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

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.

💡 Research Summary

The paper introduces FLRONet, a nested spatio‑temporal deep operator network designed to reconstruct high‑fidelity fluid flow fields from sparse sensor measurements. Traditional reconstruction techniques struggle because the measurement operator that maps the high‑dimensional flow state to a low‑dimensional sensor space is ill‑conditioned and non‑invertible. Moreover, most data‑driven models (CNNs, POD, auto‑encoders) are tied to a specific mesh and sensor layout, requiring costly retraining when discretization changes. FLRONet overcomes these limitations by learning a mapping between infinite‑dimensional function spaces, i.e., a true operator, which makes it discretization‑independent in both space and time.

The architecture follows the DeepONet branch‑trunk paradigm. The branch network consists of N independent two‑dimensional Fourier Neural Operator (FNO) blocks. Sensor readings at each observation time τ_i are first processed by a Voronoi embedding that encodes irregular sensor locations into a regular tensor. Each FNO block applies a discrete Fourier transform, retains only low‑frequency modes (thus filtering high‑frequency noise), and outputs a latent spatial field b_i. The trunk network handles temporal information: both the target reconstruction time t and the observation times τ_i are embedded with sinusoidal positional encodings, combined via a dot product that reflects temporal proximity, and fed into a three‑layer MLP to produce temporal coefficients q_i. The final reconstructed field at any query point (x, t) is obtained by summing the inner products ⟨b_i, q_i⟩ over all i, yielding a continuous space‑time representation.

Key contributions are: (1) Dual‑domain zero‑shot super‑resolution – the model trained on a coarse 140 × 240 grid can predict flow structures on a fine 1120 × 1920 grid and interpolate time steps smaller than 10⁻⁶ s without any retraining. (2) Robustness to sensor sparsity and noise – Voronoi embedding together with the spectral filtering of FNO allows accurate reconstructions even when up to 50 % of sensors fail or when measurements are corrupted by up to 20 % Gaussian noise. (3) Real‑time inference – on an NVIDIA A100 GPU the network processes a frame in roughly 16 ms, outperforming comparable 3‑D FNO models in both speed and accuracy. Experiments on the CFDBench dataset, covering a range of Reynolds numbers and complex vortex dynamics, demonstrate that FLRONet reduces the mean L2 error by 15‑30 % relative to state‑of‑the‑art baselines while preserving fine‑scale flow features.

Overall, FLRONet showcases how operator learning can deliver mesh‑agnostic, data‑efficient, and computationally cheap surrogates for inverse problems in fluid dynamics. The authors suggest future extensions to out‑of‑window temporal extrapolation, multi‑physics reconstruction (e.g., pressure and temperature), and the incorporation of physics‑based constraints directly into the loss function, further broadening the applicability of deep operator networks in real‑world engineering and scientific contexts.