Learning to correct spectral methods for simulating turbulent flows
Abstract: 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 ($\xspace2-4\times$) than standard spectral solvers at the same resolution but have longer overall runtimes ($\xspace{\sim 2\times}$), 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.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: Thanks again to the reviewers for their feedback and their patience in waiting for this updated manuscript. We addressed almost all issues *except* for a few points, below.
> page 3: equation 2: in the vector of K+1 frequencies, shouldn't the subscript be 0 in the middle instead of k? [ZMNQ]
We could include a zero index, but we find it more useful to define the k-th index at this time, since it is used in the main text of the paper.
> Could there be any 3-D problems included in the experiments? Or it is not applicable/necessary? [ziVT]
This is an interesting direction for future work. Spectral methods are often used for high-resolution 3D problems. The goal of such an experiment would be to compare to the state-of-the-art in large-scale numerical simulation, which is beyond our scope.
> One of the bottlenecks for spectral methods is the CFL condition. It is an interesting question whether ML can stretch the stability region of the spectral method. The reviewer suggests including additional experiments to test the hybrid ML model for the longer-that-allowed time steps. As other publications indicate, ML approaches often allow for improved stability region. [6u4j]
Unfortunately, if we were to try this, the physics component of our model (“Physics-Step”) would quickly become unstable. This would make training impossible. Indeed, your observation is correct and this may be an interesting direction for future work: Can we slowly increase the step size of hybrid models to overcome the traditional CFL bottleneck of spectral methods?
Assigned Action Editor: ~Ivan_Oseledets1
Submission Number: 227
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