## 5 Application
### Results  
For the target application of optimal manufacturing design, we leveraged dimensionality reduction to gain insights into the morphological data. First, we aimed to discover the latent variables governing the associated dynamic process. Second, we aimed to unravel the topology of the manifold in order to explore the input parameter space and ultimately identify the manufacturing variant that leads to the desired morphology.  
Using Entropy-Isomap, we performed the analysis for a set of morphological pathways. In Figures 8 and 9, we depict the discovered three-dimensional manifold for the set of 16 pathways. In the discovered manifold, the individual pathways are ordered according to process variables that were varied to generate the data. In each figure, the same manifold is depicted eight times. For easier inspection, each variant is individually color coded. The panels in the top row of Figure 8 depict the pathways for fixed \(\phi\) and varying \(\chi\). We observe that the pathways for increasing \(\phi\) are ordered from front to back. The panels in the bottom row of Figure 8 highlight the pathways for increasing \(\chi\). Similarly to the top row, we observe the pathways to be ordered from right to left.  
Figure 8 depicts the manifold for the early stages of morphology evolution, while Figure 9 depicts the manifold for a longer evolution time. However, in both cases, the observed ordering is consistent. The pathways for increasing \(\chi\) are ordered from the right (dark) to the left (light), while the pathways for increasing \(\phi\) are ordered from the front (green) to back (blue).  
Figure 8: The manifold of the early stage of the morphology evolution with the first 30 points per trajectory. To better illustrate the discovered ordering by two variables, we color coded the same manifold according to increasing \(\phi\) (**top**) and \(\chi\) (**bottom**). (Please view in color).  
The observed ordering of pathways strongly indicates that the input variables are also the latent variables controlling the dynamics of the process. The discovered manifold also reveals that denser sampling is required along the blend ratio space variable (\(\phi\)). Specifically, the pathways sharing the same \(\chi\) but varying \(\phi\) are spread further apart on the manifold than those sharing the same \(\chi\) value. This observation has important implications for the exploration of the design space. In particular, adding more sampling points in the \(\phi\) space offers higher exploration benefits, while adding more points in the \(\chi\) space improves exploitation chance. This indicates that the \(\phi\) space should be explored first, followed by a potential exploitation phase.  
Finally, using Entropy-Isomap, we identified two regimes in the manifold. Morphologies in the early stages of the process are mapped to evolve in the radial direction, while morphologies in late stages are mapped to evolve parallel to each other. This is interesting as the underlying process indeed has two inherent time scales. In the early stage, the fast and dynamic phase separation between the two polymers occurs. During this stage, the composition of the individual phases changes significantly. These changes mostly increase composition amplitude. In the second stage, the equilibrium composition is already established. The coarsening between already formed domains dominates the dynamics of the process. Here, the amplitude of the composition (signal) does not change significantly. The changes mostly occur in the frequency space of the domain size, with the domain sizes increasing over time.