A novel implementation of Yau-Yau filter for time-variant nonlinear problems

TL;DR

This paper introduces a new numerical algorithm combining physics-informed neural networks and PCA to efficiently implement the Yau-Yau filter for nonlinear, time-varying systems, improving speed and storage over existing methods.

Contribution

The paper presents a novel implementation of the Yau-Yau filter using PINN and PCA, enabling efficient offline training and online application for time-variant nonlinear filtering.

Findings

Accurate and efficient filtering demonstrated on three examples.

Outperforms extended Kalman and particle filters in speed and storage.

Applicable to practical nonlinear time-variant problems.

Abstract

Nonlinear filter has long been an important problem in practical industrial applications. The Yau-Yau method is a highly versatile framework that transforms nonlinear filtering problems into initial-value problems governed by the Forward Kolmogorov Equation (FKE). Previous researches have shown that the method can be applied to highly nonlinear and high dimensional problems. However, when time-varying coefficients are involved in the system models, developing an implementation of the method with high computational speed and low data storage still presents a challenge. To address these limitations, this paper proposes a novel numerical algorithm that incorporates physics-informed neural network (PINN) and principal component analysis (PCA) to solve the FKE approximately. Equipped with this algorithm, the Yau-Yau filter can be implemented by an offline stage for the training of a solver for the approximate solution of FKE and an online stage for its execution. Results of three examples indicate that this implementation is accurate, both time-efficient and storage-efficient for online computation, and is superior than existing nonlinear filtering methods such as extended Kalman filter and particle filter. It is capable of applications to practical nonlinear time-variant filtering problems.