We propose the use of physics-informed neural networks for solving the shallow-water equations on the sphere in the meteorological context. Physics-informed neural networks are trained to satisfy the differential equations along with the prescribed initial and boundary data, and thus can be seen as an alternative approach to solving differential equations compared to traditional numerical approaches such as finite difference, finite volume or spectral methods. We discuss the training difficulties of physics-informed neural networks for the shallow-water equations on the sphere and propose a simple multi-model approach to tackle test cases of comparatively long time intervals. Here we train a sequence of neural networks instead of a single neural network for the entire integration interval. We also avoid the use of a boundary value loss by encoding the boundary conditions in a custom neural network layer. We illustrate the abilities of the method by solving the most prominent test cases proposed by Williamson et al. [J. Comput. Phys. 102 (1992), 211-224].
翻译:我们提议利用物理-知情神经网络解决气象领域浅水方程式问题。物理-知情神经网络经过培训,与规定的初始和边界数据一起满足差异方程式,因此可以被视为解决差异方程式的替代方法,与传统数字方法相比,如有限差异、有限体积或光谱方法。我们讨论了物理-知情神经网络在空间浅水方程式方面的培训困难,并提出了解决相对较长间隔的测试案例的简单多模式方法。我们在这里培训了神经网络序列,而不是整个整合间隔的单一神经网络。我们还避免通过在定制神经网络层对边界条件进行编码而造成边界值损失。我们通过解决Williamson等人等人提出的最突出的测试案例来说明这一方法的能力。[J.Comput. Phys. 102(1992), 211-224]。我们通过解决Williamson 等人提出的最突出的测试案例来说明这一方法的能力。[J. Comput. Phys. 102(1992), 211-224]。