Solver-in-the-loop: Schematic diagram showing how an integrated ML model can be used to correct a “low-resolution” simulation to recover the properties of a “high-resolution” simulation at significantly reduced computational cost. We visualize the evolution of the system as a path on the solution manifold ﹩{ \mathcal M }﹩, with each point representing a system state. The corrector network, ﹩{{ \mathcal B }}_{\theta }﹩, can be viewed as a geodesic flow on the manifold that pushes the low-resolution solution closer to either the reference high-resolution solution or the underlying truth (if suitable data are available). The augmentation network, ﹩{{ \mathcal B }}_{\theta }﹩ can be trained dynamically by running the low-resolution simulation to the end state, ﹩{\hat{U}}_{f}^{{\prime} }﹩ and then snapshots along the simulation can be compared with the reference simulation U_i. The derivatives of the loss can be used to update the parameters of the correction network, θ.