  Despite their ubiquity throughout science and engineering, only a handful of
partial differential equations (PDEs) have analytical, or closed-form
solutions. This motivates a vast amount of classical work on numerical
simulation of PDEs and more recently, a whirlwind of research into data-driven
techniques leveraging machine learning (ML). A recent line of work indicates
that a hybrid of classical numerical techniques and machine learning can offer
significant improvements over either approach alone. In this work, we show that
the choice of the numerical scheme is crucial when incorporating physics-based
priors. We build upon Fourier-based spectral methods, which are known to be
more efficient than other numerical schemes for simulating PDEs with smooth and
periodic solutions. Specifically, we develop ML-augmented spectral solvers for
three common PDEs of fluid dynamics. Our models are more accurate (2-4x) than
standard spectral solvers at the same resolution but have longer overall
runtimes (~2x), due to the additional runtime cost of the neural network
component. We also demonstrate a handful of key design principles for combining
machine learning and numerical methods for solving PDEs.
