  FNO and DeepONet are by far the most popular neural operator learning
algorithms. FNO seems to enjoy an edge in popularity due to its ease of use,
especially with high dimensional data. However, a lesser-acknowledged feature
of DeepONet is its modularity. This feature allows the user the flexibility of
choosing the kind of neural network to be used in the trunk and/or branch of
the DeepONet. This is beneficial because it has been shown many times that
different types of problems require different kinds of network architectures
for effective learning. In this work, we will take advantage of this feature by
carefully designing a more efficient neural operator based on the DeepONet
architecture. We introduce U-Net enhanced DeepONet (U-DeepONet) for learning
the solution operator of highly complex CO2-water two-phase flow in
heterogeneous porous media. The U-DeepONet is more accurate in predicting gas
saturation and pressure buildup than the state-of-the-art U-Net based Fourier
Neural Operator (U-FNO) and the Fourier-enhanced Multiple-Input Operator
(Fourier-MIONet) trained on the same dataset. In addition, the proposed
U-DeepONet is significantly more efficient in training times than both the
U-FNO (more than 18 times faster) and the Fourier-MIONet (more than 5 times
faster), while consuming less computational resources. We also show that the
U-DeepONet is more data efficient and better at generalization than both the
U-FNO and the Fourier-MIONet.
