Energy network for state estimation with random sensors and sparse labels

State estimation is imperative while dealing with high-dimensional dynamical systems due to the unavailability of complete measurements. It plays a pivotal role in gaining insights, executing control, or optimizing design tasks. However, many deep learning approaches are constrained by the requirement for high-resolution labels and fixed sensor locations, limiting their practical applicability. To address these limitations, we propose a novel approach featuring an implicit optimization layer and a physics-based loss function capable of learning from sparse labels. This approach operates by minimizing the energy of neural network predictions, thereby accommodating varying sensor counts and locations. Our methodology is validated through the application of these models to two high-dimensional fluid problems: Burgers' equation and Flow Past Cylinder. Notably, our model exhibits robustness against noise in measurements, underscoring its effectiveness in practical scenarios.