Engineering applications typically require a mathematical reduction of complex physical model to a more simplistic representation, unfortunately this simplification typically leads to a missing physics problem. In this work we introduce a state space solution to recovering the hidden physics by sharing information between different operating scenarios, referred to as tasks''. We introduce an approximation that ensures the resulting model scales linearly in the number of tasks, and provide theoretical guarantees that this solution will exist for sufficiently small time-steps. Finally we demonstrate how this framework may be used to improve the prediction of Lithium-ion concentration in electric batteries.