Operator learning frameworks, because of their ability to learn nonlinear maps between two infinite dimensional functional spaces and utilization of neural networks in doing so, have recently emerged as one of the more pertinent areas in the field of applied machine learning. Although these frameworks are extremely capable when it comes to modeling complex phenomena, they require an extensive amount of data for successful training which is often not available or is too expensive. However, this issue can be alleviated with the use of multi-fidelity learning, where a model is trained by making use of a large amount of inexpensive low-fidelity data along with a small amount of expensive high-fidelity data. To this end, we develop a new framework based on the wavelet neural operator which is capable of learning from a multi-fidelity dataset. The developed model's excellent learning capabilities are demonstrated by solving different problems which require effective correlation learning between the two fidelities for surrogate construction. Furthermore, we also assess the application of the developed framework for uncertainty quantification. The results obtained from this work illustrate the excellent performance of the proposed framework.

翻译：算子学习框架因其能学习两个无限维函数空间之间的非线性映射并利用神经网络实现这一过程，近期已成为应用机器学习领域中最为重要的方向之一。尽管这些框架在模拟复杂现象方面极具能力，但其成功训练需要大量数据，而这些数据往往难以获取或代价过高。然而，这一问题可通过多保真度学习得到缓解——该方法的模型训练利用大量廉价的低保真度数据和少量昂贵的高保真度数据。为此，我们基于小波神经算子开发了一种新框架，使其能够从多保真度数据集中进行学习。通过求解多个需要有效学习两种保真度相关性以构建代理模型的问题，验证了所开发模型卓越的学习能力。此外，我们还评估了该框架在不确定性量化中的应用。本研究结果表明，所提出的框架具有优异性能。