# Reconstructing unsteady flows from sparse, noisy measurements with a physics-constrained convolutional neural network

arXiv: https://arxiv.org/abs/2409.00260 · HTML: https://mlsanity.com/paper/2409.00260/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 52 references that are in the map.
- First posted on arXiv 30 Aug 2024.

## Builds on (most cited first, 9 of 9)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Super-Resolution Analysis via Machine Learning: A Survey for Fluid Flows](https://mlsanity.com/paper/2301.10937/) (2023)
- [Physics-informed deep-learning applications to experimental fluid mechanics](https://mlsanity.com/paper/2203.15402/) (2022)
- [Turbulence Enrichment using Physics-informed Generative Adversarial Networks](https://mlsanity.com/paper/2003.01907/) (2020)
- [Multi-scale data reconstruction of turbulent rotating flows with Gappy POD, Extended POD and Generative Adversarial Networks](https://mlsanity.com/paper/2210.11921/) (2022)
- [Physics-Informed CNNs for Super-Resolution of Sparse Observations on Dynamical Systems](https://mlsanity.com/paper/2210.17319/) (2022)
- [Inferring unknown unknowns: Regularized bias-aware ensemble Kalman filter](https://mlsanity.com/paper/2306.04315/) (2023)
- [Deep learning fluid flow reconstruction around arbitrary two-dimensional objects from sparse sensors using conformal mappings](https://mlsanity.com/paper/2202.03798/) (2022)
- [Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations](https://mlsanity.com/paper/2401.10306/) (2024)

## Built on by (most cited first, 3 of 3)

- [Generalized Neural Operator for Parametric and Boundary-Value Problems](https://mlsanity.com/paper/2607.21932/) (2026)
- [Hybrid Lagrangian-Eulerian Model for Lagrangian Fluid Simulation](https://mlsanity.com/paper/2608.01164/) (2026)
- [Inferring physical fields in coupled systems with unknown parameters from incomplete observations using physics-constrained attentive neural operators](https://mlsanity.com/paper/2610.05723/) (2026)

## Abstract (arXiv)

Data from fluid flow measurements are typically sparse, noisy, and heterogeneous, often from mixed pressure and velocity measurements, resulting in incomplete datasets. In this paper, we develop a physics-constrained convolutional neural network, which is a deterministic tool, to reconstruct the full flow field from incomplete data. We explore three loss functions, both from machine learning literature and newly proposed: (i) the softly-constrained loss, which allows the prediction to take any value; (ii) the snapshot-enforced loss, which constrains the prediction at the sensor locations; and (iii) the mean-enforced loss, which constrains the mean of the prediction at the sensor locations. The proposed methods do not require the full flow field during training, making it suitable for reconstruction from incomplete data. We apply the method to reconstruct a laminar wake of a bluff body and a turbulent Kolmogorov flow. First, we assume that measurements are not noisy and reconstruct both the laminar wake and the Kolmogorov flow from sensors located at fewer than 1% of all grid points. The snapshot-enforced loss reduces the reconstruction error of the Kolmogorov flow by approximately 25% compared to the softly-constrained loss. Second, we assume that measurements are noisy and propose the mean-enforced loss to reconstruct the laminar wake and the Kolmogorov flow at three different signal-to-noise ratios. We find that, across the ratios tested, the loss functions with harder constraints are more robust to both the random initialization of the networks and the noise levels in the measurements. At high noise levels, the mean-enforced loss can recover the instantaneous snapshots accurately, making it the suitable choice when reconstructing flows from data corrupted with an unknown amount of noise. The proposed method opens opportunities for physical flow reconstruction from sparse, noisy data.