{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,21]],"date-time":"2025-12-21T06:23:59Z","timestamp":1766298239442,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,4,25]],"date-time":"2025-04-25T00:00:00Z","timestamp":1745539200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union\u2019s Horizon 2020 research and innovation program under the Maria Sk\u0142odowska-Curie","award":["101034371","101092950"],"award-info":[{"award-number":["101034371","101092950"]}]},{"name":"European Union under the project EDGELESS","award":["101034371","101092950"],"award-info":[{"award-number":["101034371","101092950"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Probabilistic Boolean Networks (PBN) can model the dynamics of complex biological systems, as well as other non-biological systems like manufacturing systems and smart grids. In this proof-of-concept paper, we propose a PBN architecture with a learning process that significantly enhances fault and failure prediction in manufacturing systems. This concept was tested using a PBN model of an ultrasound welding process and its machines. Through various experiments, the model successfully learned to maintain a normal operating state. Leveraging the complex properties of PBNs, we utilize them as an adaptive learning tool with positive feedback, demonstrating that these networks may have broader applications than previously recognized. This multi-layered PBN architecture offers substantial improvements in fault detection performance within a positive feedback network structure that shows greater noise tolerance than other methods.<\/jats:p>","DOI":"10.3390\/e27050463","type":"journal-article","created":{"date-parts":[[2025,4,25]],"date-time":"2025-04-25T04:08:24Z","timestamp":1745554104000},"page":"463","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Learning Probabilistic Boolean Network Model of a Manufacturing Process with Applications in System Asset Maintenance"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3507-1821","authenticated-orcid":false,"given":"Pedro Juan","family":"Rivera Torres","sequence":"first","affiliation":[{"name":"Departmento de Inform\u00e1tica y Autom\u00e1tica, Universidad de Salamanca, Patio de las Escuelas 1, 37006 Salamanca, Spain"},{"name":"Escuela T\u00e9cnica Superior de Ingenier\u00eda Industrial de Barcelona, Universidad Polit\u00e9cnica de Catalu\u00f1a, Av. Diagonal, 647, 08028 Barcelona, Spain"},{"name":"St. Edmund\u2019s College, University of Cambridge, Mount Pleasant, Cambridge CB3 0BN, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5178-1191","authenticated-orcid":false,"given":"Chen","family":"Chen","sequence":"additional","affiliation":[{"name":"St. Edmund\u2019s College, University of Cambridge, Mount Pleasant, Cambridge CB3 0BN, UK"},{"name":"Department of Computer Science and Technology, University of Cambridge, Cambridge CB3 0FD, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3081-5177","authenticated-orcid":false,"given":"Sara","family":"Rodr\u00edguez Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Departmento de Inform\u00e1tica y Autom\u00e1tica, Universidad de Salamanca, Patio de las Escuelas 1, 37006 Salamanca, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6864-9629","authenticated-orcid":false,"given":"Orestes","family":"Llanes Santiago","sequence":"additional","affiliation":[{"name":"Departamento de Autom\u00e1tica y Computaci\u00f3n, Universidad Tecnol\u00f3gica de La Habana Jos\u00e9 Antonio Echeverr\u00eda (CUJAE), Marianao, La Habana 11500, Cuba"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Shmulevich, I., and Dougherty, E.R. (2010). Probabilistic Boolean Networks: Modeling and Control of Gene Regulatory Networks, SIAM.","DOI":"10.1137\/1.9780898717631"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1093\/bioinformatics\/18.2.261","article-title":"Probabilistic Boolean Networks: A Rule-Based Uncertainty Model for Gene Regulatory Networks","volume":"18","author":"Shmulevich","year":"2002","journal-title":"Bioinformatics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1016\/0022-5193(69)90015-0","article-title":"Metabolic stability and epigenesis in randomly constructed genetic nets","volume":"22","author":"Kauffman","year":"1969","journal-title":"J. Theor. 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