Machine learning (ML) methods, which fit to data the parameters of a given
parameterized model class, have garnered significant interest as potential
methods for learning surrogate models for complex engineering systems for which
traditional simulation is expensive. However, in many scientific and
engineering settings, generating high-fidelity data on which to train ML models
is expensive, and the available budget for generating training data is limited.
ML models trained on the resulting scarce high-fidelity data have high variance
and are sensitive to vagaries of the training data set. We propose a new
multifidelity training approach for scientific machine learning that exploits
the scientific context where data of varying fidelities and costs are
available; for example high-fidelity data may be generated by an expensive
fully resolved physics simulation whereas lower-fidelity data may arise from a
cheaper model based on simplifying assumptions. We use the multifidelity data
to define new multifidelity Monte Carlo estimators for the unknown parameters
of linear regression models, and provide theoretical analyses that guarantee
the approach's accuracy and improved robustness to small training budgets.
Numerical results verify the theoretical analysis and demonstrate that
multifidelity learned models trained on scarce high-fidelity data and
additional low-fidelity data achieve order-of-magnitude lower model variance
than standard models trained on only high-fidelity data of comparable cost.
This illustrates that in the scarce data regime, our multifidelity training
strategy yields models with lower expected error than standard training
approaches.

在科学机器学习中，我们提出了一种新的多保真度训练方法，利用可用的不同保真度和成本的数据，通过多保真度数据定义线性回归模型的未知参数的新的多保真度蒙特卡罗估计器，并提供理论分析以保证该方法的准确性和对较小训练预算的改进鲁棒性。数值结果验证了理论分析，并表明与仅具有相似成本的高保真度数据训练的标准模型相比，我们用稀缺高保真度数据和额外低保真度数据训练的多保真度学习模型具有数量级更低的模型方差，说明在稀缺数据环境下，我们的多保真度训练策略能够产生具有较低期望误差的模型。