For the 2D incompressible Navier-Stokes equations, with given hypothetical non smooth data at time $T > 0 $that may not correspond to an actual solution at time $T$, a previously developed stabilized backward marching explicit leapfrog finite difference scheme is applied to these data, to find initial values at time $t = 0$ that can evolve into useful approximations to the given data at time $T$. That may not always be possible. Similar data assimilation problems, involving other dissipative systems, are of considerable interest in the geophysical sciences, and are commonly solved using computationally intensive methods based on neural networks informed by machine learning. Successful solution of ill-posed time-reversed Navier-Stokes equations is limited by uncertainty estimates, based on logarithmic convexity, that place limits on the value of $T > 0$. In computational experiments involving satellite images of hurricanes and other meteorological phenomena, the present method is shown to produce successful solutions at values of $T > 0$, that are several orders of magnitude larger than would be expected, based on the best-known uncertainty estimates. However, unsuccessful examples are also given. The present self-contained paper outlines the stabilizing technique, based on applying a compensating smoothing operator at each time step, and stresses the important differences between data assimilation, and backward recovery, in ill-posed time reversed problems for dissipative equations. While theorems are stated without proof, the reader is referred to a previous paper, on Navier-Stokes backward recovery, where these proofs can be found.

翻译：针对二维不可压Navier-Stokes方程，在给定时刻$T > 0$处可能存在非光滑且未必对应真实解的假设数据时，应用先前发展的稳定反向推进显式蛙跳有限差分格式处理这些数据，以寻找在时刻$t = 0$处的初始值，使其能演化成对时刻$T$给定数据的有用近似。该目标并非总能实现。涉及其他耗散系统的类似数据同化问题在地球科学领域具有重要价值，通常采用基于机器学习神经网络的计算密集型方法求解。不适定时间反演Navier-Stokes方程的成功求解受限于基于对数凸性的不确定性估计，该估计对$T > 0$的取值设定了限制。在涉及飓风及其他气象现象卫星影像的计算实验中，本方法在$T > 0$的取值上取得了成功解，其数值比基于最知名不确定性估计的预期值高出数个数量级。然而，文中也给出了失败案例。本独立论文概述了基于每时间步施加补偿平滑算子的稳定化技术，并强调了数据同化与耗散方程不适定时间反问题中反向恢复的重要区别。虽然定理陈述未附证明，但读者可参阅先前关于Navier-Stokes反向恢复的论文以获取完整证明。