This paper considers the filtering problem which consists in reconstructing the state of a dynamical system with partial observations coming from sensor measurements, and the knowledge that the dynamics are governed by a physical PDE model with unknown parameters. We present a filtering algorithm where the reconstruction of the dynamics is done with neural network approximations whose weights are dynamically updated using observational data. In addition to the estimate of the state, we also obtain time-dependent parameter estimations of the PDE parameters governing the observed evolution. We illustrate the behavior of the method in a one-dimensional KdV equation involving the transport of solutions with local support. Our numerical investigation reveals the importance of the location and number of the observations. In particular, it suggests to consider dynamical sensor placement.

翻译：本文考虑滤波问题，该问题旨在通过传感器测量提供的部分观测数据，以及已知动力学由具有未知参数的物理偏微分方程模型支配的知识，来重构动力系统的状态。我们提出了一种滤波算法，在该算法中，动力学的重构通过神经网络逼近实现，其权重利用观测数据动态更新。除了状态估计外，我们还获得了控制观测演化的偏微分方程参数的时变估计。我们通过一维KdV方程（涉及局部支撑解的输运）展示了该方法的行为。数值研究揭示了观测位置和数量的重要性，特别地，该方法建议考虑动态传感器布置。