Perdikaris P, Raissi M, Damianou A, Lawrence N D, Karniadakis G E
Department of Mechanical Engineering , Massachusetts Institute of Technology , Cambridge, MA 02139, USA.
Division of Applied Mathematics , Brown University , Providence, RI 02912, USA.
Proc Math Phys Eng Sci. 2017 Feb;473(2198):20160751. doi: 10.1098/rspa.2016.0751.
Multi-fidelity modelling enables accurate inference of quantities of interest by synergistically combining realizations of low-cost/low-fidelity models with a small set of high-fidelity observations. This is particularly effective when the low- and high-fidelity models exhibit strong correlations, and can lead to significant computational gains over approaches that solely rely on high-fidelity models. However, in many cases of practical interest, low-fidelity models can only be well correlated to their high-fidelity counterparts for a specific range of input parameters, and potentially return wrong trends and erroneous predictions if probed outside of their validity regime. Here we put forth a probabilistic framework based on Gaussian process regression and nonlinear autoregressive schemes that is capable of learning complex nonlinear and space-dependent cross-correlations between models of variable fidelity, and can effectively safeguard against low-fidelity models that provide wrong trends. This introduces a new class of multi-fidelity information fusion algorithms that provide a fundamental extension to the existing linear autoregressive methodologies, while still maintaining the same algorithmic complexity and overall computational cost. The performance of the proposed methods is tested in several benchmark problems involving both synthetic and real multi-fidelity datasets from computational fluid dynamics simulations.
多保真度建模通过将低成本/低保真度模型的实现与少量高保真观测值协同结合，实现对感兴趣量的准确推断。当低保真度模型和高保真度模型表现出强相关性时，这种方法特别有效，并且与仅依赖高保真度模型的方法相比，可以带来显著的计算收益。然而，在许多实际感兴趣的情况下，低保真度模型仅在特定的输入参数范围内才能与其高保真度对应模型具有良好的相关性，如果在其有效范围之外进行探测，可能会返回错误的趋势和错误的预测。在此，我们提出了一种基于高斯过程回归和非线性自回归方案的概率框架，该框架能够学习可变保真度模型之间复杂的非线性和空间相关的交叉相关性，并能有效防范提供错误趋势的低保真度模型。这引入了一类新的多保真度信息融合算法，它对现有的线性自回归方法进行了根本性扩展，同时仍保持相同的算法复杂度和总体计算成本。所提出方法的性能在几个基准问题中进行了测试，这些问题涉及来自计算流体动力学模拟的合成和真实多保真度数据集。
