Fast multi-fidelity Gaussian processes with derivatives for complex system modeling

Abstract Accurately obtaining physics model information is essential for comprehending the mechanisms of physical dynamics. However, the inherent complexity of these models, especially in scenarios involving turbulence and irregular terrains, often requires significant computational resources to acquire high-fidelity (HF) data. While deep learning-based approaches have demonstrated considerable potential, they generally necessitate large volumes of training data and often face challenges in efficiently integrating physical information, such as derivatives. Leveraging the concept of multi-fidelity (MF) surrogate model and Gaussian processes (GPs), we introduce a novel approach, fast MF GPs with derivatives, in which the posterior mean of the low-fidelity (LF) model replaces the prior mean of the HF model, with the relationship between the two encapsulated by an unknown function. To speed up convergence during likelihood function optimization, we also initialize the kernel of the HF model using the best hyperparameters from the LF model. This model seamlessly integrates derivative information, facilitating accurate reconstruction of general physical models by synthesizing data from multiple fidelities, including derivatives. To address computational demands, we incorporate kernel approximation techniques based on structured kernel interpolation with derivatives. The effectiveness of our proposed method is validated through three case studies: (1) solving a partial differential equation describing fluid or scalar transport in a 1D domain, (2) reconstructing a high-resolution local turbulence velocity field using the Johns Hopkins forced isotropic turbulence dataset, and (3) predicting the surface of a rugged mountainous area while finding the peak via Bayesian optimization. Comparative analysis against four other algorithms underscores the superiority of our approach across multiple fields and highlights its significant potential for engineering applications.