Hybrid numerical–learning methods for cosmological radiative transfer and reionization

Supervision

Emmanuel Franck (IRMA, Strasbourg)

Laboratory and team

IRMA, Strasbourg - Team MOCO 

Subject description

Radiative transfer is a key ingredient in models describing the formation of large-scale structures and the epoch of reionization, during which the first astrophysical sources progressively ionized the intergalactic medium. Accurate modeling of this phenomenon requires solving a kinetic equation posed in a high-dimensional space, depending on position, propagation direction, and time. In current astrophysical codes, this complexity is usually reduced by using moment models, in particular the M1 closure, which provide a good compromise between computational cost and accuracy in isotropic regimes involving isolated sources. However, recent work has highlighted the fundamental limitations of these approaches whenever the radiation field exhibits complex angular structures, such as in the presence of shadows cast by dense structures or when ionization fronts from nearby sources meet each other or interact with a photon background. These situations are common in reionization simulations and require angular descriptions richer than those accessible through classical closures. In parallel, several exploratory studies have shown that scientific learning approaches, and in particular physics-informed neural networks (PINNs), are capable of efficiently representing solutions of high-dimensional partial differential equations while capturing fine-scale structures that are difficult to approximate with traditional methods. However, a lack of guarantees and error control limits the applicability of these approaches.

We therefore propose to consider hybrid approaches combining numerical discretizations and neural models, in order to overcome the current limitations of moment models while maintaining a computational cost compatible with large-scale simulations. The objective of this thesis is to develop a numerical and algorithmic framework for cosmological radiative transfer based on such hybridization. A first step will consist in designing a semi-Lagrangian scheme for transport in phase space using a hybrid representation. The spatio-angular domain will be decomposed into macro-cells in which the solution is approximated by small neural networks. The macro-cells will be coupled using a discontinuous Galerkin approach. This strategy should preserve the locality, conservation, and parallelization properties of DG methods while benefiting from the ability of neural networks to represent complex functions with a reduced number of degrees of freedom. In a second step, the thesis will introduce a micro–macro formulation of radiative transfer. The idea will be to describe the global dynamics using a robust macroscopic model of M1 type, responsible for capturing near-equilibrium regimes, while deviations from this equilibrium, which carry fine angular information, will be represented by a DG–PINN approximation. Such a decomposition should make it possible to concentrate the most expensive computations only in regions where they are needed. Finally, a central perspective of the project will be to dynamically learn equilibrium distributions, which amounts to learning models that generalize M1, becoming increasingly rich in order to progressively reduce the microscopic part of the solution. These M1-type models, which rely on only a reduced number of angular moments, should further reduce computational cost and will be introduced as new closures for the community.

The thesis lies at the interface between scientific computing, numerical astrophysics, and machine learning for partial differential equations. It will combine numerical analysis, scheme design, scientific learning, and high-performance implementation, with the final goal of integrating these new methods into representative configurations of reionization simulations and, in particular, inverse problems. 

References:

C. D. Levermore, Relating Eddington factors to flux limiters, Journal of Quantitative Spectroscopy and Radiative Transfer (1984). 
J. Rosdahl, J. Blaizot, Dominique Aubert, Timothy Stranex, Romain Teyssier, RAMSES-RT: radiation hydrodynamics in the RAMSES code.
M. Palanque, P. Ocvirk, E. Franck, P. Gerhard, D. Aubert, O. Marchalt, Higher order methods for Radiative Transfer in Astrophysical simulations: Pn vs M1, arXiv preprint 2025.
M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, JCP 2019.

Related mathematical skills

Numerical analysis
Learning
Python

Other public funding

Candidates recruited as PhDs will benefit from IRMIA++ funding and will have to follow the Graduate Program "Mathematics and Applications: Research and Interactions" (https://irmiapp.unistra.fr/training/presentation).

IRMIA++ is one of the 15 Interdisciplinary Thematic Institute (ITI) of the University of Strasbourg. It brings together a research cluster and a master-doctorate training program, relying on 12 research teams and 9 master tracks.

It encompasses all mathematicians at Université de Strasbourg, with partners in computer science and physics. ITI IRMIA++ builds on the internationally renowned research in mathematics in Strasbourg, and its well-established links with the socio-economic environment. It promotes interdisciplinary academic collaborations and industrial partnerships.

A core part of the IRMIA++ mission is to realize high-level training through integrated master-PhD tracks over 5 years, with common actions fostering an interdisciplinary culture, such as joint projects, new courses and workshops around mathematics and its interactions.

Selection will rely on the professional project of the candidate, his/her interest for interdisciplinarity and academic results.