Energy networks for state estimation with random sensors using sparse labels

State estimation is required whenever we deal with high-dimensional dynamical systems, as the complete measurement is often unavailable. It is key to gaining insight, performing control or optimizing design tasks. Most deep learning-based approaches require high-resolution labels and work with fixed sensor locations, thus being restrictive in their scope. Also, doing Proper orthogonal decomposition (POD) on sparse data is nontrivial. To tackle these problems, we propose a technique with an implicit optimization layer and a physics-based loss function that can learn from sparse labels. It works by minimizing the energy of the neural network prediction, enabling it to work with a varying number of sensors at different locations. Based on this technique we present two models for discrete and continuous prediction in space. We demonstrate the performance using two high-dimensional fluid problems of Burgers' equation and Flow Past Cylinder for discrete model and using Allen Cahn equation and Convection-diffusion equations for continuous model. We show the models are also robust to noise in measurements.