Invited speakers
Margarita Chasapi (RWTH Aachen) : “Projection-based and data-driven methods in model order reduction”
Emmanuel Franck (Inria Center at Université de Lorraine): “From SciML approaches to hybrid methods”
Physics-informed Neural Networks (PINNs), Neural Operators (NOs) and Reduced Order Models (ROMs) are among the dominant approaches in learning for PDEs (SciML). This talk begins with classical numerical methods to show how these approaches can be naturally linked to them, as generalisations or reformulations. We will discuss their strengths in terms of expressivity and flexibility, as well as their limitations, which are often linked to a lack of theoretical guarantees and numerical control. In the second part, we will introduce hybrid methods, which fit within this SciML framework whilst seeking a compromise between expressivity and guarantees, by combining numerical structure and learning.
Filippo Gatti (CentraleSupelec, Paris): “Generative strategies to empower physics-based simulations with deep learning. Applications to earthquake engineering and computational fluid dynamics”
In this work, we provide a quantitative assessment of how largely physics-based simulation can benefit from deep-learning generative techniques. Two main frameworks are addressed: conditional generative approaches and neural operators. On one hand, diffusion models are employed to overcome the high-frequency spectral bias that affects both numerical simulations and their neural-operator surrogates. On the other hand, the successful use of neural operators to entirely replace cumbersome 3D elastic wave propagation numerical simulations is described, alongside an active-learning algorithm to improve their performance, showing how this approach can pave the way to real-time, large-scale digital twins of earthquake-prone regions. Beyond seismology, the same generative strategies extend naturally to other physics-based simulation domains. Conditional, score-based diffusion models—regularized by physical constraints such as turbulent-energy statistics or the divergence-free (Leray) spectral projector, and stabilized through autoregressive conditioning—yield stable, physically faithful forecasts of incompressible and transonic turbulent flows, including roll-out and out-of-distribution Kolmogorov-flow regimes. This confirms the generality of blending physics-based numerical simulation with generative deep learning across computational mechanics.
Siddhartha Mishra (ETH Zurich): “AI for Data-driven simulations in Physics”
Partial Differential Equations (PDEs) are often described as the language of Physics as they describe a wide array of physical phenomena over a vast range of scales. Despite their remarkable success over many decades, numerical methods for approximating PDEs can incur a very high computational cost. This limitation has provided the impetus for the design of fast and accurate Machine Learning/AI based neural PDE surrogates which can learn the PDE solution operator from data. In this talk, we review some latest developments in the field of Neural Operators, which are widely used as an ML paradigm for PDEs and discuss state of the art neural operators based on convolutions or attention. We will discuss graph and transformer based architectures for PDEs on arbitrary domains and conditional Diffusion models for PDEs with chaotic multiscale solutions. Finally, the issue of sample complexity is addressed by the design of general purpose Foundation models for PDEs.
Olivier Roustant (INSA, Toulouse): “Multifidelity Gaussian process regression for solving nonlinear partial differential equations”
Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work, as the available information is inherently multifidelity, we propose a kernel learning approach based on cokriging, leveraging empirical information from multifidelity simulations. In the first step, we fit a differentiable non-stationary kernel to an empirical kernel obtained from low-fidelity simulations. In the second step, we derive a high-fidelity kernel with estimated hyperparameters, and construct a corresponding high-fidelity mean using the multifidelity framework. These components can then be used within a Gaussian process framework for solving PDEs. Finally, we demonstrate the performance of the proposed physics-informed method on the Burgers’ equation. This is a joint work with Fatima-Zahrae El-Boukkouri and Josselin Garnier.
Lorenzo Sala (INRAE, Université Paris – Saclay): TBA