Theoretical physics has traditionally relied on human intelligence to discover the laws of nature.  Artificial intelligence is beginning to challenge this paradigm, but still struggles to learn any physical “laws” valid far beyond the training dataset.  This talk presents a hybrid approach of solving PDE-constrained inverse problems to learn electrochemical physics directly from image data.  From x-ray images of lithium iron phosphate nanoparticles during battery cycling, we learn the free energy landscape of the material, the reaction kinetics of coupled ion-electron transfer, and the nanoscale profile of surface reactivity (correlated with carbon coating thickness). From optical images of graphite anodes during fast charging, we learn the dynamics of staging phase transformations and the conditions for parasitic lithium plating.  Incorporating this knowledge in multiphase porous electrode theory (MPET) enables predictive simulations of Li-ion batteries, which can be used to optimize of fast-charging protocols and formation cycling for extended battery lifetime.