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：大多数物理学中的反问题都被表述为偏微分方程约束的优化问题。这类问题通过优化模型，使生成的偏微分方程解能精确匹配测量数据，从而辨识方程中的未知参数。这种表述方法强大且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。但实际上，许多问题本质上具有随机性，未知参数是随机的。此时，挑战就转变为恢复这个未知随机参数的完整分布，这是一项复杂得多的任务。在本文中，我们解析了这种问题在一般情况下的建模：特别地，我们还将偏微分方程求解器概念化为一个前推映射，它将参数分布映射到生成的数据分布上。这样，随机微分方程约束的优化就转化为最小化生成分布与测量分布之间的距离。然后，我们推导出一个梯度流方程来寻找真实参数概率分布。这为将偏微分方程约束优化中的许多技术扩展到随机系统开辟了一个新的范式。