Deep learning for the optimal filtering problem

Deep learning has revolutionized the fields of natural language processing and computer vision. It is the complexity and high dimensional nature of these problems which makes classical methods infeasible. Deep learning has proved to scale very well in increasing dimensions and this is one reason why scalable and well performing deep learning algorithms are now standard in these fields. In the field of high dimensional partial differential equations (PDE) deep learning has recently seen a similar success. Nonlinear equations in 10000 space dimensions has been solved approximately with neural networks. This result should be compared, e.g., with finite element or finite difference methods that are feasible in up to four dimensions and adaptive wavelet methods which has been used to solve PDE in up to 50 dimensions.

One numerical method relies on utilizing a stochastic representation of the solution to the PDE in terms of a backward stochastic differential equation. This equation is solved approximately with deep neural networks. The training is not supervised in the sense that approximate solutions with classical methods are required. Instead, the network architecture contains a discretization of the equation. We are currently running a parallel project using an extension of this methods to stochastic control problems.

The field of nonlinear filtering is concerned with computing the conditional probability distribution of a hidden state, given noisy observations. The state is modeled by a stochastic differential equation. Classical filtering algorithms, such as particle filters, suffer from the curse of dimensionality and it is desirable to obtain scalable algorithms. A fact that has not yet resulted in practical algorithms is that the solution solves a stochastic PDE, the Zakai equation. This equation, similar to certain PDE, also has a stochastic representation. Encouraged by the success for deep learning in solving PDE numerically, in this project we approach the filtering problem from this viewpoint and novel algorithms based on it will be developed and investigated. The project will consist of experimental studies and theoretical analysis.

We seek one PhD student for the project. The student must have a strong background in mathematical analysis, stochastic analysis and computational mathematics. Knowledge and skills in deep learning are meriting but not absolutely necessary.

Last modified: Mon Jan 13 15:12:58 CET 2020