  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.
