  Although very successfully used in conventional machine learning, convolution
based neural network architectures -- believed to be inconsistent in function
space -- have been largely ignored in the context of learning solution
operators of PDEs. Here, we present novel adaptations for convolutional neural
networks to demonstrate that they are indeed able to process functions as
inputs and outputs. The resulting architecture, termed as convolutional neural
operators (CNOs), is designed specifically to preserve its underlying
continuous nature, even when implemented in a discretized form on a computer.
We prove a universality theorem to show that CNOs can approximate operators
arising in PDEs to desired accuracy. CNOs are tested on a novel suite of
benchmarks, encompassing a diverse set of PDEs with possibly multi-scale
solutions and are observed to significantly outperform baselines, paving the
way for an alternative framework for robust and accurate operator learning. Our
code is publicly available at
https://github.com/bogdanraonic3/ConvolutionalNeuralOperator
