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.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Yannou, B. , Moreno, F. , Thevenot, H. J. , and Simpson, T. W. , 2005, “ Faster Generation of Feasible Design Points,” ASME Paper No. DETC2005-85449.
Devanathan, S. , and Ramani, K. , 2010, “ Creating Polytope Representations of Design Spaces for Visual Exploration Using Consistency Techniques,” ASME J. Mech. Des., 132(8), p. 081011. [CrossRef]
Larson, B. J. , and Mattson, C. A. , 2012, “ Design Space Exploration for Quantifying a System Model's Feasible Domain,” ASME J. Mech. Des., 134(4), p. 041010. [CrossRef]
Lee, T. H. , and Jung, J. J. , 2008, “ A Sampling Technique Enhancing Accuracy and Efficiency of Metamodel-Based RBDO: Constraint Boundary Sampling,” Comput. Struct., 86(13), pp. 1463–1476. [CrossRef]
Zhuang, X. , and Pan, R. , 2012, “ A Sequential Sampling Strategy to Improve Reliability-Based Design Optimization With Implicit Constraint Functions,” ASME J. Mech. Des., 134(2), p. 021002. [CrossRef]
Huang, Y.-C. , and Chan, K.-Y. , 2010, “ A Modified Efficient Global Optimization Algorithm for Maximal Reliability in a Probabilistic Constrained Space,” ASME J. Mech. Des., 132(6), p. 061002. [CrossRef]
Ren, Y. , and Papalambros, P. Y. , 2011, “ A Design Preference Elicitation Query as an Optimization Process,” ASME J. Mech. Des., 133(11), p. 111004. [CrossRef]
Chen, W. , Fuge, M. , and Chazan, J. , 2017, “ Design Manifolds Capture the Intrinsic Complexity and Dimension of Design Spaces,” ASME J. Mech. Des., 139(5), p. 051102. [CrossRef]
Chen, W. , Chazan, J. , and Fuge, M. , 2016, “ How Designs Differ: Non-Linear Embeddings Illuminate Intrinsic Design Complexity,” ASME Paper No. DETC2016-60112.
Regier, J. C. , and Stark, P. B. , 2015, “ Mini-Minimax Uncertainty Quantification for Emulators,” SIAM/ASA J. Uncertainty Quantif., 3(1), pp. 686–708. [CrossRef]
Rasmussen, C. , and Williams, C. , 2006, Gaussian Processes for Machine Learning, MIT Press, Cambridge, MA.
Williams, C. K. , and Barber, D. , 1998, “ Bayesian Classification With Gaussian Processes,” IEEE Trans. Pattern Anal. Mach. Intell., 20(12), pp. 1342–1351. [CrossRef]
Minka, T. P. , 2001, “ A Family of Algorithms for Approximate Bayesian Inference,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
Hastings, W. K. , 1970, “ Monte Carlo Sampling Methods Using Markov Chains and Their Applications,” Biometrika, 57(1), pp. 97–109. [CrossRef]
Settles, B. , 2009, “ Active Learning Literature Survey,” University of Wisconsin–Madison, Madison, WI, Report No. 1648.
Lewis, D. D. , and Catlett, J. , 2012, “ Heterogeneous Uncertainty Sampling for Supervised Learning,” 11th International Conference on Machine Learning and Applications (ICMLA), Boca Raton, FL, Dec. 12–15, pp. 148–156.
Lewis, D. D. , and Gale, W. A. , 1994, “ A Sequential Algorithm for Training Text Classifiers,” 17th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, Dublin, Ireland, July 3–6, pp. 3–12.
Bryan, B. , Schneider, J. , Nichol, R. , Miller, C. , Genovese, C. R. , and Wasserman, L. , 2005, “ Active Learning for Identifying Function Threshold Boundaries,” Advances in Neural Information Processing Systems 18, MIT Press, Cambridge, MA, pp. 163–170.
Kapoor, A. , Grauman, K. , Urtasun, R. , and Darrell, T. , 2007, “ Active Learning With Gaussian Processes for Object Categorization,” IEEE 11th International Conference on Computer Vision (ICCV), Rio de Janeiro, Brazil, Oct. 14–21, pp. 1–8.
Kapoor, A. , Grauman, K. , Urtasun, R. , and Darrell, T. , 2010, “ Gaussian Processes for Object Categorization,” Int. J. Comput. Vision, 88(2), pp. 169–188. [CrossRef]
Gotovos, A. , Casati, N. , Hitz, G. , and Krause, A. , 2013, “ Active Learning for Level Set Estimation,” 23rd International Joint Conference on Artificial Intelligence (IJCAI), Beijing, China, Aug. 3–9, pp. 1344–1350.
Hoang, T. N. , Low, B. , Jaillet, P. , and Kankanhalli, M. , 2014, “ Nonmyopic ε-Bayes-Optimal Active Learning of Gaussian Processes,” 31st International Conference on Machine Learning (ICML), Beijing, China, June 21–26, pp. 739–747.
Schreiter, J. , Nguyen-Tuong, D. , Eberts, M. , Bischoff, B. , Markert, H. , and Toussaint, M. , 2015, “ Safe Exploration for Active Learning With Gaussian Processes,” Joint European Conference on Machine Learning and Knowledge Discovery in Databases, Porto, Portugal, Sept. 7–11, pp. 133–149.
Freytag, A. , Rodner, E. , Bodesheim, P. , and Denzler, J. , 2013, “ Labeling Examples That Matter: Relevance-Based Active Learning With Gaussian Processes,” German Conference on Pattern Recognition (GCPR), Hannover, Germany, Sept. 12–15, pp. 282–291.
Freytag, A. , Rodner, E. , and Denzler, J. , 2014, “ Selecting Influential Examples: Active Learning With Expected Model Output Changes,” European Conference on Computer Vision (ECCV), Amsterdam, The Netherlands, Oct. 8–16, pp. 562–577.
Basudhar, A. , and Missoum, S. , 2010, “ An Improved Adaptive Sampling Scheme for the Construction of Explicit Boundaries,” Struct. Multidiscip. Optim., 42(4), pp. 517–529. [CrossRef]
Basudhar, A. , Dribusch, C. , Lacaze, S. , and Missoum, S. , 2012, “ Constrained Efficient Global Optimization With Support Vector Machines,” Struct. Multidiscip. Optim., 46(2), pp. 201–221. [CrossRef]
Singh, P. , van der Herten, J. , Deschrijver, D. , Couckuyt, I. , and Dhaene, T. , 2017, “ A Sequential Sampling Strategy for Adaptive Classification of Computationally Expensive Data,” Struct. Multidiscip. Optim., 55(4), pp. 1425–1438. [CrossRef]
Bouneffouf, D. , Laroche, R. , Urvoy, T. , Féraud, R. , and Allesiardo, R. , 2014, “ Contextual Bandit for Active Learning: Active Thompson Sampling,” International Conference on Neural Information Processing (ICONIP), Kyoto, Japan, Oct. 16–21, pp. 405–412.
Hsu, W.-N. , and Lin, H.-T. , 2015, “ Active Learning by Learning,” 29th Conference on Association for the Advancement of Artificial Intelligence (AAAI), Austin, TX, Jan. 25–30, pp. 2659–2665.
Bellman, R. , 1957, Dynamic Programming, Princeton University Press, Princeton, NJ.
Averkiou, M. , Kim, V. G. , Zheng, Y. , and Mitra, N. J. , 2014, “ ShapeSynth: Parameterizing Model Collections for Coupled Shape Exploration and Synthesis,” Comput. Graphics Forum, 33(2), pp. 125–134. [CrossRef]
Yumer, M. E. , Asente, P. , Mech, R. , and Kara, L. B. , 2015, “ Procedural Modeling Using Autoencoder Networks,” 28th Annual ACM Symposium on User Interface Software & Technology (UIST), Charlotte, NC, Nov. 11–15, pp. 109–118.
Burnap, A. , Liu, Y. , Pan, Y. , Lee, H. , Gonzalez, R. , and Papalambros, P. Y. , 2016, “ Estimating and Exploring the Product Form Design Space Using Deep Generative Models,” ASME Paper No. DETC2016-60091.
Yumer, M. E. , Chaudhuri, S. , Hodgins, J. K. , and Kara, L. B. , 2015, “ Semantic Shape Editing Using Deformation Handles,” ACM Trans. Graphics, 34(4), pp. 86:1–86:12. [CrossRef]
Van Der Maaten, L. , Postma, E. , and Van den Herik, J. , 2009, “ Dimensionality Reduction: A Comparative Review,” Tilburg University, Tilburg, The Netherlands, Technical Report No. TiCC-TR 2009-005.
Bengio, Y. , Courville, A. , and Vincent, P. , 2013, “ Representation Learning: A Review and New Perspectives,” IEEE Trans. Pattern Anal. Mach. Intell., 35(8), pp. 1798–1828. [CrossRef] [PubMed]
Rodrigues, F. , Pereira, F. , and Ribeiro, B. , 2014, “ Gaussian Process Classification and Active Learning With Multiple Annotators,” 31st International Conference on Machine Learning (ICML), Beijing, China, June 21–26, pp. 433–441.
Sharmanska, V. , Hernández-Lobato, D. , Miguel Hernandez-Lobato, J. , and Quadrianto, N. , 2016, “ Ambiguity Helps: Classification With Disagreements in Crowdsourced Annotations,” IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, June 27–30, pp. 2194–2202.


Grahic Jump Location
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).

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
Fig. 11

F1 scores for the two-sphere example

Grahic Jump Location
Fig. 12

F1 scores for the airfoil example

Grahic Jump Location
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.

Grahic Jump Location
Fig. 14

Some of the initial designs used in the stemless glassware example

Grahic Jump Location
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.

Grahic Jump Location
Fig. 16

Some of the initial designs used in the bottle example

Grahic Jump Location
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.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In