A theoretical analysis on the inversion of matrices via Neural Networks designed with Strassen algorithm

TL;DR

This paper presents a neural network approach for matrix inversion using the Strassen algorithm, enabling efficient approximation of inverse matrices and application to parametric PDEs, with theoretical analysis and improvements over prior methods.

Contribution

It introduces a neural network architecture based on the Strassen algorithm for efficient matrix inversion and applies it to parametric PDEs, providing a theoretical foundation.

Findings

Neural network can approximate matrix and scalar multiplication functions.

Using Strassen algorithm reduces network complexity for matrix operations.

Neural networks can solve parametric elliptic PDEs efficiently.

Abstract

We construct a Neural Network that approximates the matrix multiplication operator for any activation function such that there exists a Neural Network which can approximate the scalar multiplication function. In particular, we use the Strassen algorithm to reduce the number of weights and layers needed for such Neural Networks. This allows us to define another Neural Network for approximating the inverse matrix operator. Also, by relying on the Galerkin method, we apply those Neural Networks to solve parametric elliptic PDEs for a whole set of parameters. Finally, we discuss improvements with respect to the prior results.