Learning representations by back-propagating errors.

doi: 10.1016/j.advengsoft.2019.04.002, He, K., Zhang, X., Ren, S., and Sun, J. neurips graphs learning machine medium hyperbolic embedding 2d space source tree (2013) explained that solving the quasi-convex symmetric optimization problem may yield highly asymmetric solution. Optim. (2017). Solids Struct. The topology two steps before the terminal state contains successive V-shaped braces and is stable and statically determinate. J. Mecan. The truss is optimized for two loading conditions. GA is one of the most prevalent metaheuristic approach for binary optimization problems, which is inspired by the process of natural selection (Mitchell, 1998). Minimum weight design of elastic redundant trusses under multiple static loading conditions. Optim. 32, 33413357.

Since removal of any remaining member will cause violation of the displacement constraint, there is no unnecessary member in the sub-optimal topology. This work was kindly supported by Grant-in-Aid for JSPS Research Fellow No.JP18J21456 and JSPS KAKENHI No. The proposed method for training agent is expected to become a supporting tool to instantly feedback the sub-optimal topology and enhance our design exploration. (2015). Methods Appl. From these results, the agent is confirmed to behave well for a different loading condition. Optim. embeddings 49, 554563. 47, 783794. Dorn, W. S. (1964). The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Deep learning for topology optimization design. https://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf (accessed April 23, 2020). embeddings Eng. 10, 155162. Nakamura, S., and Suzuki, T. (2018). Topology optimization of trusses with stress and local constraints on nodal stability and member intersection. Load and support conditions are randomly provided according to a rule so that the agent can be trained to have good performance for various boundary conditions. Methods Appl. Training workflow utilizing RL and graph embedding. Figure 4 plots the history of cumulative rewards in the test simulation recorded at every 10 episodes. embedding It is notable that the agent was able to optimize the structure with the unforeseen boundary conditions which the agent has never experienced during the training. Similarly to the boundary condition B1, the agent eliminates members that do not bear forces as shown in Figure 9B. doi: 10.1007/s00158-004-0480-2, Papadrakakis, M., Lagaros, N. D., and Tsompanakis, Y. KH and MO approved the final version of the manuscript, and agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. Rev. Optim. Built Environ. Data Eng. Loading condition L1 of Example 3; (A) initial GS, (B) removal sequence of members. Genetic algorithms in truss topological optimization. Once in 10-episode training, the performance of is tested for prescribed loading and boundary conditions. doi: 10.1007/s00158-008-0237-4, Hajela, P., and Lee, E. (1995). Optimal topologies of truss structures. Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. Best scored removal process of members for loading condition L1 of Example 1; (A) initial GS, (B) removal sequence to the terminal state. Neural message passing for quantum chemistry.

It took about 3.9 h for training through about 235,000 linear structural analyses. Int. Comput. Automatic design of optimal structures. IEEE Trans. Table 3. Figure 10. Loading condition L2 of Example 3; (A) initial GS, (B) removal sequence of members. The score rapidly improves in the first 1,000 episodes and the score mostly keeps above 35.0 after 2,000 episodes. The initial GS is illustrated in Figure 3.

A branch and bound algorithm for topology optimization of truss structures. Eng. The training method for tuning the parameters is described below. Genetic algorithm used in this study. One of the loads applied at node 4 is an irregular case where pin-supports and the loaded node aligns on the same straight line. Structural optimization using evolution strategies and neural networks. Deep residual learning for image recognition, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (Las Vegas, NV), 770778. J. (2017). 72, 1528. (1996). doi: 10.1512/iumj.1957.6.56038, Bellman, R. (1961). (1989). Math. Because the optimization problem (Equation 3) contains constraint functions, the cost function F used in GA is defined using the penalty term as: where 1 and 2 are penalty coefficients for stress and displacement constraints; both are set to be 1000 in this study. A., Veness, J., Bellemare, M. G., et al. Eng. Imagenet classification with deep convolutional neural networks, in Proceedings of the 25th International Conference on Neural Information Processing Systems - Vol. Eng. doi: 10.1038/323533a0, Sheu, C. Y., and Schmit, L. A. Jr. (1972). The number of loading conditions is fixed as nload= 2, and accordingly, the sizes of inputs from nodes and members are 5 and 6, respectively. AIAA J.

doi: 10.1007/s00158-012-0877-2, Hagishita, T., and Ohsaki, M. (2009). Note that the total CPU time t[s] for obtaining this removal sequence of members includes initialization of the truss structure, import of the trained RL agent, and computing the removal sequence. Each grid is a square whose side length is 1 m. The intersection of bracing members is not connected. Yu, Y., Hur, T., and Jung, J. doi: 10.1007/s00366-019-00753-w. [Epub ahead of print]. Symmetry properties in structural optimization: Some extensions. Although nload! It is confirmed from this history that the agent successfully improves its policy to eliminate unnecessary members as the training proceeds. To reduce the required capacity of a storage device, 1,000 sets of observed transitions (s, a, s, r) are stored at the maximum. In other words, should be closer to 1 if future and instant rewards are equivalently important, and 0 if only instant reward is important. Cambridge, MA: MIT Press.

Softw. The datasets generated for this study are available on request to the corresponding author.

Multidiscip. Using the features, the method to estimate the action value with respect to removal of the member is further formulated.

Krizhevsky, A., Sutskever, I., and Hinton, G. E. (2012).

doi: 10.1038/nature24270, Sutton, R. S., and Barto, A. G. (1998). Knowl. doi: 10.1038/nature14236. Nature 518, 529533. The final truss of removal process of members presented in Figure 5 is a terminal state, where displacement constraint is violated at the nodes highlighted in red. The cumulative reward until terminal state is recorded using the greedy policy without randomness (i.e., -greedy policy with = 0) during the test. (1998). 57, 219225. doi: 10.1016/0020-7683(94)00306-H, Hayashi, K., and Ohsaki, M. (2019). Indiana Univ. Figure 12. Deepwalk: online learning of social representations. Figure 11. doi: 10.1109/TNN.1998.712192, Tamura, T., Ohsaki, M., and Takagi, J. Figure 5. In Equation (10), the action value is updated so as to minimize the difference between the sum of observed reward and estimated action value at the next state r(s)+maxaQ(s,a) and estimated action value at the previous state Q(s, a). Multidiscip. Optim. A Markovian decision process. Optim. Comput. 6, 679684. Machine learning prediction errors better than DFT accuracy. Learn. 6:59. doi: 10.3389/fbuil.2020.00059. Figure 7. Mech.

The upper-bound displacement for each boundary condition is computed by multiplying 100 to the maximum absolute value of displacement among the all DOFs of the initial GS with the same loading and boundary conditions; hence, varies depending on the structure and the loading and boundary conditions. Figure 2. Topping, B., Khan, A., and Leite, J. Copyright 2020 Hayashi and Ohsaki. The statistical data with respect to the maximum test score for each training are as follows; the average is 43.38, the standard deviation is 0.16, and the coefficient of variation is only 3.80 103. J. Arch. 44, 315341. Although stress and displacement bounds have the same value and , respectively, for each member and DOF in this study, it should be noted that each member could have a different stress bound and each DOF could have a different displacement bound for each load case, which provides a versatility to the proposed method. 8, 279292. MO contributed to problem formulation and interpretation of data, and assisted in the preparation of the manuscript. Adv. Eng. Nodes 1 and 5 are pin-supported and nodes 22 and 24 are subjected to 1 kN load in positive x direction separately as two load cases. Furthermore, the trained agent is applicable to a truss with different topology, geometry and loading and boundary conditions after it is trained for a specific truss with various loading and boundary conditions. doi: 10.1109/TKDE.2018.2807452, Cheng, G., and Guo, X. The GS consists of 6 6 grids and the number of members is more than twice of the 4 4-grid truss. doi: 10.1016/0045-7949(94)00617-C, Ohsaki, M., and Hayashi, K. (2017). Figure 4. Ohsaki, M. (1995). From this result, it is confirmed that the agent is capable of eliminating unnecessary members properly for a different-scale truss.

In the same manner as neural networks, a back-propagation method (Rumelhart et al., 1986), which is a gradient based method to minimize the loss function, can be used for solving Equation (11). Cambridge, MA: MIT Press. Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. (2017). (Princeton, NJ: Princeton University Press). These results imply that the proposed method is robust against randomness of boundary conditions and actions during the training. Jpn. doi: 10.1016/0045-7825(89)90119-9. Global optimization of truss topology with discrete bar areas-part ii: implementation and numerical results.

Arxiv:1801.05463. The inputs are the initial GS, the bounds for stress and displacement, and the graph embedding class that contains trainable parameters initialized by the vectors with the sizes defined by nL and nf. The other solutions are assumed to be global optima which have not been verified through enumeration due to extremely high computational cost. (1986). Eng. doi: 10.1007/s10589-007-9152-7, Bellman, R. (1957). Mach Learn. 4, 2631.

Example 2: 3 2-grid truss (V = 0.0340 [m3]). Nature 550, 354359. Multidiscip. doi: 10.1007/BF00992698. The performance is also tested for other different trusses in sections 4.24.4 without re-training the results in section 4.1. ArXiv:1403.6652. doi: 10.1145/2623330.2623732. Machine learning for combinatorial optimization of brace placement of steel frames.

High-speed calculation in structural analysis by reinforcement learning, in the 32nd Annual Conference of the Japanese Society for Artificial Intelligence, JSAI2018:3K1OS18a01 (in Japanese), (Kagoshima). Although the use of CNN-based convolution method is difficult to apply to trusses as they cannot be handled as pixel-wise data, the convolution is successfully implemented for trusses by introducing graph embedding, which has been extended in this paper from the standard node-based formulation to a member(edge)-based formulation. Construct. Figure 3. doi: 10.1007/s00158-017-1710-8, Ohsaki, M., and Katoh, N. (2005). 60, 231244. Faber, F. A., Hutchison, L., Huang, B., Gilmer, J., Schoenholz, S. S., Dahl, G. E., et al.

Optim. Note that it is possible that the two load cases are identical, or applied to different nodes but in the same direction. This way, features of each member considering connectivity can be extracted. Mater. Optim. Force density method for simultaneous optimization of geometry and topology of trusses. Comput. The same material property and constraints as the examples of RL are applied to the following problems. Similarly to loading condition L1 in Figure 5, several symmetric topologies are observed during the removal process, and the sub-optimal topology is a well-converged solution that does not contain unnecessary members. arXiv:1704.01212. (2018). Boundary condition B2 of Example 2; (A) initial GS, (B) removal sequence of members. Struct. Rev. A comprehensive survey of graph embedding: problems, techniques and applications. Goodfellow, I. J., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., et al. doi: 10.1007/s11831-017-9237-0, Liew, A., Avelino, R., Moosavi, V., Van Mele, T., and Block, P. (2019). Mach.

] is a concatenation operator of two vectors in the column direction. Although the agent is applied to a larger-scale truss, a sparse optimal solution is successfully obtained. Following this scheme, the parameters are trained by solving the following optimization problem (Mnih et al., 2015): In Equation (11), the training can be stabilized by using parameters ~ at the previous state for estimation of the action value at the next state s (Mnih et al., 2015). If stress and displacement constraints are satisfied, penalty terms become zero and the cost function becomes equivalent to the total structural volume V(A). Furthermore, the robustness of the proposed method is also investigated by implementing 2,000-episode training using different random seeds for 20 times. As shown in Figure 8B, the agent utilizes an reasonable policy to eliminate obviously unnecessary members connecting to supports at first, non-load-bearing members around the supports next, and members in the load path at last. The left two corners 1 and 7 are pin-supported and rightward and downward unit loads are separately applied at the bottom-right corner 43, as shown in Figure 11A in the loading condition L1. doi: 10.2514/3.50078, Silver, D., Schrittwieser, J., Simonyan, K., Antonoglou, I., Huang, A., Guez, A., et al. Eng. KH designed the study, implemented the program, and wrote the initial draft of the manuscript.

The edge length of each grid is 1 m also for this Example 2.

(2018). Right tip nodes are candidates to apply loading, and a horizontal or a vertical load with the fixed magnitude of 1.0 kN is applied at a randomly chosen node. Lecun, Y., Bottou, L., Bengio, Y., and Haffner, P. (1998). Struct. Therefore, the topology just before the terminal state shall be a sub-optimal topology, which is a truss with 12 members as shown in Figure 5B. Q-learning. different removal sequences can be obtained for different order of the same set of load cases in node state data vk; for example, exchanging the values at indices 2 and 4 and those at indices 3 and 5 in vk maintains the original loading condition but may lead to different action to be taken during each member removal process, because the neural network outputs different Q values due to the exchange. doi: 10.1515/9781400874668, PubMed Abstract | CrossRef Full Text | Google Scholar, Cai, H., Zheng, V. W., and Chang, K. C. (2017). Tieleman, T., and Hinton, G. (2012). The size of embedded member feature nf is 100. Loading condition L2 of Example 1; (A) initial GS, (B) removal sequence of members. Moreover, during the removal process, there is almost no isolated member apart from load-bearing ones and existing members efficiently transmit forces to the supports. Learning combinatorial optimization algorithms over graphs, in Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS'17, (Long Beach, CA), 63516361. (2018). Algorithm 1. The above examples using the proposed method are further investigated in view of efficiency and accuracy through comparison with genetic algorithm (GA). Comparison between proposed method (RL+GE) and GA in view of total structural volume V[m3] and CPU time for one optimization t[s] using benchmark solutions. This algorithm is terminated if the best cost function value fb is not updated for ns = 10 consecutive generations. 156, 309333. Struct. An episode is defined as a sequence of member removal process from the initial GS to the terminal state violating constraints. The one just before the terminal state is a sub-optimal truss; however, instability exists at the loaded node. It is also advantageous that the agent is easily replicated and available in other computers by importing the trained parameters. Optimising the load path of compression-only thrust networks through independent sets. The parameters are tuned using a method based on 1-step Q-learning method, which is a frequently used RL method. 1, 419430. Struct. Example 3: 6 6-grid truss (V = 0.1858 [m3]).



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