Adjoint-based Gradient Estimation Using the Space-time Solutions of Unknown Conservation Law Simulations

Many control applications can be formulated as optimization constrained by conservation laws. Such optimization can be efficiently solved by gradient-based methods, where the gradient is obtained through the adjoint method. Traditionally, the adjoint method has not been able to be implemented in "gray-box" conservation law simulations. In gray-box simulations, the analytical and numerical form of the conservation law is unknown, but the space-time solution of relevant flow quantities is available. Without the adjoint gradient, optimization can be challenging for problems with many control variables. However, much information about the gray-box simulation is contained in its space-time solution, which motivates us to estimate the adjoint gradient by leveraging the space-time solution. This article considers a type of gray-box simulations where the flux function is partially unknown. A method is introduced to estimate the adjoint gradient at a cost independent of the number of control variables. The method firstly infers a conservation law, named the twin model, from the space-time solution, and then applies the adjoint method to the inferred twin model to estimate the gradient. The method is demonstrated to achieve good gradient estimation accuracies in several numerical examples. The main contributions of this paper are: a twin model method that enables the adjoint gradient computation for gray-box conservation law simulations; and an adaptive basis construction scheme that fully exploits the information of gray-box solutions.