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

Sammanfattning

In filtering the problem is to find the conditional distribution of a dynamicallyevolving state given noisy measurements. Critically, designing accurate filters fornonlinear problems that scale well with the state dimension is exceedingly difficult.In this thesis, a novel filtering method based on deep learning solutions to theFokker–Planck partial differential equation is treated. Training can be performedoffline, which results in a computationally efficient algorithm online, even in highdimensions. 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 presentedin detail and implemented. The performance of EBDS on different exampleproblems is then investigated and compared to benchmark filters, such as variantsof the Kalman filter and particle filters. In one dimension EBDS seems to performsuperbly, especially considering how fast the filter is at evaluation. In higher dimensionsthe method performs worse in comparison to the benchmarks, although it stillyields sensible density estimates in most cases. Additionally, convergence for EBDSin the number of prediction steps is investigated empirically for two of the exampleproblems. 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 tostandard fully-connected networks.In summary, the results indicate that the method is promising and should be examinedfurther. This thesis can be viewed as a reference for future works that aim toapply EBDS in more specific settings or that aim to improve the method further.