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.

翻译：尽管卷积神经网络架构在传统机器学习中取得了巨大成功，但由于被认为在函数空间上缺乏一致性，其在学习偏微分方程解算子方面长期被忽视。本文提出针对卷积神经网络的新颖改进方法，证明其能够将函数作为输入和输出进行处理。由此产生的架构——卷积神经算子（CNOs）——专门设计用于在计算机上以离散化形式实现时仍保留其潜在的连续本质。我们证明了通用性定理，表明CNOs能以期望精度逼近偏微分方程中的算子。通过在涵盖多尺度解的一系列多样化偏微分方程基准测试上进行评估，CNOs显著优于基线方法，为鲁棒且精确的算子学习开辟了新框架。