Oceanographers are interested in predicting ocean currents and identifying divergences in a current vector field based on sparse observations of buoy velocities. Since we expect current dynamics to be smooth but highly non-linear, Gaussian processes (GPs) offer an attractive model. But we show that applying a GP with a standard stationary kernel directly to buoy data can struggle at both current prediction and divergence identification -- due to some physically unrealistic prior assumptions. To better reflect known physical properties of currents, we propose to instead put a standard stationary kernel on the divergence and curl-free components of a vector field obtained through a Helmholtz decomposition. We show that, because this decomposition relates to the original vector field just via mixed partial derivatives, we can still perform inference given the original data with only a small constant multiple of additional computational expense. We illustrate the benefits of our method on synthetic and real ocean data.
翻译:海洋学家对预测洋流和根据浮标速度的微弱观测确定当前矢量场的差异感兴趣。由于我们期望目前的动态是平滑的,但高度非线性,高斯进程提供了一种吸引人的模型。但我们表明,由于一些物理上不切实际的先前假设,应用带有标准固定内核的GP直接用于浮标数据的浮标数据,在目前的预测和差异识别两方面都可能难以做到。为了更好地反映已知的海流物理特性,我们提议对通过Helmholtz脱形法获得的矢量场的偏差和无曲线组成部分设置一个标准固定内核。我们表明,由于这种分解仅仅通过混合部分衍生物与原始矢量场有关,因此我们仍可以进行推断,因为原始数据只有少量的固定多倍的额外计算费用。我们展示了我们在合成和真实海洋数据上的方法的好处。