Spatio-temporal dynamical systems often pose important challenges to machine learning applications: real-world observations are scarce, partial, and expensive to acquire. How can existing physical knowledge for those systems, usually formalized as Partial Differential Equations, be leveraged to improve both predictive and inverse problem solving, when measurements cover only a fraction of the spatio-temporal field, and when numerical solvers introduce systematic errors that accumulate over time? Conversely, how can measured data help improving the physics described by PDEs?

This seminar explores the interplay between physical models and learning-based methods, building on the solver-in-the-loop framework and its extensions. We present in particular an instantiation of the Deep Finite Difference Method (Deep FDM) on two-dimensional heterogeneous acoustic wave propagation. By pairing a fourth-order finite difference proxy solver with a pseudo-spectral reference, we cast numerical dispersion correction as an operator learning problem and systematically benchmark numerous architectures, from linear regression to Fourier Neural Operators. Results highlight the superiority of neural operators in capturing the residual non-linear relationship between the two solutions, and raise broader questions about the transferable aspect of this benchmark to the simulation-to-reality gap, as well as to other spatio-temporal dynamical systems.

Biography: Pascal Tribel is a doctoral student at the Machine Learning Group (ULB), under the supevision of Gianluca Bontempi. He holds a position as a teaching assistant in the Computer Science Department, for courses of Programming, Modelisation and Simulation, Artificial Intelligence and Machine Learning. His research focuses on applying machine learning techniques to forecasting and analysing temporal and spatio-temporal data that are traditionally described by partial differential equations. Additionally, Pascal Tribel is a pianist who graduated from the Royal Conservatory of Brussels.

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