A Deep Learning Method for Nonlinear Stochastic Filtering: Energy-Based Deep Splitting for Fast and Accurate Estimation of Filtering Densities
Typ
Examensarbete för masterexamen
Master's Thesis
Master's Thesis
Program
Engineering mathematics and computational science (MPENM), MSc
Publicerad
2024
Författare
Rydin, Filip
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
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.
Beskrivning
Ämne/nyckelord
Nonlinear filtering, Scalable filter, Deep learning, Kalman filter, Particle filter, Fokker–Planck equation, Neural networks, Long short-term memory