The study of operator learning involves the utilization of neural networks to approximate operators. Traditionally, the focus has been on single-operator learning (SOL). However, recent advances have rapidly expanded this to include the approximation of multiple operators using foundation models equipped with millions or billions of trainable parameters, leading to the research of multi-operator learning (MOL). In this paper, we present a novel distributed training approach aimed at enabling a single neural operator with significantly fewer parameters to effectively tackle multi-operator learning challenges, all without incurring additional average costs. Our method is applicable to various neural operators, such as Deep Operator Neural Networks (DON). The core idea is to independently learn the output basis functions for each operator using its dedicated data, while simultaneously centralizing the learning of the input function encoding shared by all operators using the entire dataset. Through a systematic study of five numerical examples, we compare the accuracy and cost of training a single neural operator for each operator independently versus training a MOL model using our proposed method. Our results demonstrate enhanced efficiency and satisfactory accuracy. Moreover, our approach illustrates that some operators with limited data can be more effectively constructed with the aid of data from analogous operators through MOL learning. This highlights another MOL's potential to bolster operator learning.

翻译：算子学习研究涉及利用神经网络逼近算子。传统上，研究重点集中在单算子学习（SOL）。然而，近期进展迅速扩展到利用配备数百万或数十亿可训练参数的基础模型逼近多个算子，由此催生了多算子学习（MOL）研究。本文提出一种新颖的分布式训练方法，旨在使参数显著较少的单一神经算子能够有效应对多算子学习挑战，且无需额外增加平均成本。该方法适用于各类神经算子（如深度算子神经网络DON）。其核心思想是：利用各算子的专属数据独立学习其输出基函数，同时利用完整数据集集中学习所有算子共享的输入函数编码。通过五个数值算例的系统性研究，我们比较了独立为每个算子训练单一神经算子的精度与成本，以及使用本文方法训练MOL模型的相应指标。结果表明，所提方法在实现满意精度的同时提升了效率。此外，该方法表明，通过MOL学习借助相似算子的数据，可更有效地构建部分数据受限的算子。这凸显了MOL在强化算子学习方面的另一潜力。