  Many engineering applications can be formulated as optimizations constrained
by conservation laws. Such optimizations can be efficiently solved by the
adjoint method, which computes the gradient of the objective to the design
variables. Traditionally, the adjoint method has not been able to be
implemented in many "gray-box" conservation law simulators. In gray-box
simulators, the analytical and numerical form of the conservation law is
unknown, but the full solution of relevant flow quantities is available. In
this paper, we consider the case where the flux function is unknown. This
article introduces a method to estimate the gradient by inferring the flux
function from the solution, and then solving the adjoint equation of the
inferred conservation law. This method is demonstrated in the sensitivity
analysis of two flow problems.
