We consider a class of infinite-dimensional dynamical systems driven by non-linear parabolic partial differential equations with initial condition $\theta$ modelled by a Gaussian process `prior' probability measure. Given discrete samples of the state of the system evolving in space-time, one obtains updated `posterior' measures on a function space containing all possible trajectories. We give a general set of conditions under which these non-Gaussian posterior distributions are approximated, in Wasserstein distance for the supremum-norm metric, by the law of a Gaussian random function. We demonstrate the applicability of our results to periodic non-linear reaction diffusion equations \begin{align*} \frac{\partial}{\partial t} u - \Delta u &= f(u) \\ u(0) &= \theta \end{align*} where $f$ is any smooth and compactly supported reaction function. In this case the limiting Gaussian measure can be characterised as the solution of a time-dependent Schr\"odinger equation with `rough' Gaussian initial conditions whose covariance operator we describe.

翻译：我们考虑一类由非线性抛物型偏微分方程驱动的无穷维动力系统，其初始条件 $\theta$ 由高斯过程“先验”概率测度建模。给定系统在时空中演化的状态离散样本，可在包含所有可能轨迹的函数空间上获得更新的“后验”测度。我们给出了一组通用条件，在这些条件下，这些非高斯后验分布可由高斯随机函数的律，在关于上确界范数度量的 Wasserstein 距离意义下近似。我们证明了我们的结果适用于周期性非线性反应扩散方程 \begin{align*} \frac{\partial}{\partial t} u - \Delta u &= f(u) \\ u(0) &= \theta \end{align*} 其中 $f$ 是任意光滑且具有紧支撑的反应函数。在此情况下，极限高斯测度可表征为一个具有“粗糙”高斯初始条件的含时薛定谔方程的解，我们描述了其协方差算子。