Temporally-Consistent Koopman Autoencoders for Forecasting Dynamical
  Systems
Indranil Nayak Debdipta Goswami Mrinal Kumar Fernando Teixeira
Absence of sufficiently high-quality data often poses a key challenge in
data-driven modeling of high-dimensional spatio-temporal dynamical systems.
Koopman Autoencoders (KAEs) harness the expressivity of deep neural networks
(DNNs), the dimension reduction capabilities of autoencoders, and the spectral
properties of the Koopman operator to learn a reduced-order feature space with
simpler, linear dynamics. However, the effectiveness of KAEs is hindered by
limited and noisy training datasets, leading to poor generalizability. To
address this, we introduce the Temporally-Consistent Koopman Autoencoder
(tcKAE), designed to generate accurate long-term predictions even with
constrained and noisy training data. This is achieved through a consistency
regularization term that enforces prediction coherence across different time
steps, thus enhancing the robustness and generalizability of tcKAE over
existing models. We provide analytical justification for this approach based on
Koopman spectral theory and empirically demonstrate tcKAE's superior
performance over state-of-the-art KAE models across a variety of test cases,
including simple pendulum oscillations, kinetic plasmas, fluid flows, and sea
surface temperature data.