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

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