Abstract
Nonlinear filtering can be formulated as the propagation and update of conditional probability densities, but direct numerical propagation of the associated Forward Kolmogorov equation (FKE) over a large fixed domain is computationally expensive. This paper proposes a Recentered-Domain Yau–Yau Filter (RD-YYF) with reduced-order FKE propagation for nonlinear state estimation. The method solves the FKE on a fixed-size local computational window centered at the latest state estimate, thereby concentrating numerical resolution near the dominant posterior density. In the offline stage, physics-informed neural networks (PINNs) generate FKE solution snapshots, principal component analysis constructs a low-dimensional representation of density evolution, and a lightweight residual surrogate maps initial-condition coefficients and the domain center to terminal-solution coefficients. In the online stage, the pretrained surrogate performs per-timestep density prediction within the recentered window, followed by observation update and state estimation. Numerical experiments on two geometrically constrained target-tracking models show that RD-YYF achieves lower tracking errors than the extended Kalman filter and particle filter under matched online evaluation conditions. A fixed-domain ablation further shows that recentering improves density approximation in high-probability regions and reduces offline PINN training epochs. These results indicate that recentered-domain reduced-order FKE propagation is a practical computational strategy for nonlinear density-based filtering.