  The future space mission LISA will observe a wealth of gravitational-wave
sources at millihertz frequencies. Of these, the extreme-mass-ratio inspirals
of compact objects into massive black holes are the only sources that combine
the challenges of strong-field complexity with that of long-lived signals. Such
signals are found and characterized by comparing them against a large number of
accurate waveform templates during data analysis, but the rapid generation of
such templates is hindered by computing the $\sim10^3$-$10^5$ harmonic modes in
a fully relativistic waveform. We use order-reduction and deep-learning
techniques to derive a global fit for these modes, and implement it in a
complete waveform framework with hardware acceleration. Our high-fidelity
waveforms can be generated in under $1\,\mathrm{s}$, and achieve a mismatch of
$\lesssim 5\times 10^{-4}$ against reference waveforms that take $\gtrsim 10^4$
times longer. This marks the first time that analysis-length waveforms with
full harmonic content can be produced on timescales useful for direct
implementation in LISA analysis algorithms.
