# Energy networks for state estimation with random sensors using sparse labels

arXiv: https://arxiv.org/abs/2203.06456 · HTML: https://mlsanity.com/paper/2203.06456/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 57 references that are in the map.
- First posted on arXiv 12 Mar 2022.

## Builds on (most cited first, 8 of 8)

- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Composing graphical models with neural networks for structured representations and fast inference](https://mlsanity.com/paper/1603.06277/) (2016)
- [Generalized Inner Loop Meta-Learning](https://mlsanity.com/paper/1910.01727/) (2019)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [End-to-End Learning for Structured Prediction Energy Networks](https://mlsanity.com/paper/1703.05667/) (2017)
- [State estimation with limited sensors -- A deep learning based approach](https://mlsanity.com/paper/2101.11513/) (2021)
- [Shallow Neural Networks for Fluid Flow Reconstruction with Limited Sensors](https://mlsanity.com/paper/1902.07358/) (2019)

## Abstract (arXiv)

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.