Inverse learning of material physics through in-situ image data and continuum modeling

With the availability of microscopic spatio-temporal image data of materials, there is a tremendous amount of hidden information about the material properties. Using a framework of PDE-constrained optimization, we demonstrate that multiple constitutive relations can be extracted simultaneously from a small set of images of pattern formation. Examples include state-dependent properties such as the diffusivity, kinetic prefactor, free energy, and direct correlation function in the Cahn-Hilliard equation, Allen-Cahn equation, or dynamical density functional theory (Phase-Field Crystal Model). Compared to the data-driven modeling approach and the recent work on PDE discovery, our approach provides clear physical interpretability by prescribing a general governing equation while achieving a high expressive power through nonlinear and/or nonlocal (integro-differential) constitutive relations.
We demonstrate that the inversion technique can be applied to experimental images of lithium-iron phosphate (LFP) particles. By mapping the evolution of lithium concentration in the particles during charge and discharge, we are able to extract its free energy and reaction kinetics, which are difficult to obtain through traditional electrochemical measurements alone due to phase separation.