{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,7]],"date-time":"2026-06-07T06:41:57Z","timestamp":1780814517208,"version":"3.54.1"},"reference-count":95,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,21]],"date-time":"2024-12-21T00:00:00Z","timestamp":1734739200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61801319"],"award-info":[{"award-number":["61801319"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["1321002"],"award-info":[{"award-number":["1321002"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2021RZJ01"],"award-info":[{"award-number":["2021RZJ01"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["SUSE652A011"],"award-info":[{"award-number":["SUSE652A011"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["Y2024301"],"award-info":[{"award-number":["Y2024301"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Engineering Research Center of Ministry of Education for Integration and Application of Digital Learning Technology","award":["61801319"],"award-info":[{"award-number":["61801319"]}]},{"name":"Engineering Research Center of Ministry of Education for Integration and Application of Digital Learning Technology","award":["1321002"],"award-info":[{"award-number":["1321002"]}]},{"name":"Engineering Research Center of Ministry of Education for Integration and Application of Digital Learning Technology","award":["2021RZJ01"],"award-info":[{"award-number":["2021RZJ01"]}]},{"name":"Engineering Research Center of Ministry of Education for Integration and Application of Digital Learning Technology","award":["SUSE652A011"],"award-info":[{"award-number":["SUSE652A011"]}]},{"name":"Engineering Research Center of Ministry of Education for Integration and Application of Digital Learning Technology","award":["Y2024301"],"award-info":[{"award-number":["Y2024301"]}]},{"name":"Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province","award":["61801319"],"award-info":[{"award-number":["61801319"]}]},{"name":"Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province","award":["1321002"],"award-info":[{"award-number":["1321002"]}]},{"name":"Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province","award":["2021RZJ01"],"award-info":[{"award-number":["2021RZJ01"]}]},{"name":"Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province","award":["SUSE652A011"],"award-info":[{"award-number":["SUSE652A011"]}]},{"name":"Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province","award":["Y2024301"],"award-info":[{"award-number":["Y2024301"]}]},{"name":"Scientific Research and Innovation Team Program of Sichuan University of Science and Engineering","award":["61801319"],"award-info":[{"award-number":["61801319"]}]},{"name":"Scientific Research and Innovation Team Program of Sichuan University of Science and Engineering","award":["1321002"],"award-info":[{"award-number":["1321002"]}]},{"name":"Scientific Research and Innovation Team Program of Sichuan University of Science and Engineering","award":["2021RZJ01"],"award-info":[{"award-number":["2021RZJ01"]}]},{"name":"Scientific Research and Innovation Team Program of Sichuan University of Science and Engineering","award":["SUSE652A011"],"award-info":[{"award-number":["SUSE652A011"]}]},{"name":"Scientific Research and Innovation Team Program of Sichuan University of Science and Engineering","award":["Y2024301"],"award-info":[{"award-number":["Y2024301"]}]},{"name":"Postgraduate Innovation Fund Project of Sichuan University of Science and Engineering","award":["61801319"],"award-info":[{"award-number":["61801319"]}]},{"name":"Postgraduate Innovation Fund Project of Sichuan University of Science and Engineering","award":["1321002"],"award-info":[{"award-number":["1321002"]}]},{"name":"Postgraduate Innovation Fund Project of Sichuan University of Science and Engineering","award":["2021RZJ01"],"award-info":[{"award-number":["2021RZJ01"]}]},{"name":"Postgraduate Innovation Fund Project of Sichuan University of Science and Engineering","award":["SUSE652A011"],"award-info":[{"award-number":["SUSE652A011"]}]},{"name":"Postgraduate Innovation Fund Project of Sichuan University of Science and Engineering","award":["Y2024301"],"award-info":[{"award-number":["Y2024301"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>With the rapid development of artificial intelligence (AI), the design and implementation of very large-scale integrated circuits (VLSI) based on traditional binary computation are facing challenges of high complexity, computational power, and high power consumption. The development of Moore\u2019s law has reached the limit of physical technology, and there is an urgent need to explore new computing architectures to make up for the shortcomings of traditional binary computing. To address the existing problems, Stochastic Computing (SC) is an unconventional stochastic sequence that converts binary numbers into a coded stream of digital pulses. It has a remarkable symmetry with binary computation. It uses logic gate circuits in the probabilistic domain to implement complex arithmetic operations at the expense of computational accuracy and time. It has low power and logic resource consumption and a small circuit area. This paper analyzes the basic concepts and development history of SC and neural networks (NNs), summarizes the development progress of SC with NN at home and abroad, and discusses the development trend of SC and the future challenges and prospects of NN. Through systematic summarization, this paper provides new learning ideas and research directions for developing AI chips.<\/jats:p>","DOI":"10.3390\/sym16121701","type":"journal-article","created":{"date-parts":[[2024,12,23]],"date-time":"2024-12-23T04:33:29Z","timestamp":1734928409000},"page":"1701","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Stochastic Computing Architectures: Modeling, Optimization, and Applications"],"prefix":"10.3390","volume":"16","author":[{"given":"Lin","family":"Wang","sequence":"first","affiliation":[{"name":"School of Automation and Information Engineering, Sichuan University of Science and Engineering, Yibin 644000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1767-1831","authenticated-orcid":false,"given":"Zhongqiang","family":"Luo","sequence":"additional","affiliation":[{"name":"School of Automation and Information Engineering, Sichuan University of Science and Engineering, Yibin 644000, China"},{"name":"Artificial Intelligence Key Laboratory of Sichuan Province, Sichuan University of Science and Engineering, Yibin 644000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Li","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Automation and Information Engineering, Sichuan University of Science and Engineering, Yibin 644000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Burger, W., and Burge, M.J. (2022). Digital Image Processing: An Algorithmic Introduction, Springer Nature.","DOI":"10.1007\/978-3-031-05744-1"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bailey, D.G. (2023). Design for Embedded Image Processing on FPGAs, John Wiley & Sons.","DOI":"10.1002\/9781119819820"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"122836","DOI":"10.1016\/j.eswa.2023.122836","article-title":"Autonomous driving system: A comprehensive survey","volume":"242","author":"Zhao","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3692","DOI":"10.1109\/TIV.2023.3274536","article-title":"Motion planning for autonomous driving: The state of the art and future perspectives","volume":"8","author":"Teng","year":"2023","journal-title":"IEEE Trans. Intell. Veh."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"e8","DOI":"10.1561\/116.00000050","article-title":"Recent advances in end-to-end automatic speech recognition","volume":"11","author":"Li","year":"2022","journal-title":"Apsipa Trans. Signal Inf. Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"23367","DOI":"10.1007\/s11042-023-16438-y","article-title":"A comprehensive survey on automatic speech recognition using neural networks","volume":"83","author":"Dhanjal","year":"2024","journal-title":"Multimed. Tools Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"101567","DOI":"10.1016\/j.csl.2023.101567","article-title":"Towards inclusive automatic speech recognition","volume":"84","author":"Feng","year":"2024","journal-title":"Comput. Speech Lang."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Da Silva, I.N., Hernane Spatti, D., Andrade Flauzino, R., Liboni, L.H.B., dos Reis Alves, S.F., da Silva, I.N., Hernane Spatti, D., Andrade Flauzino, R., Liboni, L.H.B., and dos Reis Alves, S.F. (2017). Artificial Neural Network Architectures and Training Processes, Springer.","DOI":"10.1007\/978-3-319-43162-8"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Hassaballah, M., and Awad, A.I. (2020). Deep Learning in Computer Vision: Principles and Applications, CRC Press.","DOI":"10.1201\/9781351003827"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Aggarwal, C.C. (2018). Neural Networks and Deep Learning, Springer.","DOI":"10.1007\/978-3-319-94463-0"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gerstner, W., and Kistler, W.M. (2002). Spiking Neuron Models: Single Neurons, Populations, Plasticity, Cambridge University Press.","DOI":"10.1017\/CBO9780511815706"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hertz, J.A. (2018). Introduction to the Theory of Neural Computation, CRC Press.","DOI":"10.1201\/9780429499661"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2465787.2465794","article-title":"Survey of stochastic computing","volume":"12","author":"Alaghi","year":"2013","journal-title":"Acm Trans. Embed. Comput. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1109\/TNANO.2024.3373499","article-title":"From Multipliers to Integrators: A Survey of Stochastic Computing Primitives","volume":"23","author":"Liu","year":"2024","journal-title":"IEEE Trans. Nanotechnol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1515","DOI":"10.1109\/TCAD.2017.2778107","article-title":"The promise and challenge of stochastic computing","volume":"37","author":"Alaghi","year":"2017","journal-title":"IEEE Trans.-Comput.-Aided Des. Integr. Circuits Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"e309","DOI":"10.7717\/peerj-cs.309","article-title":"Stochastic computing in convolutional neural network implementation: A review","volume":"6","author":"Lee","year":"2020","journal-title":"PeerJ Comput. Sci."},{"key":"ref_17","first-page":"257","article-title":"Digital implementation of radial basis function neural networks based on stochastic computing","volume":"13","author":"Parrilla","year":"2022","journal-title":"IEEE J. Emerg. Sel. Top. Circuits Syst."},{"key":"ref_18","unstructured":"Li, P., Qian, W., Riedel, M.D., Bazargan, K., and Lilja, D.J. (February, January 30). The synthesis of linear finite state machine-based stochastic computational elements. Proceedings of the 17th Asia and South Pacific Design Automation Conference, Sydney, Australia."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1007\/s00339-022-06365-4","article-title":"Unconventional computing based on magnetic tunnel junction","volume":"129","author":"Cai","year":"2023","journal-title":"Appl. Phys. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1109\/LES.2023.3298734","article-title":"Hardware-software co-optimization of long-latency stochastic computing","volume":"15","author":"Aygun","year":"2023","journal-title":"IEEE Embed. Syst. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Yang, J., Graening, A., Romaszkan, W., Jacob, V.K., Gupta, P., and Pamarti, S. (2024, January 16\u201320). A 278-514M Event\/s ADC-Less Stochastic Compute-In-Memory Convolution Accelerator for Event Camera. Proceedings of the 2024 IEEE Symposium on VLSI Technology and Circuits (VLSI Technology and Circuits), Honolulu, HI, USA.","DOI":"10.1109\/VLSITechnologyandCir46783.2024.10631484"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4423","DOI":"10.1109\/TCAD.2023.3284289","article-title":"REX-SC: Range-extended stochastic computing accumulation for neural network acceleration","volume":"42","author":"Li","year":"2023","journal-title":"IEEE Trans.-Comput.-Aided Des. Integr. Circuits Syst."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Wang, D., Wang, Z., Yu, L., Wu, Y., Yang, J., Mei, K., and Wang, J. (October, January 30). A Survey of Stochastic Computing in Energy-Efficient DNNs On-Edge. Proceedings of the 2021 IEEE Intl Conf on Parallel & Distributed Processing with Applications, Big Data & Cloud Computing, Sustainable Computing & Communications, Social Computing & Networking (ISPA\/BDCloud\/SocialCom\/SustainCom), New York City, NY, USA.","DOI":"10.1109\/ISPA-BDCloud-SocialCom-SustainCom52081.2021.00209"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"89663","DOI":"10.1109\/ACCESS.2022.3200454","article-title":"A survey of intelligent chip design research based on spiking neural networks","volume":"10","author":"Chen","year":"2022","journal-title":"IEEE Access"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2809","DOI":"10.1109\/TNNLS.2020.3009047","article-title":"A survey of stochastic computing neural networks for machine learning applications","volume":"32","author":"Liu","year":"2020","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Poppelbaum, W., Afuso, C., and Esch, J. (1967, January 14\u201316). Stochastic computing elements and systems. Proceedings of the Fall Joint Computer Conference, Anaheim, CA, USA.","DOI":"10.1145\/1465611.1465696"},{"key":"ref_27","first-page":"37","article-title":"Stochastic computing systems","volume":"2","author":"Gaines","year":"1969","journal-title":"Adv. Inf. Syst. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1109\/12.954505","article-title":"Stochastic neural computation. I. Computational elements","volume":"50","author":"Brown","year":"2001","journal-title":"IEEE Trans. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"966","DOI":"10.1587\/transinf.2023LOP0006","article-title":"Delta-Sigma Domain Signal Processing Revisited with Related Topics in Stochastic Computing","volume":"107","author":"Waho","year":"2024","journal-title":"Ieice Trans. Inf. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Lin, Z., Xie, G., Wang, S., Han, J., and Zhang, Y. (2021, January 8\u201310). A review of deterministic approaches to stochastic computing. Proceedings of the 2021 IEEE\/ACM International Symposium on Nanoscale Architectures (NANOARCH), Vancouver, BC, Canada.","DOI":"10.1109\/NANOARCH53687.2021.9642242"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1109\/TC.1968.226901","article-title":"R68-18 random pulse machines","volume":"100","author":"Gaines","year":"1968","journal-title":"IEEE Trans. Comput."},{"key":"ref_32","first-page":"43","article-title":"Probabilistic logics and the synthesis of reliable organisms from unreliable components","volume":"34","year":"1956","journal-title":"Autom. Stud."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/S0065-2458(08)60452-0","article-title":"Statistical processors","volume":"Volume 14","author":"Poppelbaum","year":"1976","journal-title":"Advances in Computers"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/18.335883","article-title":"Generating binary sequences for stochastic computing","volume":"40","author":"Jeavons","year":"1994","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1109\/72.410355","article-title":"Architecture and statistical model of a pulse-mode digital multilayer neural network","volume":"6","author":"Kim","year":"1995","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_36","unstructured":"Gu, Y., and Li, H. (2021). Enhancing the Decoding of Short LDPC Codes with Stochastic Sequences. [Master\u2019s Thesis, Lund University]."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2023","DOI":"10.1109\/TPDS.2024.3453310","article-title":"SC-CGRA: An Energy-Efficient CGRA Using Stochastic Computing","volume":"35","author":"Mou","year":"2024","journal-title":"IEEE Trans. Parallel Distrib. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"110166","DOI":"10.1016\/j.asoc.2023.110166","article-title":"FPGA-based implementation of deep neural network using stochastic computing","volume":"137","author":"Nobari","year":"2023","journal-title":"Appl. Soft Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"942","DOI":"10.1109\/TBCAS.2020.2995869","article-title":"Low-cost adaptive exponential integrate-and-fire neuron using stochastic computing","volume":"14","author":"Xiao","year":"2020","journal-title":"IEEE Trans. Biomed. Circuits Syst."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Xie, T., Hu, Y., Wei, R., Li, M., Wang, Y., Wang, R., and Huang, R. (2024, January 25\u201327). ASCEND: Accurate yet Efficient End-to-End Stochastic Computing Acceleration of Vision Transformer. Proceedings of the 2024 Design, Automation & Test in Europe Conference & Exhibition (DATE), Valencia, Spain.","DOI":"10.23919\/DATE58400.2024.10546691"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2277","DOI":"10.1109\/TCAD.2018.2858338","article-title":"Architecture considerations for stochastic computing accelerators","volume":"37","author":"Lee","year":"2018","journal-title":"IEEE Trans.-Comput.-Aided Des. Integr. Circuits Syst."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Vatsavai, S.S., Karempudi, V.S.P., Thakkar, I., Salehi, A., and Hastings, T. (June, January 30). Sconna: A stochastic computing based optical accelerator for ultra-fast, energy-efficient inference of integer-quantized cnns. Proceedings of the 2023 IEEE International Parallel and Distributed Processing Symposium (IPDPS), Lyon, France.","DOI":"10.1109\/IPDPS54959.2023.00061"},{"key":"ref_43","unstructured":"Li, M., Hu, Y., Zhang, T., Wei, R., Zhang, Y., Huang, R., and Wang, R. (2024). Efficient yet Accurate End-to-End SC Accelerator Design. arXiv."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"12913","DOI":"10.1109\/TNNLS.2023.3265533","article-title":"An energy-efficient Bayesian neural network implementation using stochastic computing method","volume":"35","author":"Jia","year":"2023","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_45","first-page":"2158","article-title":"Design space exploration of stochastic computing architectures implemented using integrated optics","volume":"9","author":"Tahar","year":"2020","journal-title":"IEEE Trans. Emerg. Top. Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1109\/MCAS.2020.3005483","article-title":"Introduction to dynamic stochastic computing","volume":"20","author":"Liu","year":"2020","journal-title":"IEEE Circuits Syst. Mag."},{"key":"ref_47","unstructured":"Qian, W. (2011). Digital Yet Deliberately Random: Synthesizing Logical Computation on Stochastic Bit Streams, University of Minnesota."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Riahi Alam, M., Najafi, M.H., TaheriNejad, N., Imani, M., and Peng, L. (2023, January 5\u20137). Stochastic Computing for Reliable Memristive In-Memory Computation. Proceedings of the Great Lakes Symposium on VLSI 2023, Knoxville, TN, USA.","DOI":"10.1145\/3583781.3590307"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.vlsi.2017.11.002","article-title":"Normalization and dropout for stochastic computing-based deep convolutional neural networks","volume":"65","author":"Li","year":"2019","journal-title":"Integration"},{"key":"ref_50","first-page":"1","article-title":"Neural network classifiers using a hardware-based approximate activation function with a hybrid stochastic multiplier","volume":"15","author":"Li","year":"2019","journal-title":"ACM J. Emerg. Technol. Comput. Syst."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Song, Y., Li, H., Zhu, X., and Chen, Y. (2024, January 1\u20133). Design of Multiplier Circuit Based on Signed-Digit Hybrid Stochastic Computing. Proceedings of the 2024 IEEE Computer Society Annual Symposium on VLSI (ISVLSI), Knoxville, TN, USA.","DOI":"10.1109\/ISVLSI61997.2024.00022"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Hussein, A.K., and Artan, N.S. (2024, January 11\u201314). Low-Latency Deterministic Multiplier for Stochastic Computing. Proceedings of the 2024 IEEE 67th International Midwest Symposium on Circuits and Systems (MWSCAS), Springfield, MA, USA.","DOI":"10.1109\/MWSCAS60917.2024.10658788"},{"key":"ref_53","first-page":"147","article-title":"New divider design for stochastic computing","volume":"67","author":"Chu","year":"2019","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2277","DOI":"10.1109\/TCAD.2024.3365426","article-title":"Fast and Scaled Counting-Based Stochastic Computing Divider Design","volume":"43","author":"Liu","year":"2024","journal-title":"IEEE Trans.-Comput.-Aided Des. Integr. Circuits Syst."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"992","DOI":"10.1109\/TVLSI.2019.2963678","article-title":"Low-cost stochastic number generators for stochastic computing","volume":"28","author":"Salehi","year":"2020","journal-title":"IEEE Trans. Very Large Scale Integr. Syst."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1109\/TNANO.2014.2300342","article-title":"A native stochastic computing architecture enabled by memristors","volume":"13","author":"Knag","year":"2014","journal-title":"IEEE Trans. Nanotechnol."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Temenos, N., Ntinas, V., Sotiriadis, P.P., and Sirakoulis, G.C. (2023, January 21\u201325). Time-based Memristor Crossbar Array Programming for Stochastic Computing Parallel Sequence Generation. Proceedings of the 2023 IEEE International Symposium on Circuits and Systems (ISCAS), Monterey, CA, USA.","DOI":"10.1109\/ISCAS46773.2023.10181967"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"2450027","DOI":"10.1142\/S0218127424500275","article-title":"A lightweight cnn based on memristive stochastic computing for electronic nose","volume":"34","author":"Yang","year":"2024","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1326","DOI":"10.1109\/TVLSI.2018.2812214","article-title":"Toward energy-efficient stochastic circuits using parallel Sobol sequences","volume":"26","author":"Liu","year":"2018","journal-title":"IEEE Trans. Very Large Scale Integr. Syst."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"3620","DOI":"10.1109\/TED.2019.2920401","article-title":"Spin-hall-effect-based stochastic number generator for parallel stochastic computing","volume":"66","author":"Hu","year":"2019","journal-title":"IEEE Trans. Electron. Devices"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"10408","DOI":"10.1109\/TNNLS.2022.3166799","article-title":"Fully parallel stochastic computing hardware implementation of convolutional neural networks for edge computing applications","volume":"34","author":"Frasser","year":"2022","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"2707","DOI":"10.1109\/TCSI.2022.3166524","article-title":"Hybrid stochastic-binary computing for low-latency and high-precision inference of CNNs","volume":"69","author":"Chen","year":"2022","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1273","DOI":"10.1109\/TC.2018.2817237","article-title":"A stochastic computational multi-layer perceptron with backward propagation","volume":"67","author":"Liu","year":"2018","journal-title":"IEEE Trans. Comput."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1474","DOI":"10.1109\/TC.2012.231","article-title":"Logical computation on stochastic bit streams with linear finite-state machines","volume":"63","author":"Li","year":"2012","journal-title":"IEEE Trans. Comput."},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Li, J., Yuan, Z., Li, Z., Ding, C., Ren, A., Qiu, Q., Draper, J., and Wang, Y. (2017, January 14\u201319). Hardware-driven nonlinear activation for stochastic computing based deep convolutional neural networks. Proceedings of the 2017 International Joint Conference on Neural Networks (IJCNN), Anchorage, AK, USA.","DOI":"10.1109\/IJCNN.2017.7965993"},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Chen, K.C.J., and Syu, W.R. (2024, January 19\u201322). High Reliable and Accurate Stochastic Computing-based Artificial Neural Network Architecture Design. Proceedings of the 2024 IEEE International Symposium on Circuits and Systems (ISCAS), Singapore.","DOI":"10.1109\/ISCAS58744.2024.10558634"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"e12687","DOI":"10.1111\/exsy.12687","article-title":"P-SCADA-a novel area and energy efficient FPGA architectures for LSTM prediction of heart arrthymias in BIoT applications","volume":"39","author":"Varadharajan","year":"2022","journal-title":"Expert Syst."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"1543","DOI":"10.1109\/TCAD.2018.2852752","article-title":"HEIF: Highly efficient stochastic computing-based inference framework for deep neural networks","volume":"38","author":"Li","year":"2018","journal-title":"IEEE Trans.-Comput.-Aided Des. Integr. Circuits Syst."},{"key":"ref_69","doi-asserted-by":"crossref","unstructured":"Li, B., Najafi, M.H., and Lilja, D.J. (2015, January 27\u201329). An FPGA implementation of a restricted boltzmann machine classifier using stochastic bit streams. Proceedings of the 2015 IEEE 26th International Conference on Application-Specific Systems, Architectures and Processors (ASAP), Toronto, ON, Canada.","DOI":"10.1109\/ASAP.2015.7245709"},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"Chen, K.C., and Wu, C.H. (2021, January 22\u201328). High-accurate stochastic computing for artificial neural network by using extended stochastic logic. Proceedings of the 2021 IEEE International Symposium on Circuits and Systems (ISCAS), Daegu, Republic of Korea.","DOI":"10.1109\/ISCAS51556.2021.9401418"},{"key":"ref_71","doi-asserted-by":"crossref","unstructured":"Chen, K.C.J., and Chen, C.T. (2021, January 6\u20139). High-accuracy and Low-latency Hybrid Stochastic Computing for Artificial Neural Network. Proceedings of the 2021 18th International SoC Design Conference (ISOCC), Jeju, Republic of Korea.","DOI":"10.1109\/ISOCC53507.2021.9613856"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1109\/TCSI.2020.3039346","article-title":"Neural synaptic plasticity-inspired computing: A high computing efficient deep convolutional neural network accelerator","volume":"68","author":"Xia","year":"2020","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"4538","DOI":"10.1109\/TCSII.2022.3191670","article-title":"Efficient compression methods for wire-spread-based stochastic computing deep neural networks","volume":"69","author":"Li","year":"2022","journal-title":"IEEE Trans. Circuits Syst. I Express Briefs"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"2667","DOI":"10.1109\/TCSII.2020.2969691","article-title":"Hardware implementation of an improved stochastic computing based deep neural network using short sequence length","volume":"67","author":"Xiong","year":"2020","journal-title":"IEEE Trans. Circuits Syst. I Express Briefs"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"104505","DOI":"10.1016\/j.micpro.2022.104505","article-title":"Optimization for deep convolutional neural network of stochastic computing on MLC-PCM-based system","volume":"90","author":"Wang","year":"2022","journal-title":"Microprocess. Microsyst."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"102810","DOI":"10.1016\/j.sysarc.2022.102810","article-title":"Hardware-aware neural architecture search for stochastic computing-based neural networks on tiny devices","volume":"135","author":"Song","year":"2023","journal-title":"J. Syst. Archit."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.neucom.2021.10.105","article-title":"Accurate and compact convolutional neural network based on stochastic computing","volume":"471","author":"Abdellatef","year":"2022","journal-title":"Neurocomputing"},{"key":"ref_78","doi-asserted-by":"crossref","unstructured":"Li, J., Ren, A., Li, Z., Ding, C., Yuan, B., Qiu, Q., and Wang, Y. (2017, January 16\u201319). Towards acceleration of deep convolutional neural networks using stochastic computing. Proceedings of the 2017 22nd Asia and South Pacific Design Automation Conference (ASP-DAC), Chiba\/Tokyo, Japan.","DOI":"10.1109\/ASPDAC.2017.7858306"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1145\/3093336.3037746","article-title":"Sc-dcnn: Highly-scalable deep convolutional neural network using stochastic computing","volume":"52","author":"Ren","year":"2017","journal-title":"ACM Sigplan Not."},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"Yu, J., Kim, K., Lee, J., and Choi, K. (2017, January 5\u20138). Accurate and efficient stochastic computing hardware for convolutional neural networks. Proceedings of the 2017 IEEE International Conference on Computer Design (ICCD), Boston, MA, USA.","DOI":"10.1109\/ICCD.2017.24"},{"key":"ref_81","doi-asserted-by":"crossref","unstructured":"Li, S., Wang, Q., Liu, X., and Chen, J. (2018, January 26\u201330). Low cost LSTM implementation based on stochastic computing for channel state information prediction. Proceedings of the 2018 IEEE Asia Pacific Conference on Circuits and Systems (APCCAS), Chengdu, China.","DOI":"10.1109\/APCCAS.2018.8605569"},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Maor, G., Zeng, X., Wang, Z., and Hu, Y. (2019, January 17\u201320). An FPGA implementation of stochastic computing-based LSTM. Proceedings of the 2019 IEEE 37th International Conference on Computer Design (ICCD), Abu Dhabi, United Arab Emirates.","DOI":"10.1109\/ICCD46524.2019.00014"},{"key":"ref_83","doi-asserted-by":"crossref","unstructured":"Sengupta, R., Polian, I., and Hayes, J.P. (2022, January 6\u20138). Stochastic computing architectures for lightweight LSTM neural networks. Proceedings of the 2022 25th International Symposium on Design and Diagnostics of Electronic Circuits and Systems (DDECS), Prague, Poland.","DOI":"10.1109\/DDECS54261.2022.9770167"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"2213","DOI":"10.1109\/TVLSI.2019.2920152","article-title":"An energy-efficient and noise-tolerant recurrent neural network using stochastic computing","volume":"27","author":"Liu","year":"2019","journal-title":"IEEE Trans. Very Large Scale Integr. Syst."},{"key":"ref_85","doi-asserted-by":"crossref","unstructured":"Smithson, S.C., Boga, K., Ardakani, A., Meyer, B.H., and Gross, W.J. (2016, January 26\u201328). Stochastic computing can improve upon digital spiking neural networks. Proceedings of the 2016 IEEE International Workshop on Signal Processing Systems (SiPS), Dallas, TX, USA.","DOI":"10.1109\/SiPS.2016.61"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"7155","DOI":"10.1007\/s11063-023-11255-8","article-title":"Hardware Spiking Neural Networks with Pair-Based STDP Using Stochastic Computing","volume":"55","author":"Liu","year":"2023","journal-title":"Neural Process. Lett."},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Song, Z., Katti, P., Simeone, O., and Rajendran, B. (2024). Stochastic Spiking Attention: Accelerating Attention with Stochastic Computing in Spiking Networks. arXiv.","DOI":"10.1109\/AICAS59952.2024.10595893"},{"key":"ref_88","doi-asserted-by":"crossref","unstructured":"Chen, Z., Ma, Y., and Wang, Z. (2020, January 2\u20135). Optimizing stochastic computing for low latency inference of convolutional neural networks. Proceedings of the 39th International Conference on Computer-Aided Design, Storrs, CT, USA.","DOI":"10.1145\/3400302.3415697"},{"key":"ref_89","doi-asserted-by":"crossref","unstructured":"Sim, H., Kenzhegulov, S., and Lee, J. (2018, January 24\u201329). DPS: Dynamic precision scaling for stochastic computing-based deep neural networks. Proceedings of the 55th Annual Design Automation Conference, San Francisco, CA, USA.","DOI":"10.1145\/3195970.3196028"},{"key":"ref_90","doi-asserted-by":"crossref","unstructured":"Hojabr, R., Givaki, K., Tayaranian, S.R., Esfahanian, P., Khonsari, A., Rahmati, D., and Najafi, M.H. (2019, January 2\u20136). SkippyNN: An embedded stochastic-computing accelerator for convolutional neural networks. Proceedings of the 56th Annual Design Automation Conference 2019, Las Vegas, Nevada, USA.","DOI":"10.1145\/3316781.3317911"},{"key":"ref_91","doi-asserted-by":"crossref","unstructured":"Wu, D., Li, J., Yin, R., Hsiao, H., Kim, Y., and San Miguel, J. (June, January 30). UGEMM: Unary computing architecture for GEMM applications. Proceedings of the 2020 ACM\/IEEE 47th Annual International Symposium on Computer Architecture (ISCA), Virtual Event.","DOI":"10.1109\/ISCA45697.2020.00040"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"4169","DOI":"10.1109\/TCAD.2022.3197503","article-title":"SASCHA\u2014Sparsity-aware stochastic computing hardware architecture for neural network acceleration","volume":"41","author":"Romaszkan","year":"2022","journal-title":"IEEE Trans.-Comput.-Aided Des. Integr. Circuits Syst."},{"key":"ref_93","doi-asserted-by":"crossref","unstructured":"Frasser, C.F., Linares-Serrano, P., Mor\u00e1n, A., Font-Rossell\u00f3, J., Canals, V., Roca, M., Serrano-Gotarredona, T., and Rossell\u00f3, J.L. (2021, January 24\u201326). Exploiting correlation in stochastic computing based deep neural networks. Proceedings of the 2021 XXXVI Conference on Design of Circuits and Integrated Systems (DCIS), Vila do Conde, Portugal.","DOI":"10.1109\/DCIS53048.2021.9666159"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"2278","DOI":"10.1109\/5.726791","article-title":"Gradient-based learning applied to document recognition","volume":"86","author":"LeCun","year":"1998","journal-title":"Proc. IEEE"},{"key":"ref_95","first-page":"1","article-title":"Imagenet classification with deep convolutional neural networks","volume":"25","author":"Krizhevsky","year":"2012","journal-title":"Adv. Neural Inf. Process. Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1701\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:57:38Z","timestamp":1760115458000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1701"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,21]]},"references-count":95,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["sym16121701"],"URL":"https:\/\/doi.org\/10.3390\/sym16121701","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,21]]}}}