MELISSAMEthodological contributions in statistical Learning InSpired by SurfAce engineering

The underlying dynamics of many physical problems are governed by parameterized partial differential equations (PDEs). Despite important scientific advances in numerical simulation, solving efficiently PDEs remains complex and often prohibitively expensive. Physics-informed Machine Learning (PiML) has recently emerged as a promising way to learn efficient surrogate solvers, and augment the physical laws by leveraging knowledge extracted from data. From a machine learning perspective, ignoring the fundamental principles of the underlying physics may lead to ill-posed problems and thus to implausible solutions yielding poor generalization.

Numerous algorithmic contributions in deep-learning have recently exploited domain knowledge for (i) designing suitable physics-regularized loss functions, (ii) initializing neural networks with meaningful parameters, (iii) guiding the design of consistent architectures, or (iv) building hybrid models.

Despite indisputable advances, PiML remains an emerging topic with several open problems that remain to be addressed: (i) Deriving generalization/approximation guarantees; (ii) Learning with a limited amount of data; (iii) Augmenting partially known physical laws; (v) Modeling uncertainty; (vi) Building foundation models for physics.

Developing suited solutions that tackle these interrelated challenges is crucial for the usability of PiML in realistic scenarios. This is the goal of MELISSA which gathers 3 teams (Inria MALICE, Inria MAGNET and MLIA) with a strong expertise at the interface of machine learning, optimization and physics. By conducting this project from both theoretical and algorithmic perspectives, the objective is to design the next generation of provably accurate PIML algorithms in the challenging context of laser-matter interaction where data is scarce and the available physical laws only partially explain the observed dynamics.