Learning to correct spectral methods for simulating turbulent flows

Kavli Affiliate: Michael P. Brenner

| First 5 Authors: Gideon Dresdner, Dmitrii Kochkov, Peter Norgaard, Leonardo Zepeda-Núñez, Jamie A. Smith

| Summary:

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 with 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 considerably
more efficient than other numerical schemes for simulating PDEs with smooth and
periodic solutions. Specifically, we develop ML-augmented spectral solvers for
three model PDEs of fluid dynamics, which improve upon the accuracy of standard
spectral solvers at the same resolution. We also demonstrate a handful of key
design principles for combining machine learning and numerical methods for
solving PDEs.

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