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Manifold learning for the emulation of spatial fields from computational models

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Figure

Figure 1: Illustration of the pre-image method for data lying on a conical spiral. 500 pointswere randomly sampled from the spiral, shown in Fig
Figure 2: Temperature boundary conditions for the free-convection example. δ is a variablethat represents the relative length of a boundary segment and goes from 0 to 1 along thesegment as x2 increases.The cut-off shown by the horizontal dash along x1 = 10 cm islocated at x2 = 1 cm.
Figure 3: Tukey box plots of the relative error ||example using Algorithm 3 with increasing approximate manifold dimensionpoints for: (a) kPCA with 40 training points; (b) diffusion maps with 40 training points; (c)y(i)p− y(i)||2/||y(i)||2 in the free-convection r on the 300 testkPCA with 120 training points; (d) diffusion maps with 120 training points.
Figure 4: Predictions of the velocity field using 120 training points and r10 = 5 coefficientsin the free-convection example.Figure (a) is the test point corresponding to ξξξ = (3.18 ×−9[K−1], 56.7[oC])T , while Figs
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