Review of multi-fidelity models

Figure 1.  Connection between high-fidelity and low-fidelity models is commonly attributed to one or more of the following factors: dimensionality reduction, grid coarsening, linearization, partial convergence, reduced geometry complexity, and simplified physics

Figure 2.  Within the frame of multi-fidelity modeling, surrogate models are commonly used to integrate information from different fidelities. When constructing a surrogate model that combines fidelities explicitly, such as co-Kriging, the resulting approach is referred to as a multi-fidelity surrogate model. In contrast, multi-fidelity hierarchical models combine fidelities without requiring to build an explicit multi-fidelity surrogate model architecture. Methods such as importance sampling fall under the multi-fidelity hierarchical category

Figure 3.  Proportions of different attributes found in the reviewed multi-fidelity literature [78]

Figure 4.  Main differences between fidelities found in the literature

Figure 5.  Prevalence of multi-fidelity surrogate models over multi-fidelity hierarchical models in surveyed literature up to 2016. Recent trends suggest a growing shift towards hierarchical models due to advancements in computing and algorithms

Figure 6.  Multi-fidelity surrogate models' parameters are inferred utilizing either deterministic or non-deterministic methodologies contingent upon the underlying presumptions of the unknown parameters

Figure 7.  Proportion of deterministic and non-deterministic methods utilized for constructing multi-fidelity surrogate models based on the literature. The chart also displays the distribution of the combination methods introduced in Section 4.2 within each category, deterministic and non-deterministic methods

Figure 8.  Frequency of publication over time for deterministic and non-deterministic methods

Figure 9.  The cost ratio for a single analysis using a low-fidelity model compared to a high-fidelity model (known as the LFA/HFA cost ratio) is measured against the cost ratio for completing an optimization process with a multi-fidelity surrogate model versus a high-fidelity model (termed the MFO/HFO cost ratio). The dashed line in the graph serves as a threshold; points below this line suggest that using multi-fidelity models does not result in speed-ups

Figure 10.  Examples of sampling strategies

Figure 11.  Example of a nested sampling design, where the teal-colored bubbles represent LFM points, and the pink-colored bubbles represent HFM points selected using the D-optimal design criterion

Figure 12.  Nearest neighbor sampling. High-fidelity model points (teal bubbles) and low-fidelity model points (pink bubbles) are sampled independently, and then the low-fidelity model nearest neighbor point to each high-fidelity model point is moved on top of it (black bubbles)

Figure 13.  One-dimensional analytic example illustrating the performance of additive and multiplicative correction approach

Figure 14.  One-dimensional analytic example illustrating the concept of comprehensive correction [242]

Figure 15.  Effect of the LFM constants A, B, C on co-Kriging performance. The variables were modified one at a time, setting the ones not being considered to their default value

Figure 16.  One-dimensional analytic example illustrating the concept of comprehensive correction [242]

Figure 17.  One-dimensional co-Kriging example. [242]

Figure 18.  Co-Kriging sensitivity to initial sampling points

Figure 19.  Predictive landscape of the Forrester function. Gray regions delineate areas falling within one standard deviation

Figure 20.  Comparative performance of Branin function predictions employing both additive and multiplicative corrections. The associated MAPE is an evaluative metric, signaling comparable efficacy between the approaches