Gonzalo Romera | Searcharxiv

arXiv2026

A theoretical analysis on the resolution of parametric PDEs via Neural Networks designed with Strassen algorithm

We construct a family of Neural Networks that approximate 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 bound 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. Finally, we discuss how it can be applied to numerically solve elliptic PDEs.

math.NA↗

arXiv2024

Neural Networks in Numerical Analysis and Approximation Theory

In this Master Thesis, we study the approximation capabilities of Neural Networks in the context of numerical resolution of elliptic PDEs and Approximation Theory. First of all, in Chapter 1, we introduce the mathematical definition of Neural Networks and perform some basic estimates on their composition and parallelization. Then, we implement in Chapter 2 the Galerkin method using Neural Network. In particular, we manage to build a Neural Network that approximates the inverse of positive-definite symmetric matrices, which allows to get a Garlerkin numerical solution of elliptic PDEs. Finally, in Chapter 3, we introduce the approximation space of Neural Networks, a space which consists of functions in $L^p$ that are approximated at a certain rate when increasing the number of weights of Neural Networks. We find the relation of this space with the Besov space: the smoother a function is, the faster it can be approximated with Neural Networks when increasing the number of weights.

math.NA↗