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 $lesssim 5times 10^{-4}$ 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.
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