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Enhanced DeepONet for Modeling Partial Differential Operators Considering Multiple Input Functions

Machine learning, especially deep learning is gaining much attention due to the breakthrough performance in various cognitive applications. Recently, neural networks (NN) have been intensively explored to model partial differential equations as NN can be viewed as universal approximators for nonlinear functions. A deep network operator (DeepONet) architecture was proposed to model the general non-linear continuous operators for partial differential equations (PDE) due to its better generalization capabilities than existing mainstream deep neural network architectures. However, existing DeepONet can only accept one input function, which limits its application. In this work, we explore the DeepONet architecture to extend it to accept two or more input functions. We propose new Enhanced DeepONet or EDeepONet high-level neural network structure, in which two input functions are represented by two branch DNN sub-networks, which are then connected with output truck network via inner product to generate the output of the whole neural network. The proposed EDeepONet structure can be easily extended to deal with multiple input functions. Our numerical results on modeling two partial differential equation examples shows that the proposed enhanced DeepONet is about 7X-17X or about one order of magnitude more accurate than the fully connected neural network and is about 2X-3X more accurate than a simple extended DeepONet for both training and test.