Leveraging arbitrary mobile sensor trajectories with shallow recurrent decoder networks for full-state reconstruction

Sensing is one of the most fundamental tasks for the monitoring, forecasting and control of complex, spatio-temporal systems. In many applications, a limited number of sensors are mobile and move with the dynamics, with examples including wearable technology, ocean monitoring buoys, and weather balloons. In these dynamic systems (without regions of statistical-independence), the measurement time history encodes a significant amount of information that can be extracted for critical tasks. Most model-free sensing paradigms aim to map current sparse sensor measurements to the high-dimensional state space, ignoring the time-history all together. Using modern deep learning architectures, we show that a sequence-to-vector model, such as an LSTM (long, short-term memory) network, with a decoder network, dynamic trajectory information can be mapped to full state-space estimates. Indeed, we demonstrate that by leveraging mobile sensor trajectories with shallow recurrent decoder networks, we can train the network (i) to accurately reconstruct the full state space using arbitrary dynamical trajectories of the sensors, (ii) the architecture reduces the variance of the mean-square error of the reconstruction error in comparison with immobile sensors, and (iii) the architecture also allows for rapid generalization (parameterization of dynamics) for data outside the training set. Moreover, the path of the sensor can be chosen arbitrarily, provided training data for the spatial trajectory of the sensor is available. The exceptional performance of the network architecture is demonstrated on three applications: turbulent flows, global sea-surface temperature data, and human movement biomechanics.

Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects

Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations often result in discrepancies between the model and sensor-based measurements of the system, revealing the approximate nature of the equations and/or the signal-to-noise ratio of the sensor itself. In modern dynamical systems, such discrepancies between model and measurement can lead to poor quantification, often undermining the ability to produce accurate and precise control algorithms. We introduce a discrepancy modeling framework to resolve deterministic model-measurement mismatch with two distinct approaches: (i) by learning a model for the evolution of systematic state-space residual, and (ii) by discovering a model for the missing deterministic physics. Regardless of approach, a common suite of data-driven model discovery methods can be used. Specifically, we use four fundamentally different methods to demonstrate the mathematical implementations of discrepancy modeling: (i) the sparse identification of nonlinear dynamics (SINDy), (ii) dynamic mode decomposition (DMD), (iii) Gaussian process regression (GPR), and (iv) neural networks (NN). The choice of method depends on one's intent for discrepancy modeling, as well as quantity and quality of the sensor measurements. We demonstrate the utility and suitability for both discrepancy modeling approaches using the suite of data-driven modeling methods on three dynamical systems under varying signal-to-noise ratios. We compare reconstruction and forecasting accuracies and provide detailed comparatives, allowing one to select the appropriate approach and method in practice.