# Approximating Matrix Functions with Deep Neural Networks and Transformers

arXiv: https://arxiv.org/abs/2602.07800 · HTML: https://mlsanity.com/paper/2602.07800/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 22 references that are in the map.
- First posted on arXiv 8 Feb 2026.

## Builds on (most cited first, 8 of 8)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Linear algebra with transformers](https://mlsanity.com/paper/2112.01898/) (2021)
- [Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers](https://mlsanity.com/paper/2410.08304/) (2024)

## Abstract (arXiv)

Transformers have revolutionized natural language processing, but their use for numerical computation has received less attention. We study the approximation of matrix functions, which map scalar functions to matrices, using neural networks including transformers. We focus on functions mapping square matrices to square matrices of the same dimension. These types of matrix functions appear throughout scientific computing, e.g., the matrix exponential in continuous-time Markov chains and the matrix sign function in stability analysis of dynamical systems. In this paper, we make two contributions. First, we prove bounds on the width and depth of ReLU networks needed to approximate the matrix exponential to an arbitrary precision. Second, we show experimentally that a transformer encoder-decoder with suitable numerical encodings can approximate certain matrix functions at a relative error of 5% with high probability. Our study reveals that the encoding scheme strongly affects performance, with different schemes working better for different functions.