How Designs Differ: Non-Linear Embeddings Illuminate Intrinsic Design Complexity

Abstract

This paper shows how to measure the complexity and reduce the dimensionality of a geometric design space. It assumes that high-dimensional design parameters actually lie in a much lower-dimensional space that represents semantic attributes. Past work has shown how to embed designs using techniques like autoencoders; in contrast, this paper quantifies when and how various embeddings are better than others. It captures the intrinsic dimensionality of a design space, the performance of recreating new designs for an embedding, and the preservation of topology of the original design space. We demonstrate this with both synthetic superformula shapes of varying non-linearity and real glassware designs. We evaluate multiple embeddings by measuring shape reconstruction error, topology preservation, and required semantic space dimensionality. Our work generates fundamental knowledge about the inherent complexity of a design space and how designs differ from one another. This deepens our understanding of design complexity in general.

Beyond the Known: Detecting Novel Feasible Domains Over an Unbounded Design Space

“…When ε = 0, the proposed method is equivalent to uncertainty sampling, which only samples points on the decision boundary (f = 0), i.e., exploitation. As ε increases, the [8,9]. At a point away from the real-world stemless glass samples, the glass contours are self-intersecting; at another point, the shape becomes a stem glass.…”

“…One way to get this low-dimensional representation is to use manifold learning such as Multi-Dimensional Scaling (MDS) to map the design space to a low-dimensional embedding space, and reconstruct the original shapes or synthesize new shapes based on how shapes vary [32]. Instead of learning a one-way mapping via methods like MDS, one can also directly learn a two-way mapping between the design space and a low-dimensional space by using methods like principal component analysis (PCA) or autoencoders [33,34,8,9]. Another way is to associate designs with a few semantic attributes (e.g., compactness/luxuriousness of a car) by crowd-sourcing, and then learn how those attributes map to the design variables, such that new designs can be synthesized by editing these semantic attributes [35].…”

“…To handle real-world design spaces, our proposed method assumes that design variables original expressed in a high-dimensional design space X usually lie on a lower-dimensional design manifold ( Fig. 5) [8,9,36,37]. We can thus project designs on that lower-dimensional manifold F by a mapping g : X → F .…”

“…We call these invalid designs-designs that are unrealistic or nonfunctional in the real-world. Since real-world samples are all valid designs, normally designs lying between any two real-world samples (i.e., inside the convex hull of all the real-world samples in the embedding space) will also be valid [8,9]. However, this assumption may not hold for designs that lie on the manifold but beyond the real-world samples.…”

“…In this paper, we address these problems by proposing an adaptive sampling method that picks designs (to simulate, to show to human, or otherwise validate) such that the algorithm both sufficiently refines the boundaries for feasible designs that we know about, while also expanding the design space to discover new feasible designs (possibly in a disconnected feasible domain) that we may not know about. Compared to past work, this paper differs in that 1) the sampling process is applied on an unbounded design space and discovers multiple disconnected feasible domains; 2) given an increasingly larger budget, our method will handle unbounded design spaces without sacrificing the quality of the boundaries for known feasible designs; and 3) it works for high dimensional real world design spaces by exploiting and expanding along lowdimensional Design Manifolds [8,9]. This second item, addressed in detail by Sec.…”