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.

翻译：大多数物理科学中的逆问题都表述为偏微分方程约束的优化问题。这需要通过在优化过程中生成与测量数据高度吻合的偏微分方程解，来识别方程中的未知参数。该公式具有强大功能，被广泛应用于众多科学与工程领域。然而，一个关键假设是未知参数必须是确定性的。但在现实中，许多问题本质上具有随机性，且未知参数是随机的。此时挑战变为恢复该未知随机参数的全部概率分布——这是一项更为复杂的任务。本文在通用框架下研究该问题，特别地，我们将偏微分方程求解器概念化为推动参数分布生成数据分布的前推映射。通过这种方式，随机微分方程约束的优化转化为最小化生成分布与测量分布之间的距离。随后我们构建了梯度流方程来求解真实参数概率分布。这为将偏微分方程约束优化中的多项技术拓展至含随机性的系统开辟了新范式。