  Most inverse problems from physical sciences are formulated as
PDE-constrained optimization problems. This involves identifying unknown
parameters in equations by optimizing the model to generate PDE solutions that
closely match measured data. The formulation is powerful and widely used in
many sciences and engineering fields. However, one crucial assumption is that
the unknown parameter must be deterministic. In reality, however, many problems
are stochastic in nature, and the unknown parameter is random. The challenge
then becomes recovering the full distribution of this unknown random parameter.
It is a much more complex task. In this paper, we examine this problem in a
general setting. In particular, we conceptualize the PDE solver as a
push-forward map that pushes the parameter distribution to the generated data
distribution. This way, the SDE-constrained optimization translates to
minimizing the distance between the generated distribution and the measurement
distribution. We then formulate a gradient-flow equation to seek the
ground-truth parameter probability distribution. This opens up a new paradigm
for extending many techniques in PDE-constrained optimization to that for
systems with stochasticity.
