Prof. Bruno Després (University Paris 6) and Dr. Moreno Pintore (Inria - Paris 6)

Neural Networks from the viewpoint of Numerical Analysis

Abstract: The compositional structure of NN functions will be analyzed within a convenient functional framework inspired by the Murat-Trombetti Theorem.
Approximations properties will be reviewed, such as the Cybenko Theorem, the Yarotsky Theorem and some basic analytical formulas.
This material will be applied to the description of the DeepRitz method for the calculation of a numerical solution to the Poisson equation.
Two hand-on sessions will be dedicated to the coding of exact polynomials in Tensorflow and to the measurement of the Lipschitz constant of given NN functions.

Prof. Michael Herty (RWTH Aachen University) and Sara Veneruso (RWTH Aachen University)

Particle-based methods in high-dimensional optimization

Abstract: We will present recent advances in particle methods for optimization. Those include ensemble methods for filtering problems, consensus-based methods, and particle swarm optimization. We also discuss stochastic methods like stochastic gradient descent. Besides the presentation of the methods we will discuss theoretical properties like convergence and stability as well as recent advances and applications to new problems. Numerical results will also be presented.

Prof. George Karniadakis (Brown University) and Dr. Khemraj Shukla (Brown University)

Deep Learning for Scientists and Engineers

Abstract: This course introduces deep learning concepts and their application in tackling scientific and engineering challenges. It covers various methodologies, including the theory and implementation of Scientific Machine Learning techniques such as Physics-Informed Neural Networks (PINNs), Deep Operator Networks (DeepONet), and Fourier Neural Operators (FNO), to solve complex computational problems across fields like solid mechanics, fluid mechanics, material evaluation, systems biology, chemistry, and nonlinear dynamics. The curriculum includes topics on Neural Network Architectures, Optimizers, PINN, DeepONet, Uncertainty Quantification, and multi-GPU implementation of Scientific Machine Learning. In addition to lectures, we will demonstrate the practical implementation of these methods using the JAX/PyTorch framework in Python.

Prof. Remi Abgrall (University of Zurich)

Special lecture: Kinetic scheme for non linear hyperbolic and parabolic problems: application to compressible flows

Abstract: We present kinetic type methods able to approximate compressible type flow, with or without viscous and thermal effects. Many numerical example illustrate the methods and show effectiveness.
The work is strongly inspired from AbgrallTorlo2020, AbgrallNassajian2023, WissocqAbgrall2024a, WissocqAbgrall2024b, WissocqAbgrallLiu2025.