Neural operators generalize classical neural networks to maps between infinite-dimensional spaces, e.g. function spaces. Prior works on neural operators proposed a series of novel architectures to learn such maps and demonstrated unprecedented success in learning solution operators of partial differential equations. Due to their close proximity to fully connected architectures, these models mainly suffer from high memory usage and are generally limited to shallow deep learning models. In this paper, we propose U-shaped Neural Operator (U-NO), a U-shaped memory enhanced architecture that allows for deeper neural operators. U-NOs exploit the problem structures in function predictions and demonstrate fast training, data efficiency, and robustness with respect to hyperparameters choices. We study the performance of U-NO on PDE benchmarks, namely, Darcy's flow law and the Navier-Stokes equations. We show that U-NO results in an average of 14% and 34% prediction improvement on Darcy's flow and turbulent Navier-Stokes equations, respectively, over the state of art. On Navier-Stokes 3D spatio-temporal operator learning task, we show U-NO provides 40% improvement over the state of art methods.
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