  We present a phase autoencoder that encodes the asymptotic phase of a
limit-cycle oscillator, a fundamental quantity characterizing its
synchronization dynamics. This autoencoder is trained in such a way that its
latent variables directly represent the asymptotic phase of the oscillator. The
trained autoencoder can perform two functions without relying on the
mathematical model of the oscillator: first, it can evaluate the asymptotic
phase and phase sensitivity function of the oscillator; second, it can
reconstruct the oscillator state on the limit cycle in the original space from
the phase value as an input. Using several examples of limit-cycle oscillators,
we demonstrate that the asymptotic phase and phase sensitivity function can be
estimated only from time-series data by the trained autoencoder. We also
present a simple method for globally synchronizing two oscillators as an
application of the trained autoencoder.
