  Deep neural operators can learn operators mapping between
infinite-dimensional function spaces via deep neural networks and have become
an emerging paradigm of scientific machine learning. However, training neural
operators usually requires a large amount of high-fidelity data, which is often
difficult to obtain in real engineering problems. Here, we address this
challenge by using multifidelity learning, i.e., learning from multifidelity
datasets. We develop a multifidelity neural operator based on a deep operator
network (DeepONet). A multifidelity DeepONet includes two standard DeepONets
coupled by residual learning and input augmentation. Multifidelity DeepONet
significantly reduces the required amount of high-fidelity data and achieves
one order of magnitude smaller error when using the same amount of
high-fidelity data. We apply a multifidelity DeepONet to learn the phonon
Boltzmann transport equation (BTE), a framework to compute nanoscale heat
transport. By combining a trained multifidelity DeepONet with genetic algorithm
or topology optimization, we demonstrate a fast solver for the inverse design
of BTE problems.
