Digital Twins Benefit from Joint Parameter and State Estimation with Uncertainty Quantification

Estimating both the current state and underlying parameters of complex, evolving systems is a fundamental challenge in fields ranging from robotics to weather forecasting, and is central to the development of increasingly sophisticated digital twins. Liliang Wang and Alex Gorodetsky, from the University of Michigan, along with their colleagues, now present a new method for achieving this estimation online, meaning it can update continuously as new data arrives. Their approach tackles the difficulty of accurately representing uncertainty in these estimations by employing a variational inference framework, which provides a computationally efficient approximation of the complete probability distribution of states and parameters. The resulting technique not only matches the accuracy of more complex methods in simpler scenarios, but also demonstrates robustness in chaotic systems and scales effectively to handle high-dimensional problems, representing a significant advance in real-time system modelling and prediction.

To support prediction and decision-making, reliability and computational speed are vital for dynamic systems. Online parameter-state estimation ensures computational efficiency, while uncertainty quantification is essential for making reliable predictions and decisions. The method recursively updates inferences at each time step, assimilating new data without reprocessing historical information, a significant advancement over traditional offline approaches.

Non-Gaussian State Estimation via Unscented Kalman Filter

This supplementary section details implementation and hyperparameter settings for experiments on Bayesian filtering and state estimation, ensuring reproducibility and deeper understanding of the methods used. It addresses cases where the state distribution is not Gaussian, a common challenge in real-world systems with non-Gaussian noise or dynamics. The approach involves two stages: approximating the marginal distribution of parameters using Monte Carlo sampling, and then approximating the conditional distribution of the state using a variational filtering method that minimizes the difference between the true posterior and an approximate distribution. Specific hyperparameter values are provided for various filtering methods, including the Unscented Kalman Filter, Joint Particle Filter, Joint UKF, and Joint EnKF.

These settings balance accuracy with preventing weight degeneracy in the Particle Filter. The research also details the use of a Delayed Rejection Adaptive Metropolis (DRAM) method, a sophisticated Markov Chain Monte Carlo technique, to generate ground truth posterior distributions for experiments, emphasizing that this is an offline process. Key concepts include Bayesian filtering, which estimates system state based on noisy observations, and the posterior distribution, representing the probability of the state and parameters given the observations. Other important terms are the Evidence Lower Bound, used in variational inference, the Unscented Kalman Filter, which uses a deterministic sampling technique, and the KL Divergence, a measure of the difference between probability distributions. This appendix serves as a detailed guide for replicating the experiments and verifying the results.

Online Variational Inference for Dynamical Systems

Scientists have developed a new online variational inference framework for estimating both unknown parameters and states within dynamical systems, achieving substantial improvements in computational efficiency and reliability. This work addresses a critical need in applications like digital twins, where continuous updates to system knowledge are essential for accurate prediction and decision-making, particularly when computational resources are limited. Experiments demonstrate the proposed method accurately infers both unobserved states and unknown parameters, matching the performance of joint particle filters in low-dimensional scenarios. The research shows robustness under challenging conditions, including noisy and partial observations, even within the complex, chaotic environment of a Lorenz 96 system.

Tests revealed the method maintains stable performance despite data imperfections and model discrepancies, crucial for real-world applications. A key breakthrough lies in the method’s scalability to high-dimensional convection systems, where it demonstrably outperforms the joint ensemble Kalman filter, delivering improved accuracy and efficiency. The algorithmic design is supported by a theorem establishing upper bounds on the joint posterior approximation error, providing a theoretical foundation for its reliability. This framework effectively approximates the joint distribution of state and parameters, capturing both individual uncertainties and their interdependencies, and enabling more trustworthy predictions and risk-aware decision-making.

Online Inference for Dynamical System States

This research presents a novel online variational inference framework for simultaneously estimating both the states and unknown parameters within dynamical systems, a capability crucial for applications like digital twins requiring continuous updates and reliable predictions. The team successfully developed a method that approximates the complex joint distribution of states and parameters, enabling efficient recursive updates with each new data point, and importantly, provides a quantifiable measure of uncertainty in these estimations. A key achievement lies in establishing theoretical bounds on the accuracy of this approximation, guaranteeing a level of reliability often lacking in similar techniques. The proposed method demonstrates strong performance across a range of challenging scenarios, matching the accuracy of more computationally intensive methods in simpler systems and maintaining robustness even with noisy data and model discrepancies in complex, chaotic environments. Notably, the framework scales effectively to high-dimensional systems, outperforming existing ensemble Kalman filter approaches, suggesting its potential for real-world applications involving large-scale data. Future research directions include extending the framework to handle more complex model structures and investigating adaptive strategies for refining the approximation over time, potentially leading to even more accurate and efficient state and parameter estimation.