Scientific Machine Learning (SciML) is a relatively new research field bridging machine learning (ML) and scientific computing. Its aim is the development of new methods to solve several kinds of problems, which can be forward solution of PDEs, identification of parameters, or inverse problems. The methods that are investigated in this context must be robust, scalable, reliable and interpretable. Two main families of methods can be distinguished. On one hand, methods that approximate the solution function, i.e., the mapping from instances of the function variables to the function values, such as with Physics-Informed Neural Networks (PINNS) and their numerous variants. On the other hand, methods that approximate the solution operator, which are generally classified as Neural Operators (NOs). Each of these two families has advantages and drawbacks when one is willing to consider complex PDE models of realistic physical problems. NOs require data, and when that is limited or not available, they are unable to learn the solution operator faithfully. PINNs do not require data but are prone to failure, especially on multi-scale dynamic systems due to optimization challenges. In this postdpctoral project, we will focus on NOs in the context of time-harmonic electromagnetics wave propagation in heterogenous domains involving irregularly-shaped geometrical features. The overarching goal will be to design NOs that can efficiently deal with the system of time-harmonic Maxwell equations for the complex-valued electric and magnetic fields with different types of boundary conditions and source terms in two- and three-dimensional settings, and data from unstructured mesh-based FEM (Finite Element Method) simulators. In addiiton, these NOs shall ultimately be capable of generalization over different geometrical characteristics of scattering structures to serve as fast surrogates in inverse design strategies for finding optimal scatterer shapes driven by a performance objective.
The work will start by a detailed bibliographical review of existing operator learning methods including DeepONet, FNO (Fourier NO), PINO, etc. by addressing their viability in relation to the physical problems considered in high-frequency electromagnetism. Initial developments and assessment activities will be performed in a two-dimensional setting and considering variaous problems of increasing complexity. Further invesigations in a three-dimensional setting will be realized for the most promising approaches.
This postdoctoral project is proposed in the contexte of the collaborative projet MAXINET between Inria and Thales Research & Technology.
This position requires French or EU citizenship.
