The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines. We present a unifying framework for blending mechanistic and machine-learning approaches to identify dynamical systems from noisily and partially observed data. We compare pure data-driven learning with hybrid models which incorporate imperfect domain knowledge, referring to the discrepancy between an assumed truth model and the imperfect mechanistic model as model error. Our formulation is agnostic to the chosen machine learning model, is presented in both continuous- and discrete-time settings, and is compatible both with model errors that exhibit substantial memory and errors that are memoryless. First, we study memoryless linear (w.r.t. parametric-dependence) model error from a learning theory perspective, defining excess risk and generalization error. For ergodic continuous-time systems, we prove that both excess risk and generalization error are bounded above by terms that diminish with the square-root of T T , the time-interval over which training data is specified. Secondly, we study scenarios that benefit from modeling with memory, proving universal approximation theorems for two classes of continuous-time recurrent neural networks (RNNs): both can learn memory-dependent model error, assuming that it is governed by a finite-dimensional hidden variable and that, together, the observed and hidden variables form a continuous-time Markovian system. In addition, we connect one class of RNNs to reservoir computing, thereby relating learning of memory-dependent error to recent work on supervised learning between Banach spaces using random features. Numerical results are presented (Lorenz ’63, Lorenz ’96 Multiscale systems) to compare purely data-driven and hybrid approaches, finding hybrid methods less datahungry and more parametrically efficient. We also find that, while a continuous-time framing allows for robustness to irregular sampling and desirable domain- interpretability, a discrete-time framing can provide similar or better predictive performance, especially when data are undersampled and the vector field defining the true dynamics cannot be identified. Finally, we demonstrate numerically how data assimilation can be leveraged to learn hidden dynamics from noisy, partially-observed data, and illustrate challenges in representing memory by this approach, and in the training of such models.
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Tracking of Dynamical Processes with Model Switching Using Temporal Convolutional Networks
This paper considers the problem of tracking and predicting dynamical processes with model switching. The classical approach to this problem has been to use an interacting multiple model (IMM) which uses multiple Kalman filters and an auxiliary system to estimate the posterior probability of each model given the observations. More recently, data-driven approaches such as recurrent neural networks (RNNs) have been used for tracking and prediction in a variety of settings. An advantage of data-driven approaches like the RNN is that they can be trained to provide good performance even when the underlying dynamic models are unknown. This paper studies the use of temporal convolutional networks (TCNs) in this setting since TCNs are also data-driven but have certain structural advantages over RNNs. Numerical simulations demonstrate that a TCN matches or exceeds the performance of an IMM and other classical tracking methods in two specific settings with model switching: (i) a Gilbert-Elliott burst noise communication channel that switches between two different modes, each modeled as a linear system, and (ii) a maneuvering target tracking scenario where the target switches between a linear constant velocity mode and a nonlinear coordinated turn mode. In particular, the results show that the TCN tends to identify a mode switch as fast or faster than an IMM and that, in some cases, the TCN can perform almost as well as an omniscient Kalman filter with perfect knowledge of the current mode of the dynamical system.
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- Award ID(s):
- 1836695
- NSF-PAR ID:
- 10349348
- Date Published:
- Journal Name:
- 2021 IEEE Aerospace Conference
- Page Range / eLocation ID:
- 1 to 9
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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