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

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.