Low-dimensional data-driven modeling through temporally parameterized neural networks for geophysical forecasting

The dynamics of high-dimensional systems can often be described by low-dimensional models, as the presence of dissipation induces long-time dynamics to collapse onto a finite-dimensional manifold. Data-driven modeling though neural networks is a powerful tool for constructing such models, creating a nonlinear mapping between the high-dimensional state space and a low-dimensional latent space and forecasting the dynamics in the reduced dimensional state. For systems with time-dependent external forcing, there exist inherent difficulties in generating minimal dimensional models, as the data-driven models must learn the dynamics of both the system and the forcing. We overcome the need to learn the dynamics of the external forcing through the use of temporally parameterized neural networks, data-driven models that learn to supplement a predefined time-dependent parametrization of the dynamical system. Here, we apply our temporally parametrized neural networks the problem of forecasting sea surface temperatures, a system with inherent time dependent forcing due to seasonal variations from the orbit of the Earth around the Sun. We develop data-driven low-dimensional models using temporally parameterized neural networks and compare the dimension reduction and forecasting capabilities to those modeled with standard neural networks. We show that the temporally parameterized neural networks can reproduce the sea surface temperatures fields with a much smaller latent dimension than standard neural networks, while capturing the dynamics of the sea surface temperatures with comparable accuracy.