Reports and More Stuff

Here you can find a collection of some the stuff I’ve been working on lately (and a long time ago as well).


2022

PINNs and GaLS: An Priori Error Estimates for Shallow Physically Informed Neural Network Applied to Elliptic
Problems
Recently Physically Informed Neural Networks have gained more and more popularity to solve partial differential equations, given the fact they escape the course of dimensionality. First Physically Informed Neural Networks are viewed as an underdetermined point matching collocation method then we expose the connection between Galerkin Least Square (GALS)
and PINNs, to develop an a priori error estimate, in the context of elliptic problems. In particular, techniques that belong to the realm of the least square finite elements and Rademacher complexity analysis will be used to obtain the above-mentioned error estimate.

Arxiv


2021

NumSA – KAUST
All the stuff I developed during my MS can be found in this repo. This includes the following stuff,

  1. SLEPc wrap for NGSolve,
  2. An example of Federated Newton Learn (FEDNL) implementation,
  3. PINNs using TensorFlow.

GitHub Docs

A Priori Error Estimates For Shallow Physically Informed Neural Networks In One Dimension In this report, a connection between LSQFEM and the PINNs is drawn, furthermore we exploit such a connection in order to develop a robust error estimate for PINNs using techniques developed in the context of the FNM. I also prepared a beamer containing more general ideas on the topics of using FNN to solve PDE.

Report Beamer


2020

Python Course – Collegio Ghislieri
With a friend we taught a course on Python coding at Collegio Ghislieri in Pavia during the summer term of 2020, you can find the material of the course here.

Sound Soft Scattering Problem – BEM During a course at UniPV we studied how to use boundary elements methods (BEM) to solve the sound soft scattering problem. I decided to implement this using the FE toolbox NGSOLVE/NETGEN. Full example can be found here.

Poisson Problem – IGA During a course at UniPV we had a look at isogeometrical analysis methods for solving PDE. In particular I coded a solver for the Poisson problem using isogemoetrical analysis in Julia. Full code can be found in this repository, for example on how to use the library take look at this notebook.