Approximating Matrix Functions with Deep Neural Networks and Transformers

Abstract: 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.