Learning state and proposal dynamics in state-space models using differentiable particle filters and neural networks (2411.15638v2)

Abstract: State-space models are a popular statistical framework for analysing sequential data. Within this framework, particle filters are often used to perform inference on non-linear state-space models. We introduce a new method, StateMixNN, that uses a pair of neural networks to learn the proposal distribution and transition distribution of a particle filter. Both distributions are approximated using multivariate Gaussian mixtures. The component means and covariances of these mixtures are learnt as outputs of learned functions. Our method is trained targeting the log-likelihood, thereby requiring only the observation series, and combines the interpretability of state-space models with the flexibility and approximation power of artificial neural networks. The proposed method significantly improves recovery of the hidden state in comparison with the state-of-the-art, showing greater improvement in highly non-linear scenarios.

• The paper introduces StateMixNN, a novel framework that integrates differentiable particle filters with neural networks to learn state transition and proposal distributions in SSMs.
• The paper demonstrates superior performance over traditional methods like the bootstrap and auxiliary particle filters on challenging models such as the Lorenz 96 and Kuramoto oscillator.
• The paper employs an alternating training regime that updates transition and proposal networks separately, enhancing both training stability and estimation accuracy.

Overview of "Learning state and proposal dynamics in state-space models using differentiable particle filters and neural networks"

This paper introduces StateMixNN, a novel approach to learning both state transition and proposal distributions in general nonlinear state-space models (SSMs) using particle filters. The method centers on leveraging differentiable particle filters (DPFs) combined with neural networks to enhance accuracy and efficiency in state estimation. Specifically, the proposal and transition distributions are approximated via multivariate Gaussian mixtures, whose components are learned through neural network outputs. This methodology aligns the interpretability of SSMs with the flexibility of neural networks without requiring apriori knowledge of hidden states.

Key Contributions and Methodology

1. Differentiable Particle Filters (DPFs): The paper employs the differentiable particle filter framework to facilitate gradient-based training, a crucial aspect of optimizing the neural network parameters. By integrating differentiability into the particle filter's resampling process, the proposed method allows for backpropagation, enabling effective learning from noisy data typically characterized by SSMs.
2. StateMixNN Architecture: The StateMixNN framework employs dense neural networks to parameterize Gaussian mixture models for both transition and proposal distributions in particle filtering. These networks take historical state data and current observations as inputs, producing mean and covariance parameters for the multivariate Gaussian mixtures. This setup supports both efficient and accurate approximation of complex distributions, enhancing the model's capacity to handle non-linear and high-dimensional state spaces.
3. Training Regime: A key innovation in this paper is the alternating training procedure for the transition and proposal networks. By updating each network while holding the other constant, the method mitigates potential identifiability issues and stabilizes the training process. This approach enables adaptation to complex systems through gradual incorporation of observation data.
4. Numerical Validation: The method is validated on two challenging dynamic systems: the Lorenz 96 model, known for its chaotic dynamics, and the Kuramoto oscillator, characterized by phase coupling. StateMixNN demonstrates superior performance over traditional methods including the bootstrap particle filter (BPF) and the improved auxiliary particle filter (IAPF). The method shows significant improvements in mean square error (MSE) under various particle counts, observation lengths, noise levels, and model dimensions.

Implications and Future Directions

The introduction of StateMixNN offers substantive improvements over traditional particle filtering approaches by addressing the challenges of model flexibility and learning efficiency. Its ability to accurately represent complex dynamical systems while employing neural networks offers a compelling advancement in SSM research, particularly in scenarios with non-linear dynamics and high-dimensional state spaces.

Practically, this method can be extended and applied to real-world systems in fields such as meteorology, finance, and robotics, improving state estimation accuracy in situations where models are complex and non-linear. Moreover, the ability to train models solely from observation data lowers the barrier for applying advanced filtering techniques in environments where ground truth state data is inaccessible.

Theoretically, future work could explore extending StateMixNN to other classes of distributions beyond Gaussian mixtures, incorporating domain-specific knowledge into the network architecture, or refining the model's robustness against noise and uncertainties. Additionally, further exploration into optimizing the training paradigm, potentially including more sophisticated gradient-based methods or alternative differentiable particle filtering approaches, might yield further improvements in both convergence speed and model generalization.

In summary, the proposed StateMixNN contributes important developments toward expanding the utility and performance of particle filters in SSMs, with promising avenues for both applied and continued theoretical research in machine learning and dynamical systems analysis.