KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics (2107.10043v3)

Abstract: State estimation of dynamical systems in real-time is a fundamental task in signal processing. For systems that are well-represented by a fully known linear Gaussian state space (SS) model, the celebrated Kalman filter (KF) is a low complexity optimal solution. However, both linearity of the underlying SS model and accurate knowledge of it are often not encountered in practice. Here, we present KalmanNet, a real-time state estimator that learns from data to carry out Kalman filtering under non-linear dynamics with partial information. By incorporating the structural SS model with a dedicated recurrent neural network module in the flow of the KF, we retain data efficiency and interpretability of the classic algorithm while implicitly learning complex dynamics from data. We demonstrate numerically that KalmanNet overcomes non-linearities and model mismatch, outperforming classic filtering methods operating with both mismatched and accurate domain knowledge.

• The paper introduces KalmanNet, a hybrid framework that integrates classical Kalman filtering with RNNs to learn state estimation in partially known dynamic systems.
• It leverages data-driven techniques to compute the Kalman gain, overcoming limitations of traditional filters in handling nonlinearities and model uncertainties.
• Experiments on linear, nonlinear, and chaotic systems demonstrate KalmanNet’s superior performance and its potential for real-time localization and tracking applications.

Overview of "KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics"

The paper "KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics" introduces a novel approach to real-time state estimation in scenarios where the dynamics of a system are not fully known. This work presents KalmanNet, a framework that integrates principles of Kalman filtering with deep learning, particularly Recurrent Neural Networks (RNNs), to handle non-linear dynamics and partially uncertain environments effectively.

Key Contributions and Methodology

1. Hybrid Model-Based/Deep Learning Approach: KalmanNet is designed as a hybrid method, leveraging the interpretability and data efficiency of Kalman filtering while incorporating the flexibility and adaptability of deep learning. By fusing a structured state-space model with a dedicated RNN module within the Kalman filter's operational flow, KalmanNet maintains key advantages of classic algorithms in terms of computational efficiency and interpretability.
2. Learning to Infer with Partial Knowledge: Traditional Kalman filters rely heavily on accurate linear state-space models and known noise statistics for optimality. KalmanNet addresses these limitations by utilizing RNNs to learn the Kalman gain from data, thus bypassing the need for explicit knowledge of noise statistics. This capability enables the system to adapt to real-time data, overcoming model mismatches and nonlinearities that are common in practical scenarios.
3. State Estimation under Uncertain Dynamics: The applicability of KalmanNet is demonstrated across various state-space models, including both synthetic linear and non-linear systems and real-world datasets. The experimental results emphasize its robustness against model uncertainties and superior performance compared to traditional Model-Based (MB) methods like the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF).
4. Practical Implications and Future Prospects: The authors illustrate potential applications in localization and tracking, with experiments highlighting KalmanNet's ability to perform under dynamics that are partially known or inaccurately modeled. This work opens avenues for further exploration of hybrid methodologies in real-time applications, suggesting future developments in model-agnostic filters that continue to exploit domain knowledge for enhanced prediction accuracy.

Numerical Results and Evaluations

• Linear State-Space Models:

In scenarios where the state-space model is fully known, KalmanNet is shown to achieve minimal Mean Squared Error (MSE) comparable to the optimal performance of traditional Kalman filters. Importantly, KalmanNet demonstrates transferability in learning, applicable to varying trajectory lengths and initial conditions.

• Non-Linear State-Space Models:

For highly non-linear scenarios such as a sinusoidal state evolution with non-linear observation mappings, KalmanNet outperforms classic filters by effectively learning the complex dynamics from data.

• Chaotic Systems and Real-World Data:

In testing with chaotic systems like the Lorenz Attractor and real-world datasets from robotic tracking, KalmanNet displays resilience and adaptability, providing accurate state estimations even with approximate or mismatched system models.

Conclusion

This paper successfully marries classical signal processing with modern machine learning, proposing KalmanNet as a viable solution for dynamic systems with partially known dynamics. It evidences a significant advancement in state estimation strategies, emphasizing the efficacy of data-driven methodologies in real-time applications. Future research may explore unsupervised training paradigms and model-free systems to expand KalmanNet's applicability to a wider range of complex environments.