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DNS: Data-driven Nonlinear Smoother for Complex Model-free Process

Fredrik Cumlin, Anubhab Ghosh, Saikat Chatterjee

TL;DR

This work tackles smoothing for complex model-free dynamical processes using linear measurements by proposing DNS, a data-driven nonlinear smoother trained in an unsupervised, measurement-only setting. DNS employs a deep recurrent architecture to produce Gaussian prior parameters that yield a closed-form posterior $p(\mathbf{x}_t|\hat{\mathbf{x}}_{1:t-1}, \mathbf{y}_{1:T})$, effectively incorporating anti-causal information. Empirical results on Lorenz, Chen, and a non-Markovian double spring-pendulum demonstrate that DNS achieves superior posterior estimation, especially at low SMNR, compared with iDANSE, DKS, and even STM-based ERTSS in some regimes. The approach enables accurate state estimation for model-free systems without state data, with potential impact on denoising, tracking, and biomedical signal processing where physics-based STMs are unavailable.

Abstract

We propose data-driven nonlinear smoother (DNS) to estimate a hidden state sequence of a complex dynamical process from a noisy, linear measurement sequence. The dynamical process is model-free, that is, we do not have any knowledge of the nonlinear dynamics of the complex process. There is no state-transition model (STM) of the process available. The proposed DNS uses a recurrent architecture that helps to provide a closed-form posterior of the hidden state sequence given the measurement sequence. DNS learns in an unsupervised manner, meaning the training dataset consists of only measurement data and no state data. We demonstrate DNS using simulations for smoothing of several stochastic dynamical processes, including a benchmark Lorenz system. Experimental results show that the DNS is significantly better than a deep Kalman smoother (DKS) and an iterative data-driven nonlinear state estimation (iDANSE) smoother.

DNS: Data-driven Nonlinear Smoother for Complex Model-free Process

TL;DR

This work tackles smoothing for complex model-free dynamical processes using linear measurements by proposing DNS, a data-driven nonlinear smoother trained in an unsupervised, measurement-only setting. DNS employs a deep recurrent architecture to produce Gaussian prior parameters that yield a closed-form posterior , effectively incorporating anti-causal information. Empirical results on Lorenz, Chen, and a non-Markovian double spring-pendulum demonstrate that DNS achieves superior posterior estimation, especially at low SMNR, compared with iDANSE, DKS, and even STM-based ERTSS in some regimes. The approach enables accurate state estimation for model-free systems without state data, with potential impact on denoising, tracking, and biomedical signal processing where physics-based STMs are unavailable.

Abstract

We propose data-driven nonlinear smoother (DNS) to estimate a hidden state sequence of a complex dynamical process from a noisy, linear measurement sequence. The dynamical process is model-free, that is, we do not have any knowledge of the nonlinear dynamics of the complex process. There is no state-transition model (STM) of the process available. The proposed DNS uses a recurrent architecture that helps to provide a closed-form posterior of the hidden state sequence given the measurement sequence. DNS learns in an unsupervised manner, meaning the training dataset consists of only measurement data and no state data. We demonstrate DNS using simulations for smoothing of several stochastic dynamical processes, including a benchmark Lorenz system. Experimental results show that the DNS is significantly better than a deep Kalman smoother (DKS) and an iterative data-driven nonlinear state estimation (iDANSE) smoother.
Paper Structure (11 sections, 11 equations, 3 figures, 2 tables)

This paper contains 11 sections, 11 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem formulation and Background
  3. Problem formulation
  4. DNS: Data-driven Nonlinear Smoother
  5. DNS and its unsupervised learning
  6. DRA Architecture and Unsupervised Learning
  7. On a simple approach
  8. Experiments and results
  9. Results
  10. Conclusion
  11. Acknowledgement

Figures (3)

  • Figure 1: Deep recurrent architectures of the proposed data-driven nonlinear smoother (DNS)
  • Figure 2: Lorenz-63 demonstration with DNS at $0$ dB
  • Figure 3: Visualization of state estimation and uncertainty quantification for the stochastic Lorenz system at 0 dB SMNR. The trajectories for the first coordinate of the three-dimensional state are shown.