A long-sought "Holy Grail" in the field of rechargeable batteries is the development of stable and effective metal electrodes. Metal electrodes allow for the investigation of new, environmentally friendly electrochemical species in addition to offering greater energy density, affordability, and sustainability when compared to traditional electrode materials. The intrinsic multiscale and multi-physics complexity of metal plating and stripping phenomena means that, despite significant advancements in recent years, a fundamental understanding of metal electrode behaviour is still elusive. Our goal is to expedite the study of metal electrode systems by combining sophisticated statistical inference methods with multiscale and multi-physics simulations. We specifically create and solve numerically a system of coupled partial differential equations (PDEs) that represent the interaction of mechanical, morphological, and electrochemical processes during metal deposition. Through simulation-based inference and differentiable programming, we infer surface morphologies and their evolution from indirect experimental observables. This framework not only provides predictive insights for optimizing metal plating and electrode stability but also establishes a transferable methodology for addressing a broader class of multiscale, multi-physics problems in materials science and energy storage.