In general, low-fidelity data is easier to obtain in greater quantities, but it may be too inaccurate or not dense enough to accurately train a machine learning model. High-fidelity data is costly to obtain, so there may not be sufficient data to use in training, however, it is more accurate. A small amount of high fidelity data, such as from measurements, combined with low fidelity data, can improve predictions when used together; this has motivated geophysicists to develop cokriging [1], which is based on Gaussian process regression at two different fidelity levels by exploiting correlations-albeit only linear ones - between different levels. An example of cokriging for obtaining the sea surface temperature (as well as the associated uncertainty) is presented in [2], where satellite images are used as low-fidelity data whereas in situ measurements are used as high-fidelity data. To exploit nonlinear correlations at different levels of fidelity, a probabilistic framework based on Gaussian process regression and nonlinear autoregressive scheme was proposed in [3] that can learn complex nonlinear and space-dependent cross-correlations between multifidelity models. However, the limitation of this work is the high computational cost for big data sets, and to this end, the subsequent work in [4] was based on neural networks and provided the first method of multifidelity training of deep neural networks.

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.