{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T04:48:09Z","timestamp":1762058889708,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,12]],"date-time":"2022-07-12T00:00:00Z","timestamp":1657584000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"DFG","award":["SPP 1886"],"award-info":[{"award-number":["SPP 1886"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Topology optimisation is a mathematical approach relevant to different engineering problems where the distribution of material in a defined domain is distributed in some optimal way, subject to a predefined cost function representing desired (e.g., mechanical) properties and constraints. The computation of such an optimal distribution depends on the numerical solution of some physical model (in our case linear elasticity) and robustness is achieved by introducing uncertainties into the model data, namely the forces acting on the structure and variations of the material stiffness, rendering the task high-dimensional and computationally expensive. To alleviate this computational burden, we develop two neural network architectures (NN) that are capable of predicting the gradient step of the optimisation procedure. Since state-of-the-art methods use adaptive mesh refinement, the neural networks are designed to use a sufficiently fine reference mesh such that only one training phase of the neural network suffices. As a first architecture, a convolutional neural network is adapted to the task. To include sequential information of the optimisation process, a recurrent neural network is constructed as a second architecture. A common 2D bridge benchmark is used to illustrate the performance of the proposed architectures. It is observed that the NN prediction of the gradient step clearly outperforms the classical optimisation method, in particular since larger iteration steps become viable.<\/jats:p>","DOI":"10.3390\/a15070241","type":"journal-article","created":{"date-parts":[[2022,7,12]],"date-time":"2022-07-12T20:52:41Z","timestamp":1657659161000},"page":"241","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Topology Optimisation under Uncertainties with Neural Networks"],"prefix":"10.3390","volume":"15","author":[{"given":"Martin","family":"Eigel","sequence":"first","affiliation":[{"name":"Weierstrass Institute for Applied Analysis and Stochastics, 10117 Berlin, Germany"}]},{"given":"Marvin","family":"Haase","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Technical University Berlin, 10623 Berlin, Germany"}]},{"given":"Johannes","family":"Neumann","sequence":"additional","affiliation":[{"name":"Rafinex Ltd., Great Haseley OX44 7JQ, UK"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1016\/j.cma.2018.02.003","article-title":"Risk averse stochastic structural topology optimization","volume":"334","author":"Eigel","year":"2018","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_2","unstructured":"Eigel, M., Neumann, J., Schneider, R., and Wolf, S. (2016). Stochastic topology optimisation with hierarchical tensor reconstruction. WIAS, 2362."},{"key":"ref_3","unstructured":"Rawat, S., and Shen, M.H. (2019). A Novel Topology Optimization Approach using Conditional Deep Learning. arXiv."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Cang, R., Yao, H., and Ren, Y. (2018). One-Shot Optimal Topology Generation through Theory-Driven Machine Learning. arXiv.","DOI":"10.1016\/j.cad.2018.12.008"},{"key":"ref_5","unstructured":"Zhang, Y., Chen, A., Peng, B., Zhou, X., and Wang, D. (2019). A deep Convolutional Neural Network for topology optimization with strong generalization ability. arXiv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1515\/rnam-2019-0018","article-title":"Neural networks for topology optimization","volume":"34","author":"Sosnovik","year":"2019","journal-title":"Russ. J. Numer. Anal. Math. Model."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1080\/0305215X.2021.1902998","article-title":"A deep convolutional neural network for topology optimization with perceptible generalization ability","volume":"54","author":"Wang","year":"2022","journal-title":"Eng. Optim."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1118","DOI":"10.1016\/j.cma.2018.09.007","article-title":"Multiscale topology optimization using neural network surrogate models","volume":"346","author":"White","year":"2019","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_9","unstructured":"Dockhorn, T. (2019). A Discussion on Solving Partial Differential Equations using Neural Networks. arXiv."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/j.promfg.2020.02.273","article-title":"DL-Scale: Deep Learning for model upgrading in topology optimization","volume":"44","author":"Kallioras","year":"2020","journal-title":"Procedia Manuf."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1007\/s00158-020-02748-4","article-title":"TOuNN: Topology optimization using neural networks","volume":"63","author":"Chandrasekhar","year":"2021","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"091702","DOI":"10.1115\/1.4050105","article-title":"A parametric level set method for topology optimization based on deep neural network","volume":"143","author":"Deng","year":"2021","journal-title":"J. Mech. Des."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1927","DOI":"10.1007\/s00158-020-02788-w","article-title":"Two-stage convolutional encoder-decoder network to improve the performance and reliability of deep learning models for topology optimization","volume":"63","author":"Ates","year":"2021","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Malviya, M. (2020). A Systematic Study of Deep Generative Models for Rapid Topology Optimization. engrXiv.","DOI":"10.31224\/osf.io\/9gvqs"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"106283","DOI":"10.1016\/j.compstruc.2020.106283","article-title":"Topology optimization of 2D structures with nonlinearities using deep learning","volume":"237","author":"Abueidda","year":"2020","journal-title":"Comput. Struct."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Halle, A., Campanile, L.F., and Hasse, A. (2021). An Artificial Intelligence\u2013Assisted Design Method for Topology Optimization without Pre-Optimized Training Data. Appl. Sci., 11.","DOI":"10.3390\/app11199041"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"B90","DOI":"10.1115\/1.1497478","article-title":"Linearized Theory of Elasticity","volume":"55","author":"Slaughter","year":"2002","journal-title":"Appl. Mech. Rev."},{"key":"ref_18","unstructured":"Wallach, H., Larochelle, H., Beygelzimer, A., d\u2019Alch\u00e9-Buc, F., Fox, E., and Garnett, R. (2019). PyTorch: An Imperative Style, High-Performance Deep Learning Library. Advances in Neural Information Processing Systems 32, Curran Associates, Inc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1142\/S0218488598000094","article-title":"The Vanishing Gradient Problem During Learning Recurrent Neural Nets and Problem Solutions","volume":"6","author":"Hochreiter","year":"1998","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","article-title":"Long Short-Term Memory","volume":"9","author":"Hochreiter","year":"1997","journal-title":"Neural Comput."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ghaderpour, E., Pagiatakis, S.D., and Hassan, Q.K. (2021). A survey on change detection and time series analysis with applications. Appl. Sci., 11.","DOI":"10.3390\/app11136141"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Graves, A. (2013). Generating Sequences With Recurrent Neural Networks. arXiv.","DOI":"10.1007\/978-3-642-24797-2_3"},{"key":"ref_23","first-page":"802","article-title":"Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting","volume":"28","author":"Shi","year":"2015","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_24","first-page":"9","article-title":"The FEniCS Project Version 1.5","volume":"3","author":"Blechta","year":"2015","journal-title":"Arch. Numer. Softw."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Naushad, R., Kaur, T., and Ghaderpour, E. (2021). Deep transfer learning for land use and land cover classification: A comparative study. Sensors, 21.","DOI":"10.3390\/s21238083"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1106","DOI":"10.1137\/0733054","article-title":"A Convergent Adaptive Algorithm for Poisson\u2019s Equation","volume":"33","year":"1996","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/7\/241\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:48:48Z","timestamp":1760140128000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/7\/241"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,12]]},"references-count":26,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["a15070241"],"URL":"https:\/\/doi.org\/10.3390\/a15070241","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2022,7,12]]}}}