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

arXiv: https://arxiv.org/abs/2501.04024 · HTML: https://mlsanity.com/paper/2501.04024/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 15 references that are in the map.
- First posted on arXiv 27 Dec 2024.

## Builds on (most cited first, 7 of 7)

- [Think Globally, Act Locally: A Deep Neural Network Approach to High-Dimensional Time Series Forecasting](https://mlsanity.com/paper/1905.03806/) (2019)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Neural Network Matrix Factorization](https://mlsanity.com/paper/1511.06443/) (2015)
- [Spectral Learning on Matrices and Tensors](https://mlsanity.com/paper/2004.07984/) (2020)
- [Deep matrix factorizations](https://mlsanity.com/paper/2010.00380/) (2020)
- [Denise: Deep Robust Principal Component Analysis for Positive Semidefinite Matrices](https://mlsanity.com/paper/2004.13612/) (2020)
- [Time-Series Analysis via Low-Rank Matrix Factorization Applied to Infant-Sleep Data](https://mlsanity.com/paper/1904.04780/) (2019)

## Abstract (arXiv)

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.