulation data. At inference time, the model outputs a low-rank decomposition of the distribution field of the charged particles, and we demonstrate that this step is faster than the standard linear algebra technique. Numerical experiments show that the method effectively interpolates time-series data, generalizing to unseen test data in a manner beyond just memorizing training data; patterns in factorization also inherently followed the same numerical trend as those within algebraic methods (e.g., truncated singular-value decomposition). However, when training on the first 70% of a time-series data and testing on the remaining 30%, the method fails to meaningfully extrapolate. Despite this limiting result, the technique may have benefits for simulations in a statistical steady-state or otherwise showing temporal stability.