Online calibration of deep learning sub-models for hybrid numerical modeling systems

Defining end-to-end (or online) training schemes for the calibration of neural sub-models in hybrid systems requires working with an optimization problem that involves the solver of the physical equations. Online learning methodologies thus require the numerical model to be differentiable, which is not the case for most modeling systems. To overcome this, we present an efficient and practical online learning approach for hybrid systems. The method, called EGA for Euler Gradient Approximation, assumes an additive neural correction to the physical model, and an explicit Euler approximation of the gradients. We demonstrate that the EGA converges to the exact gradients in the limit of infinitely small time steps. Numerical experiments show significant improvements over offline learning, highlighting the potential of end-to-end learning for hybrid modeling. End-to-end learning in hybrid numerical models involves solving an optimization problem that integrates the model’s solver. In many fields, these solvers are written in low-abstraction programming languages that lack automatic differentiation. This work presents a practical approach to solving the optimization problem by efficiently approximating the gradient of the end-to-end objective function.