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MKNO: Multi-Kernel Neural Operator

Abstract

Neural operators learn resolution independent map- pings between functional spaces, and are a pop- ular way to generate solutions for an entire class of partial differential equations (PDE) as opposed to just one instance, leading to significant compu- tational gains. However, these methods rely on a continuous-discrete equivalence between the func- tional form and the samples, which may be vio- lated if the samples are not captured faithfully. We propose the multi-kernel neural operator (MKNO) which can capture different frequency components at varying levels of resolutions. MKNO accom- plishes this by using the Fourier kernels to capture lower frequency global information and graph ker- nels to capture more local and high frequency fea- ture information. MKNO is discretization invariant, and learns a general solution operator that can be applied to varying discretizations. To validate our architecture we apply MKNO to a number of differ- ent two dimensional PDEs.
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