pDANSE: Particle-based Data-driven Nonlinear State Estimation from Nonlinear Measurements

TL;DR

pDANSE introduces a particle-based extension of the data-driven nonlinear state estimation framework to handle nonlinear measurements with an unknown state-transition model. It uses an RNN to output a Gaussian prior $p(\mathbf{x}_t|\mathbf{y}_{1:t-1}) = \mathcal{N}(\mathbf{x}_t; \mathbf{m}_{t|1:t-1}(\theta), \mathbf{L}_{t|1:t-1}(\theta))$, and employs a reparameterization-based Monte Carlo approach to estimate posterior moments when $\mathbf{y}_t = \mathbf{h}(\mathbf{x}_t) + \mathbf{w}_t$ with $\mathbf{w}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{C}_w)$ renders the posterior non-Gaussian. The learning problem supports unsupervised and semi-supervised training, deriving a tractable lower bound for training via MC samples of the prior. Empirical results on the Lorenz-63 system show that pDANSE achieves competitive state estimation compared to a model-driven particle filter across four nonlinear measurement models, highlighting its potential for data-driven Bayesian filtering without explicit STM knowledge. The approach enables scalable, causally-ordered filtering by leveraging a neural prior and MC-based posterior moment estimation, with practical performance improvements achieved using limited labeled data in semi-supervised settings.

Abstract

We consider the problem of designing a data-driven nonlinear state estimation (DANSE) method that uses (noisy) nonlinear measurements of a process whose underlying state transition model (STM) is unknown. Such a process is referred to as a model-free process. A recurrent neural network (RNN) provides parameters of a Gaussian prior that characterize the state of the model-free process, using all previous measurements at a given time point. In the case of DANSE, the measurement system was linear, leading to a closed-form solution for the state posterior. However, the presence of a nonlinear measurement system renders a closed-form solution infeasible. Instead, the second-order statistics of the state posterior are computed using the nonlinear measurements observed at the time point. We address the nonlinear measurements using a reparameterization trick-based particle sampling approach, and estimate the second-order statistics of the state posterior. The proposed method is referred to as particle-based DANSE (pDANSE). The RNN of pDANSE uses sequential measurements efficiently and avoids the use of computationally intensive sequential Monte-Carlo (SMC) and/or ancestral sampling. We describe the semi-supervised learning method for pDANSE, which transitions to unsupervised learning in the absence of labeled data. Using a stochastic Lorenz-$63$ system as a benchmark process, we experimentally demonstrate the state estimation performance for four nonlinear measurement systems. We explore cubic nonlinearity and a camera-model nonlinearity where unsupervised learning is used; then we explore half-wave rectification nonlinearity and Cartesian-to-spherical nonlinearity where semi-supervised learning is used. The performance of state estimation is shown to be competitive vis-à-vis particle filters that have complete knowledge of the STM of the Lorenz-$63$ system.

Figures (10)

• Figure 1: Schematic of $\text{pDANSE}$ at inference time.
• Figure 2: Sampling from the Gaussian prior in \ref{['eq:RNNPrior']} using the reparameterization trick, to obtain samples $\lbrace \mathbf{x}^{(l)}_{t}\left(\pmb{\theta}\right)\rbrace_{l=1}^{L}$.
• Figure 3: Visual illustration of the BSE performance for cubic nonlinearity in an unsupervised learning setup at $\text{SMNR} = 20 \text{ dB}$. Top-left - A true state trajectory for the stochastic Lorenz-$63$ system from $\mathcal{D}_{\text{test}}$. Top-right - The corresponding noisy measurement trajectory. Bottom-left - The estimated trajectory of PF. Bottom-right- The estimated trajectory of pDANSE.
• Figure 4: Demonstrating the qualitative performance of $\text{pDANSE}$ ($\kappa= 0\%$) for the BSE problem using noisy, element-wise cubic measurements of the Lorenz-$63$ process, at $\text{SMNR} = 20 \text{ dB}$. A chosen realization of a true state trajectory from $\mathcal{D}_{\text{test}}$ is shown in black. The estimated state trajectories are the corresponding posterior mean estimates using PF (shown in blue) and $\text{pDANSE}$ (shown in green). The shaded region indicates a $\pm 3 \sigma$ confidence for the posterior mean estimate.
• Figure 5: NMSE (in dB) on $\mathcal{D}_{\text{test}}$ vs. SMNR (in dB), demonstrating the performance of $\text{pDANSE}$ ($\kappa=0\%$) for the BSE task using noisy, cubic measurements of the Lorenz-$63$ process. The nonlinear function $\mathbf{h}$ is defined in \ref{['eq:cubic_measurement_fn']}. $\text{pDANSE}$ was trained using $N=1000, T=200$.