Physics-informed deep operator networks (DeepONets) have emerged as a promising approach toward numerically approximating the solution of partial differential equations (PDEs). In this work, we aim to develop further understanding of what is being learned by physics-informed DeepONets by assessing the universality of the extracted basis functions and demonstrating their potential toward model reduction with spectral methods. Results provide clarity about measuring the performance of a physics-informed DeepONet through the decays of singular values and expansion coefficients. In addition, we propose a transfer learning approach for improving training for physics-informed DeepONets between parameters of the same PDE as well as across different, but related, PDEs where these models struggle to train well. This approach results in significant error reduction and learned basis functions that are more effective in representing the solution of a PDE.

翻译：物理信息深度算子网络已成为数值逼近偏微分方程解的一种有前景的方法。本研究旨在通过评估所提取基函数的普适性并展示其在谱方法模型降维方面的潜力，进一步理解物理信息深度算子网络的学习机制。研究结果通过奇异值衰减和展开系数的变化规律，为衡量物理信息深度算子网络的性能提供了清晰依据。此外，我们提出了一种迁移学习方法，用于改进物理信息深度算子网络在相同偏微分方程参数间以及在不同但相关的偏微分方程间的训练效果——这些场景下模型往往难以获得良好训练。该方法显著降低了误差，并获得了能更有效表示偏微分方程解的基函数。