Spatio-temporal process models are often used for modeling dynamic physical and biological phenomena that evolve across space and time. These phenomena may exhibit environmental heterogeneity and complex interactions that are difficult to capture using traditional statistical process models such as Gaussian processes. This work proposes the use of Fourier neural operators (FNOs) for constructing statistical dynamical spatio-temporal models for forecasting. An FNO is a flexible mapping of functions that approximates the solution operator of possibly unknown linear or non-linear partial differential equations (PDEs) in a computationally efficient manner. It does so using samples of inputs and their respective outputs, and hence explicit knowledge of the underlying PDE is not required. Through simulations from a nonlinear PDE with known solution, we compare FNO forecasts to those from state-of-the-art statistical spatio-temporal-forecasting methods. Further, using sea surface temperature data over the Atlantic Ocean and precipitation data across Europe, we demonstrate the ability of FNO-based dynamic spatio-temporal (DST) statistical modeling to capture complex real-world spatio-temporal dependencies. Using collections of testing instances, we show that the FNO-DST forecasts are accurate with valid uncertainty quantification.

翻译：时空过程模型常用于对在空间和时间上演化的动态物理与生物现象进行建模。这些现象可能表现出环境异质性和复杂的相互作用，难以使用高斯过程等传统统计过程模型来捕捉。本研究提出使用傅里叶神经算子构建用于预测的统计动态时空模型。FNO 是一种灵活的函数映射，能以计算高效的方式逼近可能未知的线性或非线性偏微分方程的解算子。它利用输入样本及其相应输出来实现这一点，因此不需要底层 PDE 的显式知识。通过从具有已知解的非线性 PDE 进行模拟，我们将 FNO 的预测结果与最先进的统计时空预测方法进行了比较。此外，利用大西洋的海表温度数据和欧洲的降水数据，我们证明了基于 FNO 的动态时空统计建模能够捕捉复杂的现实世界时空依赖性。通过使用测试实例集合，我们表明 FNO-DST 预测准确，且具有有效的不确定性量化。