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.

本研究旨在探讨物理信息深度算子网络（DeepONets）学习的内容，特别是评估提取的基函数的普遍性及其对模型降维的潜力。研究结果表明，通过奇异值和扩展系数的衰减来衡量DeepONets的性能，并提出了一种跨参数和相关偏微分方程（PDE）的迁移学习方法，以显著降低训练误差，提高基函数的有效性。

物理信息深度算子网络学习了什么？理解与改进科学计算应用的训练