Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems

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.