Invertible Neural Networks for Reduced-Order Modeling of Hyperbolic PDEs

Invertible Neural Networks for Reduced-Order Modeling of Hyperbolic PDEs Level of qualifications required : Graduate degree or equivalent Fonction : Internship Research Assignment Reduced-order models (ROMs) aim to accelerate simulations of high-dimensional dynamical systems by projecting solutions onto low-dimensional spaces. Classical ROM methods, such as Proper Orthogonal Decomposition (POD), rely on linear projections of snapshots, which may require many modes to capture sharp gradients or traveling waves typical of hyperbolic PDEs. This internship focuses on using global or local invertible neural networks to learn nonlinear transformations of snapshots that accelerate eigenvalue decay, enabling more efficient low-dimensional representations. The approach is inspired by re-registration and morphing techniques but leverages globally-defined invertible networks in physical space to align features across snapshots. A significant part of the work will be on designing and training thesenetworks, understanding their representation capacity, and comparing them to potential local alternatives. The method will first be applied to the viscous Burgers equation, where benchmark solutions provide a clear expectation for network performance, before potentially extending to more complex hyperbolic systems such as the Saint-Venant equations. The intern will implement the invertible networks, integrate them into a ROM framework, and analyze the accuracy, efficiency, and eigenvalue decay compared to classical POD. This work bridges modern machine learning techniques with physics-informed model reduction for hyperbolic problems. Main activities SOTA on reduced order modeling for hyperbolic systems ROM with invertible analytic map for viscous Burgers equation ROm with invertible neural networks Comparaison with classical approaches and validation Skills Basics of PDEs and numerical methods, especially hyperbolic problems Understanding of reduced-order modeling (POD, ROM) Experience with Python and scientific libraries (NumPy, SciPy, PyTorch/JAX) Knowledge or interest in invertible neural networks / normalizing flows Ability to analyze snapshots, eigenvalue decay, and model accuracy Benefits package Partial reimbursement of public transport costs Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.) Possibility of teleworking (after 6 months of employment) and flexible organization of working hours Professional equipment available (videoconferencing, loan of computer equipment, etc.) Social, cultural and sports events and activities Theme/Domain :Numerical schemes and simulationsScientific computing(BAP E) Warning : you must enter your e-mail address in order to save your application to Inria. Applications must be submitted online on the Inria website. Processing of applications sent from other channels is not guaranteed. Instruction to apply Defence Security :This position is likely to be situated in a restricted area (ZRR), as defined in Decree No. 2011-1425 relating to the protection of national scientific and technical potential (PPST).Authorisation to enter an area is granted by the director of the unit, following a favourable Ministerial decision, as defined in the decree of 3 July 2012 relating to the PPST. An unfavourable Ministerial decision in respect of a position situated in a ZRR would result in the cancellation of the appointment. Recruitment Policy :As part of its diversity policy, all Inria positions are accessible to people with disabilities. Inria is the French national research institute dedicated to digital science and technology. It employs 2,600 people. Its 200 agile project teams, generally run jointly with academic partners, include more than 3,500 scientists and engineers working to meet the challenges of digital technology, often at the interface with other disciplines. The Institute also employs numerous talents in over forty different professions. 900 research support staff contribute to the preparation and development of scientific and entrepreneurial projects that have a worldwide impact. #J-18808-Ljbffr