Deep Operator Learning for High-Fidelity Fluid Flow Field Reconstruction from Sparse Sensor Measurements

TL;DR

The paper addresses reconstructing high-fidelity fluid flow fields from sparse sensors by introducing FLRONet, a deep operator learning framework that combines DeepONet-style branch nets with Fourier Neural Operators to create a discretization-invariant mapping from sparse observations to continuous space-time flow fields. FLRONet decomposes the inverse measurement operator into spatial and temporal components, enabling zero-shot super-resolution in both space and time and providing robustness to missing data and noise through Voronoi embeddings and spectral filtering. The approach yields accurate reconstructions on the CFDBench cylinder dataset, outperforms 3D-FNO baselines, and achieves real-time inference on mid-range GPUs, making it suitable for mesh-free digital-twin applications. The work demonstrates substantial progress toward discretization-agnostic, data-efficient flow reconstruction and highlights future directions in temporal extrapolation and real-world generalization.

Abstract

Reconstructing high-fidelity fluid flow fields from sparse sensor measurements is vital for many science and engineering applications but remains challenging because of dimensional disparities between state and observational spaces. Due to such dimensional differences, the measurement operator becomes ill-conditioned and non-invertible, making the reconstruction of flow fields from sensor measurements extremely difficult. Although sparse optimization and machine learning address the above problems to some extent, questions about their generalization and efficiency remain, particularly regarding the discretization dependence of these models. In this context, deep operator learning offers a better solution as this approach models mappings between infinite-dimensional functional spaces, enabling superior generalization and discretization-independent reconstruction. We introduce FLRONet, a deep operator learning framework that is trained to reconstruct fluid flow fields from sparse sensor measurements. FLRONet employs a branch-trunk network architecture to represent the inverse measurement operator that maps sensor observations to the original flow field, a continuous function of both space and time. Validation performed on the CFDBench dataset has demonstrated that FLRONet consistently achieves high levels of reconstruction accuracy and robustness, even in scenarios where sensor measurements are inaccurate or missing. Furthermore, the operator learning approach endows FLRONet with the capability to perform zero-shot super-resolution in both spatial and temporal domains, offering a solution for rapid reconstruction of high-fidelity flow fields.

Figures (14)

• Figure 1: Problem formulation of FLRONet. FLRONet utilizes deep operator network to reconstruct the original flow field evolution for a given observation time window $\left[\tau_0, \tau_n \right]$. The reconstruction is continuous in both spatial and temporal domains.
• Figure 2: (a) Architectural design of FLRONet. FLRONet architecture employs a branch-trunk framework. The branch network encodes the spatial relationship among sensors in space by processing $[\mathbf{y}(\tau_1), \mathbf{y}(\tau_2), ..., \mathbf{y}(\tau_n)]$ through an Voronoi embedding layer $\Theta$. The resulting embeddings are then passed to $N$ independent two-dimensional FNO blocks, generating $N$ output fields $b$. The $\textbf{trunk network}$ models the temporal correlation between the target time $t$ and the observation time instances $[\tau_1, \tau_2, ..., \tau_n]$. It applies a sinusoid embedding layer $\Phi$ to embed $t$ and $\tau$ independently, combining them via a dot product, and finally passes the result to an MLP to produce $N$ output values $q$. A dot product combines $b$ and $q$ to produce the final construction $\mathbf{\hat{u}}(t)$. (b) Architecture of each FNO block. We adopt the standard configuration with four Fourier layers. Each layer applies a discrete Fourier transform $\mathcal{F}$ to map the physical input into the frequency domain, a complex-valued linear transformation $\mathcal{R}$ performs frequency processing. Afterward, only the lowest-frequency modes are retained before being transformed back to physical domain via the inverse Fourier transform $\mathcal{F}^{-1}$. A direct linear transformation $\mathcal{Q}$ is also added to the output before going to the non-linear activation $\sigma$.
• Figure 3: Voronoi-Tessellation-Based Embedding: (a) An original snapshot of the velocity field from the CFD dataset, with 32 randomly distributed sensors marked as white dots. (b) The corresponding Voronoi tessellation, computed from sensor locations and velocity values.
• Figure 4: The geometry of the fluid flow around the circular cylinder problem as described in the CFD dataset.
• Figure 5: Reconstructed fluid flow across architectures for the boundary condition $v_0 = 3.5$ m/s. The comparison highlights differences in reconstruction quality, with FLRONet showing the closest resemblance to the ground truth.