This paper launches a thorough discussion on the locality of local neural operator (LNO), which is the core that enables LNO great flexibility on varied computational domains in solving transient partial differential equations (PDEs). We investigate the locality of LNO by looking into its receptive field and receptive range, carrying a main concern about how the locality acts in LNO training and applications. In a large group of LNO training experiments for learning fluid dynamics, it is found that an initial receptive range compatible with the learning task is crucial for LNO to perform well. On the one hand, an over-small receptive range is fatal and usually leads LNO to numerical oscillation; on the other hand, an over-large receptive range hinders LNO from achieving the best accuracy. We deem rules found in this paper general when applying LNO to learn and solve transient PDEs in diverse fields. Practical examples of applying the pre-trained LNOs in flow prediction are presented to confirm the findings further. Overall, with the architecture properly designed with a compatible receptive range, the pre-trained LNO shows commendable accuracy and efficiency in solving practical cases.

翻译：本文深入探讨了局部神经算子（LNO）的局部性，这是LNO在求解瞬态偏微分方程（PDEs）时能够灵活适应不同计算域的核心特性。我们通过考察LNO的感受野和感受范围来研究其局部性，重点关注局部性在LNO训练和应用中的作用。在一系列针对流体动力学学习的LNO训练实验中，发现初始感受范围与学习任务相匹配对LNO的良好表现至关重要。一方面，过小的感受范围会导致致命问题，通常使LNO产生数值振荡；另一方面，过大的感受范围会阻碍LNO达到最佳精度。我们认为本文发现的规律在应用LNO学习和求解各类领域的瞬态PDEs时具有普遍性。通过展示预训练LNO在流场预测中的实际应用案例，进一步验证了这些发现。总体而言，通过设计具有合适感受范围的架构，预训练LNO在解决实际问题时展现出令人满意的准确性和效率。