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 methods 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 26% and 44% prediction improvement on Darcy's flow and turbulent Navier-Stokes equations, respectively, over the state of the art. On Navier-Stokes 3D spatiotemporal operator learning task, we show U-NO provides 37% improvement over the state of art methods.

翻译：神经算子将经典神经网络推广到无限维空间（如函数空间）之间的映射。先前关于神经算子的工作提出了一系列学习此类映射的新方法，并在求解偏微分方程的解算子方面取得了前所未有的成功。由于这些模型与全连接架构高度相似，它们主要面临内存占用高的问题，通常局限于浅层深度学习模型。本文提出U形神经算子（U-NO），一种U形记忆增强架构，可实现更深层的神经算子。U-NO利用函数预测中的问题结构，展现出训练速度快、数据高效以及对超参数选择具有鲁棒性的特点。我们在偏微分方程基准测试（即达西流动定律和纳维-斯托克斯方程）上研究U-NO的性能。结果表明，与现有最优方法相比，U-NO在达西流动和湍流纳维-斯托克斯方程上的预测改进平均分别达26%和44%。在三维时空纳维-斯托克斯算子学习任务中，U-NO相比现有最优方法实现了37%的改进。