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

TL;DR

The paper tackles reconstructing full fluid-flow fields from sparse measurements by introducing an operator-learning framework based on the Energy Transformer, a memory-inspired architecture that stores patterns as energy minima. By partitioning data into patches and using a patcher–detector pipeline, the method learns a reconstruction operator $\mathcal{R}$ that maps partial observations $\tilde{\mathcal{P}}$ to full fields $\mathcal{P}$ via a trainable tokenizer, Energy Transformer, and detokenizer. The ET combines a multi-head energy attention mechanism and a Hopfield network to form an energy function whose local minimizer yields the reconstructed tokens, enabling accurate reconstructions even with 90% missing data across multi-modal, noisy fluid datasets. Across 2D vortex street, 2D jet, and 3D turbulent jet cases, the approach demonstrates competitive accuracy with relatively low training/inference costs on a single GPU, offering a practical reconstruction tool for fluid mechanics and potentially other engineering domains.

Abstract

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.

Figures (16)

• Figure 1: Overview of the proposed workflow. The dashed arrows represent the operations that are only used in the training stage. (a) Patch and depatch operations. (b) Tokenize (blue arrow) and detokenize (green arrow) operation. (c) Reconstruction operation based on the Energy Transformer.
• Figure 2: ET block. (a) The structure of the ET block. (b) The update rule in the forward pass of the ET block.
• Figure 3: Signature of the 2D vortex street dataset: values of $u, v, p, T$ (horizontal velocity, vertical velocity, pressure and temperature) at the center of the domain versus the time. The red dashed line represents the split of training and test data. Periodicity can be observed in all components, which is characteristic of the vortex street.
• Figure 4: Training loss of the 2D vortex street example.
• Figure 5: Reconstruction results of a 2D vortex street flow at time step 90. Each row in the figure displays a different physical field of the flow. In each panel, the x-axis represents the horizontal spatial coordinate, while the y-axis represents the vertical spatial coordinate, with values in physical units relevant to the flow domain. The columns show (from left to right): (1) the full field, representing the original, high-resolution simulation data; (2) the observed field, displaying sparse measurements of the field, typically available from limited sensors or measurements; and (3) the reconstructed field, where the sparse observations have been processed by an algorithm to estimate the full field. The color in each plot represents the magnitude of each physical field, with warmer colors (e.g., red) indicating higher values and cooler colors (e.g., blue) indicating lower values. Time step 90 is chosen because it is in the test dataset and it captures a characteristic snapshot of the vortex street pattern, where complex flow structures are fully developed, making it a representative example for illustrating the algorithm's performance in reconstructing detailed flow features.