Bi-Level optimization for interpolation-based parameter estimation of differential equations

Inverse problem or parameter estimation of ordinary differential equations (ODEs), the iterative process of minimizing the mismatch between model-predicted and experimental states by tuning the parameter values within an optimization formulation, is commonplace in chemical engineering applications. A popular method for parameter estimation is sequential optimization (single-shooting), which numerically integrates the ODE in each iteration. However, computing the gradients for the optimization steps requires calculating sensitivities, i.e., the derivatives of states with respect to the parameters, through the numerical integrator, which can be computationally expensive. In this work, we use interpolation to reduce the cost of these sensitivity calculations. Leveraging this interpolation, we also propose a bi-level optimization framework that exploits the structure of the differential equations and solves a convex inner problem. We apply this framework to examples spanning conventional parameter estimation and the emerging concept of data-driven dynamic model discovery. We show that our approach not only estimates the correct parameters for benchmark problems, but can also be readily extended to delay, stiff, and partially observed differential equations without major modifications.