Convolutional Neural Operators for robust and accurate learning of PDEs
This provides a robust and accurate alternative framework for operator learning in PDEs, addressing a domain-specific bottleneck in scientific computing.
The authors tackled the problem of learning solution operators for PDEs using convolutional neural networks, which were previously considered inconsistent in function space, and demonstrated that their novel convolutional neural operators (CNOs) significantly outperform baselines on a diverse set of PDE benchmarks.
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