Research Papers: D3 Methods

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

[+] Author and Article Information
Wei Chen

Department of Mechanical Engineering,
University of Maryland,
College Park, MD 20742
e-mail: wchen459@umd.edu

Mark Fuge

Department of Mechanical Engineering,
University of Maryland,
College Park, MD 20742
e-mail: fuge@umd.edu

1Corresponding author.

Contributed by the Design Theory and Methodology Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 20, 2017; final manuscript received June 16, 2017; published online October 2, 2017. Assoc. Editor: Harrison M. Kim.

J. Mech. Des 139(11), 111405 (Oct 02, 2017) (10 pages) Paper No: MD-17-1152; doi: 10.1115/1.4037306 History: Received February 20, 2017; Revised June 16, 2017

To solve a design problem, sometimes it is necessary to identify the feasible design space. For design spaces with implicit constraints, sampling methods are usually used. These methods typically bound the design space; that is, limit the range of design variables. But bounds that are too small will fail to cover all possible designs, while bounds that are too large will waste sampling budget. This paper tries to solve the problem of efficiently discovering (possibly disconnected) feasible domains in an unbounded design space. We propose a data-driven adaptive sampling technique—ε-margin sampling, which learns the domain boundary of feasible designs and also expands our knowledge on the design space as available budget increases. This technique is data-efficient, in that it makes principled probabilistic trade-offs between refining existing domain boundaries versus expanding the design space. We demonstrate that this method can better identify feasible domains on standard test functions compared to both random and active sampling (via uncertainty sampling). However, a fundamental problem when applying adaptive sampling to real world designs is that designs often have high dimensionality and thus require (in the worst case) exponentially more samples per dimension. We show how coupling design manifolds with ε-margin sampling allows us to actively expand high-dimensional design spaces without incurring this exponential penalty. We demonstrate this on real-world examples of glassware and bottle design, where our method discovers designs that have different appearance and functionality from its initial design set.

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Fig. 1

The probability density function of the latent function f. The shaded areas represent the probability of a sample being labeled as the opposite label with some degree of certainty (controlled by the margin ε). When the predicted label ŷ=1, the probability is P(f < −ε); and when ŷ=−1, the probability is P(f > ε).

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Fig. 2

The value of pε under different ε. On the left plot, the gray area is the ground truth of the feasible domain; the thicker line is the decision boundary obtained by the GP classifier; the thinner lines are isocontours of V; and the circle points are training samples. When ε = 0, pε = 0.5 for all the points on the decision boundary, thus in this case ε-margin sampling is equivalent to uncertainty sampling. As ε increases, ε-margin sampling starts to take the variance into consideration, i.e., given two points (e.g., points 1 and 3) on the decision boundary, it will pick the one with a higher variance. Whereas for points having the same variance (e.g., points 2 and 3), it always prefers the one on the decision boundary.

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Fig. 3

The flexible pool boundary. In each active learning iteration, we set the pool boundary by expanding the bounding box of the current labeled samples by a constant α.

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Fig. 4

Queries at the exploitation stage (left) and the exploration stage (right). The gray area is the ground truth of the feasible domain. The solid line is the decision boundary obtained by the GP classifier; and the dashed line is the isocontour of pε. At the exploitation stage, the center c is the last queried positive sample, which makes the next query stay along the decision boundary. At the exploration stage, c is the centroid of the initial positive samples, which keeps the queries centered around the existing (real-world) samples rather than biasing toward some direction.

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Fig. 5

Three-dimensional visualization of high-dimensional design space showing that design variables actually lie on a two-dimensional manifold [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.

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Fig. 6

Domain expansion process for the Branin example. The points are queried samples before the current iteration; and the stars are current queries. The solid lines are decision boundaries obtained by the GP classifier; and the dashed lines are the pool boundaries.

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Fig. 7

F1 scores for the Branin example. For ε-margin sampling, during exploration stages, the exploited decision boundaries do not change, so does the F1 score; while during rest of the time (exploitation stages), the F1 score shows a fluctuant increase as the decision boundary changes.

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Fig. 8

Queried samples for random sampling and uncertainty sampling within 210 iterations. The solid lines are decision boundaries obtained by the GP classifier; and the dashed lines are the pool boundaries. (a) Random sampling and (b) uncertainty sampling.

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Fig. 9

Queried samples by ε-margin sampling with different GP kernel length scales. A large length scale accelerates exploration but may miss small feasible domains. (a) Length scale l = 0.5 and (b) length scale l = 1.0.

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Fig. 10

F1 scores for the Hosaki example. With a larger length scale, the F1 score increases faster, but may stop increasing because it cannot discover small feasible domains.

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Fig. 11

F1 scores for the two-sphere example

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Fig. 12

F1 scores for the airfoil example

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Fig. 13

The estimated feasible domains and corresponding valid airfoil designs. The top left figure shows the initial and queried samples and the estimated feasible domains in the embedding space F. The solid dots represent valid designs, while the hollow dots represent invalid ones. The bottom figure shows the airfoil designs in the estimated feasible domain.

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Fig. 14

Some of the initial designs used in the stemless glassware example

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Fig. 15

The discovered feasible domain and valid designs. The top figure shows the initial and queried samples and the estimated feasible domain in the embedding space F The solid dots represent valid designs, while the hollow dots represent invalid ones. Started with the stemless glasses shown in Fig. 14, the proposed method discovered other types of revolved objects such as vases and bowls.

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Fig. 16

Some of the initial designs used in the bottle example

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Fig. 17

The discovered feasible domains and valid designs. The solid dots represent valid designs, while the hollow dots represent invalid ones. Started with the bottles shown in Fig. 16, the proposed method discovered two feasible domains, between which there are designs with self-intersecting contours.



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