A Deep Learning Method for Nonlinear Stochastic Filtering: Energy-Based Deep Splitting for Fast and Accurate Estimation of Filtering Densities

Abstract

In filtering the problem is to find the conditional distribution of a dynamically evolving state given noisy measurements. Critically, designing accurate filters for nonlinear problems that scale well with the state dimension is exceedingly difficult. In this thesis, a novel filtering method based on deep learning solutions to the Fokker–Planck partial differential equation is treated. Training can be performed offline, which results in a computationally efficient algorithm online, even in high dimensions. This is promising for applications which require good real-time performance, such as target-tracking. The filtering method, referred to as Energy-Based Deep Splitting (EBDS), is presented in detail and implemented. The performance of EBDS on different example problems is then investigated and compared to benchmark filters, such as variants of the Kalman filter and particle filters. In one dimension EBDS seems to perform superbly, especially considering how fast the filter is at evaluation. In higher dimensions the method performs worse in comparison to the benchmarks, although it still yields sensible density estimates in most cases. Additionally, convergence for EBDS in the number of prediction steps is investigated empirically for two of the example problems. The results in both examples indicate strong convergence of order 1/2. Lastly, a neural network architecture based on Long Short-Term Memory (LSTM) encoders is proposed for EBDS. This architecture yields reduced errors compared to standard fully-connected networks. In summary, the results indicate that the method is promising and should be examined further. This thesis can be viewed as a reference for future works that aim to apply EBDS in more specific settings or that aim to improve the method further.