A Physics-guided Spectral-Fourier U-Net Operator: From velocity field to hidden tensor field in viscoelastic turbulence

Elastic stress plays a central role in viscoelastic flows and is fundamental to the formation of elastic turbulence and elasto-inertial turbulence. However, direct experimental access to stress fields remains extremely challenging, which limits the understanding of the underlying mechanisms. In this work, we propose PhySF-UNO (Physics-guided Spectral-Fourier U-Net Operator), a physics-guided neural-operator framework that reconstructs viscoelastic stress fields by predicting the polymer conformation tensor from velocity fields. High-fidelity direct numerical simulation data are used to train the model, enabling an end-to-end mapping from two-dimensional velocity fields to the conformation tensor components. The proposed framework adopts a multi-scale U-Net encoder–decoder architecture and incorporates Fourier neural operator blocks within skip connections to enhance global spectral representations while preserving local spatial structures. To ensure physical consistency, the symmetric positive-definite property of the conformation tensor is enforced during training, suppressing nonphysical predictions and improving robustness under strong elastic conditions. Extensive evaluations across multiple Weissenberg numbers demonstrate that PhySF-UNO consistently outperforms representative baselines, including convolutional neural networks, U-Net, and TransUNet, in terms of normalized root mean square error, peak signal-to-noise ratio, and structural accuracy, while exhibiting significantly improved generalization at high Wi. The proposed framework provides a data-driven approach for inferring viscoelastic stress fields from velocity measurements and offers a promising surrogate for traditional numerical solvers under complex flow conditions.