{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T22:33:40Z","timestamp":1783636420893,"version":"3.55.0"},"reference-count":28,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T00:00:00Z","timestamp":1750896000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100014718","name":"Innovative Research Group Project of the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51975253"],"award-info":[{"award-number":["51975253"]}],"id":[{"id":"10.13039\/100014718","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A significant advancement in Neural Network (NN) research is the integration of domain-specific knowledge through custom loss functions. This approach addresses a crucial challenge: How can models utilize physics or mathematical principles to enhance predictions when dealing with sparse, noisy, or incomplete data? Physics-Informed Neural Networks (PINNs) put this idea into practice by incorporating a forward model, such as Partial Differential Equations (PDEs), as soft constraints. This guidance helps the networks find solutions that align with established laws. Recently, researchers have expanded this framework to include Bayesian NNs (BNNs) which allow for uncertainty quantification. However, what happens when the governing equations of a system are not completely known? In this work, we introduce methods to automatically select PDEs from historical data in a parametric family. We then integrate these learned equations into three different modeling approaches: PINNs, Bayesian-PINNs (B-PINNs), and Physical-Informed Bayesian Linear Regression (PI-BLR). To assess these frameworks, we evaluate them on a real-world Multivariate Time Series (MTS) dataset related to electrical power energy management. We compare their effectiveness in forecasting future states under different scenarios: with and without PDE constraints and accuracy considerations. This research aims to bridge the gap between data-driven discovery and physics-guided learning, providing valuable insights for practical applications.<\/jats:p>","DOI":"10.3390\/e27070682","type":"journal-article","created":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T03:33:25Z","timestamp":1750995205000},"page":"682","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Physics-Informed Neural Networks with Unknown Partial Differential Equations: An Application in Multivariate Time Series"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4358-0768","authenticated-orcid":false,"given":"Seyedeh Azadeh Fallah","family":"Mortezanejad","sequence":"first","affiliation":[{"name":"School of Automotive and Traffic Engineering, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2014-4282","authenticated-orcid":false,"given":"Ruochen","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Automotive and Traffic Engineering, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0678-7759","authenticated-orcid":false,"given":"Ali","family":"Mohammad-Djafari","sequence":"additional","affiliation":[{"name":"International Science Consulting and Training (ISCT), 91440 Bures sur Yvette, France"},{"name":"Shanfeng Company, Shaoxing 312352, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. 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