ENACT - Opérateurs neuronaux pour la modélisation des transformations de phase dans les matériaux

Keywords

Phase transformations, Phase-field models, Partial differential equations, Neural networks, Neural operators

Subject details

SCIENTIFIC BACKGROUND Phase transformations play a crucial role in the elaboration of metallic materials, as they determine the microstructure — and therefore the properties — of the material. They occur, for example, during additive manufacturing (3D printing) of a metal part: the molten metal first solidifies into a crystalline structure, after which further solid-state phase transformations occur, giving rise to complex microstructures (dendrites, precipitates) whose morphology and spatial arrangement directly govern the part's in-service properties and their evolution under service conditions. Monitoring these transformations experimentally is of critical importance, yet remains a major challenge. Current approaches fall into two categories. The most common relies on macroscopic signals sensitive to the microstructure: volume changes (dilatometry), electrical resistivity, thermoelectric power, or X-ray diffraction (conventional or synchrotron) analysed over the whole sample. The most sophisticated approach uses direct in situ imaging by electron or X-ray microscopy, providing spatially resolved information on the evolving microstructure, but is technically demanding and difficult to implement at scale. A highly desirable intermediate approach would consist in measuring local signals at the sample surface from which the full microstructure could be reconstructed. Before such a technique can be experimentally implemented, its theoretical feasibility must first be established using a model capable of reproducing the complexity of real microstructures — the phase-field model being the best candidate, as it can readily be coupled to the physical fields relevant to such measurements. Phase-field models describe these transformations through nonlinear reaction-diffusion PDEs coupled to external fields (temperature, chemical composition, stresses, liquid flow, etc.). Today, such PDEs are generally solved numerically with classical discretization methods (finite difference, finite element, Fourier spectral, etc.), and more recently with neural network-based techniques. The multi-scale and nonlinear nature of these models, however, requires fine spatio-temporal discretizations, leading to computation times prohibitive for scaling up to the process level or conducting extensive parametric studies. OBJECTIVES OF THE THESIS This thesis aims to implement 'observers' — a tool from control theory — for phase-field models. An observer incorporates partial experimental measurements into the model in order to reconstruct the complete phase transformation field. Their effectiveness is well established for linear PDEs and some results exist for nonlinear PDEs, but their application to phase-field models of the type under consideration remains unexplored. Observer design will be investigated through the backstepping method, which seeks an invertible mapping between the observer error and a target system under appropriate hypotheses — a framework recently established for linear PDEs and partially extended to nonlinear PDEs. The practical implementation for nonlinear PDEs, however, remains suboptimal as it relies heavily on the underlying linear dynamics. The goal is to overcome this limitation by learning the nonlinear backstepping transformation directly using Invertible Neural Networks (INN). Rather than linearising the problem, INNs approximate the invertible mapping between two nonlinear PDEs, leveraging their natural ability to handle nonlinearities and parametric PDEs, using the existing simulation code. Once this mapping is obtained, the observer will enable direct use of experimental surface measurements to reconstruct the full microstructural state, without resorting to numerous simulations to identify the corresponding phase transformation.