Deep operator networks (DeepONets), a class of neural operators that learn mappings between function spaces, have recently been developed as surrogate models for parametric partial differential equations (PDEs). In this work we propose a derivative-enhanced deep operator network (DE-DeepONet), which leverages the derivative information to enhance the prediction accuracy, and provide a more accurate approximation of the derivatives, especially when the training data are limited. DE-DeepONet incorporates dimension reduction of input into DeepONet and includes two types of derivative labels in the loss function for training, that is, the directional derivatives of the output function with respect to the input function and the gradient of the output function with respect to the physical domain variables. We test DE-DeepONet on three different equations with increasing complexity to demonstrate its effectiveness compared to the vanilla DeepONet.

翻译：深度算子网络（DeepONet）是一类学习函数空间之间映射关系的神经算子，近年来已被开发为参数化偏微分方程的代理模型。本文提出一种导数增强型深度算子网络（DE-DeepONet），通过利用导数信息提升预测精度，并在训练数据有限时获得更准确的导数近似。DE-DeepONet将输入降维集成到DeepONet中，并在损失函数中引入两类导数标签用于训练：输出函数关于输入函数的方向导数，以及输出函数关于物理域变量的梯度。我们在三个复杂度递增的不同方程上测试了DE-DeepONet，结果表明其性能优于原始DeepONet。