Operator Learning for Reconstruction Problems: An Energy Transformer Approach with Applications in Fluid Mechanics

Machine learning methods have shown great success in various scientific areas, including fluid mechanics. However, reconstruction problems, where full data must be recovered from partial observations, remain challenging. In this paper, we propose a novel operator learning framework for solving reconstruction problems using the energy transformer. We formulate reconstruction as a mapping from incomplete observed data to full reconstructed data. The method is validated on three examples of fluid mechanics: 2D vortex street simulation, experimental jet impingement, and 3D turbulent jet flow. Results demonstrate the ability 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 scientific domains.