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

Yuzhong Hu, Jiayi Kang, Lei Ma, Xiaoming Zhang

TL;DR

This work tackles time-variant nonlinear filtering by marrying the Yau–Yau filter with Physics-Informed Neural Networks and PCA. The proposed PINN-based Yau–Yau Filter (PINNYYF) uses an offline PINN to solve the Forward Kolmogorov Equation and PCA to compress time-variant solutions into a compact basis, enabling an online approximate solver for real-time state estimation. Across three nonlinear examples, PINNYYF achieves competitive accuracy with substantially reduced online computation and storage compared to traditional approaches such as EKF, PF, and Legendre-based Yau–Yau filters. The results demonstrate practical potential for real-time nonlinear filtering in systems with time-varying dynamics, while outlining avenues for scaling to higher dimensions and real-world data applications.

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.

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

TL;DR

This work tackles time-variant nonlinear filtering by marrying the Yau–Yau filter with Physics-Informed Neural Networks and PCA. The proposed PINN-based Yau–Yau Filter (PINNYYF) uses an offline PINN to solve the Forward Kolmogorov Equation and PCA to compress time-variant solutions into a compact basis, enabling an online approximate solver for real-time state estimation. Across three nonlinear examples, PINNYYF achieves competitive accuracy with substantially reduced online computation and storage compared to traditional approaches such as EKF, PF, and Legendre-based Yau–Yau filters. The results demonstrate practical potential for real-time nonlinear filtering in systems with time-varying dynamics, while outlining avenues for scaling to higher dimensions and real-world data applications.

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.
Paper Structure (15 sections, 30 equations, 11 figures, 4 tables, 3 algorithms)

This paper contains 15 sections, 30 equations, 11 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonlinear Filtering Problem and the Yau-Yau Filter
  4. Physics-Informed Neural Networks
  5. Principal Component Analysis
  6. Implementation Algorithm for Time-variant Systems
  7. Stage IA (offline): Train FKE solver with PINN
  8. Stage IB (offline): Train the approximate time-variant FKE solver
  9. Stage II (online): Real-time execution for system state estimation
  10. Numerical Experiments
  11. Network Description
  12. Example 1: Cubic Sensor Problem (Time-invariant)
  13. Example 2: Time-variant Harmonic Sensor Problem
  14. Example 3: Time-variant Cubic Sensor Problem
  15. Conclusion

Figures (11)

  • Figure 1: The general structure of a Physics-Informed Neural Network (PINN).
  • Figure 2: The training process of FKE solver with PINN.
  • Figure 3: The training of the approximate FKE solver based on PCA.
  • Figure 4: Execute the approximate FKE solver and obtain the end time solutions.
  • Figure 5: The residual network structure used in Stage IB.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2