# Operator Learning for Reconstructing Flow Fields from Sparse Measurements: an Energy Transformer Approach

arXiv: https://arxiv.org/abs/2501.08339 · HTML: https://mlsanity.com/paper/2501.08339/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 36 references that are in the map.
- First posted on arXiv 2 Jan 2025.

## Builds on (most cited first, 11 of 11)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Energy Transformer](https://mlsanity.com/paper/2302.07253/) (2023)
- [AOSLO-net: A deep learning-based method for automatic segmentation of retinal microaneurysms from adaptive optics scanning laser ophthalmoscope images](https://mlsanity.com/paper/2106.02800/) (2021)
- [SMS: Spiking Marching Scheme for Efficient Long Time Integration of Differential Equations](https://mlsanity.com/paper/2211.09928/) (2022)

## Built on by (most cited first, 12 of 12)

- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Reduced Order Modeling with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2502.10930/) (2025)
- [Learning Turbulent Flows with Generative Models: Super-resolution, Forecasting, and Sparse Flow Reconstruction](https://mlsanity.com/paper/2509.08752/) (2025)
- [When do World Models Successfully Learn Dynamical Systems?](https://mlsanity.com/paper/2507.04898/) (2025)
- [Constrained Sensing and Reliable State Estimation with Shallow Recurrent Decoders on a TRIGA Mark II Reactor](https://mlsanity.com/paper/2510.12368/) (2025)
- [Ambient Physics: Training Neural PDE Solvers with Partial Observations](https://mlsanity.com/paper/2602.13873/) (2026)
- [Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force method](https://mlsanity.com/paper/2505.04875/) (2025)
- [Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization](https://mlsanity.com/paper/2604.05042/) (2026)
- [Droplet-LNO: Physics-Informed Laplace Neural Operators for Accurate Prediction of Droplet Spreading Dynamics on Complex Surfaces](https://mlsanity.com/paper/2604.20993/) (2026)
- [Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributions](https://mlsanity.com/paper/2605.06861/) (2026)
- [Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach](https://mlsanity.com/paper/2605.23712/) (2026)
- [Learning Incompressible 3D Flows from Sparse Samples: A Set Transformer Approach to Divergence-Free Kernel Fitting](https://mlsanity.com/paper/2610.16645/) (2026)

## Abstract (arXiv)

Machine learning methods have shown great success in various scientific areas, including fluid mechanics. However, reconstruction problems, where full velocity fields must be recovered from partial observations, remain challenging. In this paper, we propose a novel operator learning framework for solving reconstruction problems by using the Energy Transformer (ET), an architecture inspired by associative memory models. We formulate reconstruction as a mapping from incomplete observed data to full reconstructed fields. The method is validated on three fluid mechanics examples using diverse types of data: (1) unsteady 2D vortex street in flow past a cylinder using simulation data; (2) high-speed under-expanded impinging supersonic jets impingement using Schlieren imaging; and (3) 3D turbulent jet flow using particle tracking. The results demonstrate the ability of ET to accurately reconstruct complex flow fields from highly incomplete data (90% missing), even for noisy experimental measurements, with fast training and inference on a single GPU. This work provides a promising new direction for tackling reconstruction problems in fluid mechanics and other areas in mechanics, geophysics, weather prediction, and beyond.