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Machine learning prediction errors better than DFT accuracy. [0, 1] is a discount factor; i.e., the action value becomes closer to expected cumulative reward as is larger, and conversely, the action value becomes closer to expected instant reward as is smaller. 8, 279292. The truss is optimized for two loading conditions. Although it takes a long time for the training, the trained agent requires very low computational cost compared with GA at the application stage. Comput. Appl. Build. Another approach may be to incorporate a rule-based method to create a hybrid optimization agent. 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. Comput. Mach Learn. Struct. 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. 30, 16161637. MO contributed to problem formulation and interpretation of data, and assisted in the preparation of the manuscript. During the test, nodes 1 and 5 are pin-supported, and loads are applied at node 23 in positive x and negative y direction separately as different loading conditions, which is denoted as loading condition L1. The trainable parameters are optimized by a back-propagation method to minimize the loss function computed by estimated action value and observed reward. Since removal of any remaining member will cause violation of the displacement constraint, there is no unnecessary member in the sub-optimal topology. 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. Best scored removal process of members for loading condition L1 of Example 1; (A) initial GS, (B) removal sequence to the terminal state. Imagenet classification with deep convolutional neural networks, in Proceedings of the 25th International Conference on Neural Information Processing Systems - Vol. doi: 10.1016/0045-7949(94)00617-C, Ohsaki, M., and Hayashi, K. (2017). Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. The size of embedded member feature nf is 100. We use a PC with a CPU of Intel(R) Core(TM) i9-7900X @ 3.30GHz. The topology two steps before the terminal state contains successive V-shaped braces and is stable and statically determinate. Cambridge, MA: MIT Press. 27, 193200. The agent is trained using a 72-member truss with 4 4 grids. Each grid is a square whose side length is 1 m. The intersection of bracing members is not connected. When the number of transition steps reaches 1,000, the latest transition overrides the oldest one. Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. (2017). The initial GS is illustrated in Figure 3. Boundary condition B2 of Example 2; (A) initial GS, (B) removal sequence of members. However, in order to create a more reliable agent, it is necessary to implement the training with various topology, geometry, and loading and boundary conditions. Global optimization of truss topology with discrete bar areas-part ii: implementation and numerical results. Copyright 2020 Hayashi and Ohsaki. Struct. Genetic algorithm used in this study. Struct. 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. It is also advantageous that the agent is easily replicated and available in other computers by importing the trained parameters. Example 3: 6 6-grid truss (V = 0.1858 [m3]). Eng. Optimising the load path of compression-only thrust networks through independent sets. Solids Struct. IEEE Trans. Q-learning. A., Veness, J., Bellemare, M. G., et al. Figure 4. Figure 9. Although the agent is applied to a larger-scale truss, a sparse optimal solution is successfully obtained. (2014). Deep residual learning for image recognition, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (Las Vegas, NV), 770778. Construct. 10, 111124. It is verified from the numerical examples that the trained agent acquired a policy to reduce total structural volume while satisfying the stress and displacement constraints. Ringertz, U. T. (1986). Boundary condition B1 of Example 2; (A) initial GS, (B) removal sequence of members. Loading condition L2 of Example 1; (A) initial GS, (B) removal sequence of members. J. Mecan. According to Table 3, the proposed RL+GE method is much more efficient than GA; the CPU time exponentially increases as the number of variable increases in GA; on the other hand, the CPU time increases almost linearly in RL+GE. 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. doi: 10.1016/j.conbuildmat.2013.08.078. Nature 518, 529533. 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). 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. Comput. doi: 10.1007/s00158-012-0877-2, Hagishita, T., and Ohsaki, M. (2009). doi: 10.1109/TKDE.2018.2807452, Cheng, G., and Guo, X. Figure 4 plots the history of cumulative rewards in the test simulation recorded at every 10 episodes. Moreover, all the trained RL agents with the best parameters led to the same 12-member sub-optimal solution as Figure 5B for loading condition L1. Indiana Univ. Once in 10-episode training, the performance of is tested for prescribed loading and boundary conditions. Mech. This work was kindly supported by Grant-in-Aid for JSPS Research Fellow No.JP18J21456 and JSPS KAKENHI No. COURSERA: Neural Netw. Figure 12. Although nload! 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). It took about 3.9 h for training through about 235,000 linear structural analyses. 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. (Princeton, NJ: Princeton University Press). 32, 33413357. Background information of deep learning for structural engineering. Struct. -relaxed approach in structural topology optimization. Topologies at steps 37, 60, 84, 100, 144, and 145 in the removal sequence are illustrated in Figure 11B. History of cumulative reward of each test measured every 10 episodes. arXiv:1702.05532. Enhanced artificial intelligence for ensemble approach to predicting high performance concrete compressive strength. Ohsaki, M. (1995). doi: 10.1109/TNN.1998.712192, Tamura, T., Ohsaki, M., and Takagi, J. 47, 783794. doi: 10.1002/2475-8876.12059. Lecture 6.5RmsProp: Divide the gradient by a running average of its recent magnitude. doi: 10.1038/nature24270, Sutton, R. S., and Barto, A. G. (1998). Example 1: 4 4-grid truss (V = 0.0853 [m3]). Deepwalk: online learning of social representations. The cumulative reward until terminal state is recorded using the greedy policy without randomness (i.e., -greedy policy with = 0) during the test. Loading condition L1 of Example 3; (A) initial GS, (B) removal sequence of members. Figure 5. Training workflow utilizing RL and graph embedding. doi: 10.1038/323533a0, Sheu, C. Y., and Schmit, L. A. Jr. (1972). FDMopt: force density method for optimal geometry and topology of trusses. The GS consists of 6 6 grids and the number of members is more than twice of the 4 4-grid truss. 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. (2018). 156, 309333. Rev. (2015). The other solutions are assumed to be global optima which have not been verified through enumeration due to extremely high computational cost. J. doi: 10.1007/s00158-004-0480-2, Papadrakakis, M., Lagaros, N. D., and Tsompanakis, Y. Figure 7. (1997). doi: 10.1007/s00366-019-00753-w. [Epub ahead of print]. Even in this irregular case, the agent successfully obtained the sparse optimal solution, as shown in Figure 12. If stress and displacement constraints are satisfied, penalty terms become zero and the cost function becomes equivalent to the total structural volume V(A). Genetic algorithm for topology optimization of trusses. Human-level control through deep reinforcement learning. These results imply that the proposed method is robust against randomness of boundary conditions and actions during the training. Optim. RMSprop (Tieleman and Hinton, 2012) is adopted as the optimization method in this study. 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. (2013) explained that solving the quasi-convex symmetric optimization problem may yield highly asymmetric solution. 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. This applicability was demonstrated through both smaller-scale and larger-scale trusses and sparse sub-optimal topologies were obtained for both cases. Optim. doi: 10.1515/9781400874668, PubMed Abstract | CrossRef Full Text | Google Scholar, Cai, H., Zheng, V. W., and Chang, K. C. (2017). J. Two pin-supports are randomly chosen; one from nodes 1 and 2 and the other from nodes 4 and 5. The edge length of each grid is 1 m also for this Example 2. The training method for tuning the parameters is described below. Eng. Symmetry properties in structural optimization: Some extensions. Achtziger, W., and Stolpe, M. (2009). Topological design of truss structures using simulated annealing. The datasets generated for this study are available on request to the corresponding author. The proposed method is also comparable to GA with np = 200 in terms of proximity to the global optimum; RL+GE generally reached the feasible solutions with less total structural volume compared with the solutions obtained by GA. Faber, F. A., Hutchison, L., Huang, B., Gilmer, J., Schoenholz, S. S., Dahl, G. E., et al. 13, 258266. Optim. Received: 22 November 2019; Accepted: 09 April 2020; Published: 30 April 2020. A comprehensive survey of graph embedding: problems, techniques and applications. doi: 10.1007/s00158-019-02214-w. Mitchell, M. (1998). At the 18th step, a tower-like symmetric topology is created with extending members from upper tips to loaded nodes. Generative adversarial networks. If the cumulative reward is larger than the previous best score, at that step is saved. Knowl. As Example 3, the agent is applied to a larger-scale truss, as shown in Figure 10, without re-training. Figure 11. Consequently, the training concerns a total of 2 2 20 20 = 1, 600 combinations of support and loading conditions, and these combinations are almost equally simulated as long as the number of training episodes are sufficient. 4, 2631. Topology optimization of trusses with stress and local constraints on nodal stability and member intersection. 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. 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). arXiv:1406.2661. GA is one of the most prevalent metaheuristic approach for binary optimization problems, which is inspired by the process of natural selection (Mitchell, 1998). Yu, Y., Hur, T., and Jung, J. In GA, a set of solutions are repeatedly modified using the operations such as selection, where superior solutions at current generation are selected for new generation, crossover, where the selected solutions are combined to breed child solutions sharing the same characteristics as the parents, and mutation, where the selected solutions randomly change their values with low probability. 60, 231244. In the first boundary condition B1, as shown in Figure 8A, left tip nodes 1 and 3 are pin-supported and bottom-right nodes 7 and 10 are subjected to downward unit load of 1 kN separately as different loading cases. Neural message passing for quantum chemistry. Nature 550, 354359. Optim. 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. 1, 419430. doi: 10.1016/S0045-7825(97)00215-6, Perozzi, B., Al-Rfou', R., and Skiena, S. (2014). In this paper, a machine-learning based method combining graph embedding and Q-learning is proposed for binary truss topology optimization to minimize total structural volume under stress and displacement constraints. Optim. Comput. Multidiscip. Eng. In other words, should be closer to 1 if future and instant rewards are equivalently important, and 0 if only instant reward is important. JP18K18898. Since ^ is also computed using {1, , 6}, the action value Q(^,i) is dependent on = {1, , 9}. Combining machine learning models via adaptive ensemble weighting for prediction of shear capacity of reinforced-concrete deep beams. 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. A branch and bound algorithm for topology optimization of truss structures. Watkins, C. J. C. H., and Dayan, P. (1992). (2017). Data Eng. doi: 10.1007/s10589-007-9152-7, Bellman, R. (1957). 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. Comput. Mastering the game of go without human knowledge. doi: 10.1007/s11831-017-9237-0, Liew, A., Avelino, R., Moosavi, V., Van Mele, T., and Block, P. (2019). The program is implemented within Python 3.7 environment. Eng. The above examples using the proposed method are further investigated in view of efficiency and accuracy through comparison with genetic algorithm (GA). 6 Articles, This article is part of the Research Topic, https://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf, Creative Commons Attribution License (CC BY), Department of Architecture and Architectural Engineering, Graduate School of Engineering, Kyoto University, Kyoto, Japan. Struct. doi: 10.1016/j.advengsoft.2019.04.002, He, K., Zhang, X., Ren, S., and Sun, J. Figure 10. It is confirmed from this history that the agent successfully improves its policy to eliminate unnecessary members as the training proceeds. Note that the nodes highlighted in blue are pin-supported, those in yellow are loaded. Eng. In utilizing the trained agent in Example 1, nload! From this result, it is confirmed that the agent is capable of eliminating unnecessary members properly for a different-scale truss. The same material property and constraints as the examples of RL are applied to the following problems. No use, distribution or reproduction is permitted which does not comply with these terms. Adaptive Control Processes. Deep learning for topology optimization design. doi: 10.1007/s00158-008-0237-4, Hajela, P., and Lee, E. (1995). Goodfellow, I. J., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., et al. ArXiv:1403.6652. doi: 10.1145/2623330.2623732. 3, 2552. = 2 different removal sequences can be obtained in this way, only the better result with less total structural volume is provided in the RL+GE column in Table 3. 8, 301304. Tieleman, T., and Hinton, G. (2012). doi: 10.1109/5.726791, Lee, S., Ha, J., Zokhirova, M., Moon, H., and Lee, J. Structural optimization using evolution strategies and neural networks. Genetic algorithms in truss topological optimization. Mach. The agent trained in Example 1 is reused for a smaller 3 -grid truss without re-training. Arxiv:1801.05463. ] is a concatenation operator of two vectors in the column direction. arXiv:1704.01212. Optim. Optim. The whole training workflow is described in Figure 2. 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. The benchmark solutions for the 3 2-grid truss provided in Table 3 are global optimal solutions; this global optimality is verified through enumeration which took 44.1 h for each boundary condition. In this study, = 0.99 is adopted, because cumulative reward indicating the amount of reduction of structural volume as a result of the action is much more important than the instant reward. The number of training episodes is set as 5,000. AIAA J. 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. In addition, the accuracy of the trained agent is less dependent on the size of the problem; the trained agent reached presumable global optimum for 10 10-grid truss with L1 loading condition, although the agent was caught at the local optimum for 8 8-grid truss with the same loading condition, and even for 3 2-grid truss with B1 boundary condition. *Correspondence: Kazuki Hayashi, hayashi.kazuki.55a@st.kyoto-u.ac.jp, View all Similarly to the boundary condition B1, the agent eliminates members that do not bear forces as shown in Figure 9B. The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. To reduce the required capacity of a storage device, 1,000 sets of observed transitions (s, a, s, r) are stored at the maximum.