Deep Learning for Koopman Operator Estimation in Idealized Atmospheric Dynamics

TL;DR

This study aims to identify the limitations of existing methods, refine these models to overcome various bottlenecks, and introduce novel convolutional neural network architectures that capture simplified dynamics.

Abstract

Deep learning is revolutionizing weather forecasting, with new data-driven models achieving accuracy on par with operational physical models for medium-term predictions. However, these models often lack interpretability, making their underlying dynamics difficult to understand and explain. This paper proposes methodologies to estimate the Koopman operator, providing a linear representation of complex nonlinear dynamics to enhance the transparency of data-driven models. Despite its potential, applying the Koopman operator to large-scale problems, such as atmospheric modeling, remains challenging. This study aims to identify the limitations of existing methods, refine these models to overcome various bottlenecks, and introduce novel convolutional neural network architectures that capture simplified dynamics.

Figures (6)

• Figure 1: On the left are the un-transformed variables density, horizontal velocity, vertical velocity, and temperature, respectively. On the right are the same four variables but transformed, except density. To transform the variables, the density values are divided out of each respective variable. It is important to note that the density variable is particularly significant. The density variable defines the structure of the bubble, while the temperature variable describes the heat distribution within that structure. Thus, the temperature variable reflects the heatmap of the system, with the density providing the underlying form.
• Figure 2: Reconstructions from a preliminary fully-connected model, inspired by Lusch2018. The model is particularly limited by rapid dimensionality reduction when dealing with high-dimensional input data. It also fails to capture meaningful dynamics, resulting in crude estimates.
• Figure 3: Reconstructions from a preliminary full-CNN model, inspired by Xiao_2023. The model is hindered by the complexity of the data, resulting in significant information loss. While it achieves a rough approximation of the data, it encounters challenges with the density and temperature variables.
• Figure 4: The novel CNN-AE architecture. On the left is the encoder, which consists of transparent blocks representing residual blocks and red blocks denoting max pooling layers. In the middle is the latent space. To reconstruct the input, the output from the first latent space bypasses $\mathbf{K}^m$ and is directly passed to the subsequent latent space. To advance the state, the output from the first latent space is passed through $\mathbf{K}^m$ before being fed into the next latent space. On the right is the decoder, which mirrors the encoder with transparent blocks representing residual blocks and blue blocks indicating transposed convolutions.
• Figure 5: Partial-CNN autoencoder reconstruction and prediction of an instance from the validation set.