论文标题
U-no：U形神经操作员
U-NO: U-shaped Neural Operators
论文作者
论文摘要
神经操作员将经典神经网络推广到无限维空间之间的地图，例如功能空间。关于神经操作员的先前工作提出了一系列新的方法来学习此类地图，并在学习偏微分方程的解决方案操作员方面表现出了前所未有的成功。由于它们与完全连接的体系结构的近距离近似，这些模型主要遭受高度记忆使用的影响，并且通常仅限于浅层学习模型。在本文中，我们提出了U形神经操作员（U-NO），这是一种U形记忆增强的结构，可提供更深的神经操作员。 U-NOS在功能预测中利用问题结构，并在超参数选择方面证明了快速训练，数据效率和鲁棒性。我们研究U-NO在PDE基准测试中的性能，即达西的流量法和Navier-Stokes方程。我们表明，在最新情况下，U-NO的平均导致了对达西的流量和动荡的Navier-Stokes方程的平均预测和44％的预测。在Navier-Stokes 3D时空操作员学习任务上，我们显示U-NO可对最先进的方法提供37％的改善。
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.
