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Structured CNN designed using the prior information of problems potentiallyimproves efficiency over conventional CNNs in various tasks in solving PDEs andinverse problems in signal processing. This paper introduces BNet2, asimplified Butterfly-Net and inline with the conventional CNN. Moreover, aFourier transform initialization is proposed for both BNet2 and CNN withguaranteed approximation power to represent the Fourier transform operator.Experimentally, BNet2 and the Fourier transform initialization strategy aretested on various tasks, including approximating Fourier transform operator,end-to-end solvers of linear and nonlinear PDEs, and denoising and deblurringof 1D signals. On all tasks, under the same initialization, BNet2 achievessimilar accuracy as CNN but has fewer parameters. And Fourier transforminitialized BNet2 and CNN consistently improve the training and testingaccuracy over the randomly initialized CNN.

Butterfly-Net2: Simplified Butterfly-Net and Fourier Transform Initialization