## Events

**past events**and here for a presentation of the Rome Centre on Mathematics for Modelling and Data ScienceS.

If you want to receive updates on RoMaDS activities, write an email to salvi(at)mat.uniroma2.it to be added to our

**mailing list**.

**Upcoming events**

**22.05.2024 - Seminar: Dario Trevisan (Università di Pisa)**, Gaussian Approximation and Bayesian Posterior Distribution in Random Deep Neural Networks

14h30-15h30, Department of Mathematics, Aula Dal Passo.

Link to Teams streaming.

## Abstract

We establish novel rates for the Gaussian approximation of randomly initialized deep neural networks with Gaussian parameters and Lipschitz activation functions, in the so-called wide limit, i.e., where the sizes of all hidden layers become large. Using the Wasserstein metric and related functional analytic tools, we demonstrate the distribution of the output of a network and the corresponding Gaussian approximation are at a distance that scales inversely with the width of the network, surpassing previously established rates.

Furthermore, we extend our findings to approximate the exact Bayesian posterior distribution of the network when the likelihood is a bounded Lipschitz function of the network output, on a finite training set. This includes common cases, such as the Gaussian likelihood, which is defined as the exponential of the negative mean squared error. Our inequalities thus shed light on the network's Gaussian behavior by quantitatively capturing the distributional convergence results in the wide limit.

The exposition will aim to be self-contained, by introducing all the basic concepts related to artificial neural networks and Bayesian statistics to a mathematical audience. Based on arXiv:2203.07379 (joint with A. Basteri) and arXiv:2312.11737.

**17.06.2024 - Seminar: Tan Bui-Thanh (University of Texas at Austin)**, Learn2Solve: A Deep Learning Framework for Real-Time Solutions of Forward, Inverse, and UQ Problems

15h00-16h00, Department of Mathematics, Aula Dal Passo.

## Abstract

Digital models (DMs) are designed to be replicas of systems and processes. At the core of a digital model (DM) is a physical/mathematical model that captures the behavior of the real system across temporal and spatial scales. One of the key roles of DMs is enabling “what if” scenario testing of hypothetical simulations to understand the implications at any point throughout the life cycle of the process, to monitor the process, to calibrate parameters to match the actual process and to quantify the uncertainties. In this talk, we will present various (faster than) real-time Scientific Deep Learning (SciDL) approaches for forward, inverse, and UQ problems. Both theoretical and numerical results for various problems including transport, heat, Burgers, (transonic and supersonic) Euler, and Navier-Stokes equations will be presented.

**23-27.09.2024 - Summer school: High-Dimensional Approximation: From Theoretical Foundations to Machine Learning and PDEs**, Cetraro, Italy.

Application deadline: May 15. All further information available at https://www.mat.uniroma2.it/~speleers/cime2024/.

## Courses

Courses:

- Adaptivity in High-Dimensional Statistical Learning (Francis Bach)

- Numerics of High-Dimensional PDEs (Markus Bachmayr)

- High-Dimensional Approximation in Machine Learning (Gitta Kutyniok)

- High-Dimensional Integration (Dirk Nuyens)

- Compressive Sensing (Holger Rauhut)