Paid internship F/H Master 2, Adaptive sampling by optimal transport for PINNs in parametric problems

The Makutu project team specializes in large-scale simulations applied to the reconstruction of complex media, used to gain a better understanding of the internal dynamics of environments that are difficult or even impossible to probe. To this end, it develops advanced numerical methods that are integrated into open-source platforms deployed on state-of-the-art HPC environments.

In the framework of this Master internship, Makutu team proposes to explore a new research direction in the use of statistical numerical methods for solving PDEs. Among them, Physics-Informed Neural Networks (PINNs) have recently emerged as a promising approach for solving partial differential equations (PDEs). Their effectiveness, however, critically depends on the choice of collocation points. A uniform sampling strategy, while simple to implement, can become suboptimal when the solution exhibits locally complex features, for example, boundary layers, singularities, or regions of strong variation.

Classical adaptive methods, based on local error or residual estimation, allow the sampling density to be increased where the problem is more challenging. However, they often produce highly irregular point distributions, leading to clusters of points that can degrade both the stability of the training and the quality of the approximation.

The objective of the internship is to develop a regular adaptive sampling strategy, combining the advantages of a uniform grid with those of local adaptation. The central idea is to leverage optimal transport techniques to define a regular flow that maps a uniform point distribution to one adapted to the local complexity of the problem.

The main challenge lies in ensuring the regularity of the transport: the goal is not merely to move points, but to construct a smooth, bijective, and well-conditioned flow that avoids the formation of clusters or voids.

The internship will explore invertible neural methods designed to explicitly construct a differentiable transport map connecting a reference distribution (e.g. uniform) to a distribution of collocation points adapted to the problem.

One initial approach is to use normalizing flows, which offer a flexible framework for density estimation and allow to

• Directly learn the sampling distribution by maximizing likelihood or minimizing a suitable divergence (e.g., Kullback–Leibler, Wasserstein).
• Introduce Jacobian regularization terms to ensure a smooth and stable flow.

A second approach is inspired by Monge–Ampère's formulation: quadratic optimal transport can be written as the gradient of a convex potential, which can be parameterized using convex neural networks (ICNN). We can also explore continuous architectures such as Neural ODE flows, which define transport as the flow of a learned vector field and allow us to directly control the temporal and spatial regularity of the mapping.

In all cases, the goal is to construct a regular flow between uniform and adapted point sets, by introducing appropriate regularization criteria on the Jacobian and the transported density, tailored to parametric problems.

The internship will be conducted entirely in JAX, taking advantage of its fully differentiable and high-performance environment for scientific computing.

Test cases will involve parametric elliptic and wave equations, allowing systematic comparison between:

• uniform sampling,
• heuristic methods based on the residual,
• and the proposed regular optimal-transport-based strategy.

The resulting approach may be coupled with classical numerical solvers to obtain solutions that are both more accurate and computationally efficient. Depending on the outcomes, the internship may lead to a PhD project focusing on the coupling of learning-based and adaptive numerical methods.

Pilotage/Management :
La personne recrutée aura la responsabilité de ****.

Compétences techniques et niveau requis : the internship is intended for a Master 2 student with:

• strong skills in Python and numerical methods,
• a keen interest in algorithm implementation and scientific programming,
• some background in machine learning is a plus, but not mandatory.

Langues : French and/or english

Compétences relationnelles : teamwork is essential, autonomy is welcome

• Subsidized meals
• Partial reimbursement of public transport costs
• Possibility of teleworking and flexible organization of working hours
• Professional equipment available (videoconferencing, loan of computer equipment, etc.)
• Social, cultural and sports events and activities

Depending on the amount of the gratuity in effect.