PhD position

Applications should be submitted online on the dedicated website. For more information, please contact Emmanuel Franck, researcher at the Inria Centre at Université de Lorraine : [email protected].


Launched in 2023 for a duration of 6 years, The NumPEx PEPR aims to contribute to the design and development of numerical methods and software components that will equip future European Exascale and post-Exascale machines. NumPEx also aims to support scientific and industrial applications in fully exploiting their potentials.

Exa-MA aims to revolutionize methods and algorithms for exascale scaling: discretization, resolution, learning and order reduction, inverse problem, optimization and uncertainties. We are contributing to the software stack of future European computers.

Main activities

In light of the significant successes achieved by deep learning methods in computer-aided vision or language processing, new learning-based methods have emerged for the simulation and resolution of PDEs. We can mention PINN methods which allow solving a PDE by replacing finite-element approx- imations with neural networks [WSWP23]-[SS22] or neural operators that approximate the inverse operator of the PDE and allow for quickly predicting the solution from the source. For example, in [GPZ+23]-[CZP+24], the authors use a neural operator to predict the dynamics of a plasma in a simplified configuration in a relatively short time. Many realistic applications such as plasma physics require dealing with complicated geometries and multi-scale phenomena over long times. The challenge of this thesis is therefore to try to push these neural network-based approaches to a higher level for multi-scale problems. We would like to investigate approaches that maintain accuracy and stability over long times on general geometries.

A first approach will be to consider the Neural Galerkin method [BPVE24] which maintains an ODE structure in time but approximates the spatial part as well as the parametric dependence of the PDE by a neural network. This method allows using the good properties in high dimensions of networks to reduce the number of degrees of freedom. We propose to couple this approach with recent approaches from PINNs to deal with general geometries. Secondly, we aim to study long-term stabil- ity, which is a critical problem by incorporating the structure of the equations [Sun19], using splitting schemes to preserve the structure, or combining the scheme with ”stabilization” methods [BP24]. One of the key points will be to determine robust neural network architectures.

The second approach will focus on neural operators. Early results have shown that this is a promising direction. However, long-term stability issues remain significant. We wish to explore several methods to improve long-term approximations [MHSB23]-[LVP+23] and extend them to multi-scale configurations. In addition to these general approaches, we can also study how to incorporate the structure of the physical problem into the architecture of the operators. The obtained approaches will be coupled with methods capable of dealing with general geometries such as [LKC+24]-[BET22] which use parameterized integral kernels in the physical domain.

Purely neural methods will remain limited in precision. For this reason, ultimately, we would like to couple them with more classical numerical approaches to obtain algorithms that are faster than traditional approaches and reliable. This type of coupling has already yielded very encouraging results [FMDN23].

In sum, this topic represents a stimulating opportunity to explore recent advances in the field of deep learning and numerical simulation. We propose a balanced approach that combines traditional methods and innovative techniques to solve complex problems in the physical sciences. In particular, we will validate the approaches Fluid and MHD PDE systems with turbulence and convective mixing, with potential applications to engineering and fusion. Students who are interested in the intellectual challenges and practical applications of computational modeling are encouraged to apply.

Required skills

  • A master’s degree in applied mathematics in PDE and numerical analysis or an Machine learning but with knowledge of PDE and numerical methods.
  • Good coding experience (preferably in Python) is required.
  • Good level in english is also recommanded
More information and references