小波神经算子：参数偏微分方程的神经算子
Wavelet neural operator: a neural operator for parametric partial differential equations
摘要：
随着传感器技术和物联网的巨大进步，我们现在可以访问TB级的历史数据;然而，如何最好地利用这些数据来预测未来事件缺乏明确性。在这种情况下，一种可能的替代方案是利用算子学习算法，直接学习两个函数空间之间的非线性映射;这有助于实时预测自然产生的复杂进化动力学。在这项工作中，我们介绍了一种新的算子学习算法称为小波神经算子（WNO），融合积分核与小波变换。WNO利用小波在函数的时频局部化方面的优势，并能够在空间域中精确跟踪模式和有效学习函数映射。由于小波在时间/空间和频率上都是局部化的，WNO可以提供高的空间和频率分辨率。这提供了在复杂问题的解决方案中学习参数依赖关系的更精细细节。所提出的WNO的有效性和鲁棒性在涉及Burger方程、Darcy流、纳维-斯托克斯方程、Allen-Cahn方程和波浪平流方程的一系列问题上得到了说明。与现有的运营商学习框架的比较研究。最后，所提出的方法被用来建立一个数字孪生模型，能够根据现有的历史数据预测地球的空气温度。
英文摘要：
With massive advancements in sensor technologies and Internet-of-things, we now have access to terabytes of historical data; however, there is a lack of clarity in how to best exploit the data to predict future events. One possible alternative in this context is to utilize operator learning algorithm that directly learn nonlinear mapping between two functional spaces; this facilitates real-time prediction of naturally arising complex evolutionary dynamics. In this work, we introduce a novel operator learning algorithm referred to as the Wavelet Neural Operator (WNO) that blends integral kernel with wavelet transformation. WNO harnesses the superiority of the wavelets in time-frequency localization of the functions and enables accurate tracking of patterns in spatial domain and effective learning of the functional mappings. Since the wavelets are localized in both time/space and frequency, WNO can provide high spatial and frequency resolution. This offers learning of the finer details of the parametric dependencies in the solution for complex problems. The efficacy and robustness of the proposed WNO are illustrated on a wide array of problems involving Burger's equation, Darcy flow, Navier-Stokes equation, Allen-Cahn equation, and Wave advection equation. Comparative study with respect to existing operator learning frameworks are presented. Finally, the proposed approach is used to build a digital twin capable of predicting Earth's air temperature based on available historical data.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
