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 is a popular architecture that performs global convolutions in the Fourier space. However%2C such global operations are often prone to over-smoothing and may fail to capture local details. In contrast%2C convolutional neural networks (CNN) can capture local features but are limited to training and inference at a single resolution. In this work%2C we present a principled approach to operator learning that can capture local features under two frameworks by learning differential operators and integral operators with locally supported kernels. Specifically%2C inspired by stencil methods%2C we prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators%2C we utilize suitable basis representations for the kernels based on discrete-continuous convolutions. Both these approaches preserve the properties of operator learning and%2C hence%2C the ability to predict at any resolution. Adding our layers to FNOs significantly improves their performance%2C reducing the relative L2-error by 34-72%25 in our experiments%2C which include a turbulent 2D Navier-Stokes and the spherical shallow water equations.?quality=80&w=800)
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