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

## TL;DR

This paper introduces a neural network-based method to speed up simulations of plasma behavior by efficiently decomposing data.

## Contribution

A data-driven neural network approach for low-rank matrix factorization is proposed, offering faster computation than traditional methods.

## Key findings

- The method achieves comparable accuracy to standard techniques for interpolation tasks.
- The model generalizes to unseen data but fails to extrapolate future states effectively.
- The technique is well-suited for simulations with temporal stability but not for time-evolving systems.

## 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 achieves comparable reconstruction accuracy for interpolation tasks, 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. These results suggest that while the model offers a computationally efficient alternative for datasets with temporal stability, its current formulation is best suited for interpolation rather than for predicting future states in time-evolving systems. This study thus lays the groundwork for further refinement of neural network-based approaches to low-rank matrix factorization in high-dimensional plasma simulations.