Accurately modeling transport phenomena in shear flows is critical in diverse applications, from nutrient delivery and drug dispersion in biological systems to pollutant transport in environmental settings. Traditional operator learning methods, such as Fourier Neural Operators (FNOs), have shown promise but often demand significant computational resources. In this work, we introduce a structured neural network (SNN) that embeds the numerical discretization of the advection–diffusion equation directly into the architecture. Trained on time-averaged tracer fields with fixed Reynolds and Schmidt numbers, the SNN achieves closer agreement with ground truth fields than an FNO trained on the same data. In particular, the SNN more accurately represents boundary regions and maintains lower error across spatiotemporal predictions and one-dimensional line-slice comparisons. The computational efficiency of the SNN is also notable: training for 4000 epochs required only 1 h 7 min, compared to 13 h 46 min for the FNO. A separate test case with more turbulent flow conditions further demonstrated improved performance of the SNN relative to the FNO. These findings highlight the advantages of embedding numerical structure into neural networks, yielding both improved predictive accuracy and reduced training cost for modeling transport in shear flows.
