  Accurate long-term predictions are the foundations for many machine learning
applications and decision-making processes. However, building accurate
long-term prediction models remains challenging due to the limitations of
existing temporal models like recurrent neural networks (RNNs), as they capture
only the statistical connections in the training data and may fail to learn the
underlying dynamics of the target system. To tackle this challenge, we propose
a novel machine learning model based on Koopman operator theory, which we call
Koopman Invertible Autoencoders (KIA), that captures the inherent
characteristic of the system by modeling both forward and backward dynamics in
the infinite-dimensional Hilbert space. This enables us to efficiently learn
low-dimensional representations, resulting in more accurate predictions of
long-term system behavior. Moreover, our method's invertibility design
guarantees reversibility and consistency in both forward and inverse
operations. We illustrate the utility of KIA on pendulum and climate datasets,
demonstrating 300% improvements in long-term prediction capability for pendulum
while maintaining robustness against noise. Additionally, our method excels in
long-term climate prediction, further validating our method's effectiveness.
