Correction: A Physics Informed Neural Network (PINN) framework for fractional order modeling of Alzheimer's disease

This study presents a novel fractional order model of Alzheimer's disease (mental disorder) using the Caputo derivative to accurately capture long term memory and hereditary effects in neurodegeneration. The mathematical model incorporates key pathological constituents including neurons, amyloid beta (Aβ), tau proteins and microglial responses, allowing detailed simulation of their dynamic interactions. Fundamental properties of the model, including positivity, boundedness, invariant regions and equilibrium points, are rigorously analyzed to ensure biological feasibility. Sensitivity analysis identifies amyloid toxicity as the most influential driver of neuronal loss underscoring its central role in AD progression. Furthermore, a Physics Informed Neural Network (PINN) is developed to approximate system dynamics from noisy observations while ensuring compliance with biological and physical constraints. Compared to standard neural networks the PINN exhibits superior accuracy and robustness especially under data scarcity. By integrating fractional calculus, optimal control and machine learning, this work advances computational modeling of Alzheimer's disease and offers insights into therapeutic optimization.

[This corrects the article DOI: 10.3389/fmicb.2025.1712709.].

[This corrects the article DOI: 10.3389/fphar.2022.1007556.].

• please read through all the templates before choosing • pick the most relevant text template(s) from the following page and delete all others.• edit the text as necessary, ensuring that the original incorrect text is included for the record, please see the below. • please do not use any extra formatting when editing the templates, and only modify the red text unless absolutely necessary • submit to Frontiers following the instructions on this page.When the original text contained incorrect information, to preserve the scientific record, please include that text when editing the below templates. For example:There was a mistake in the Funding statement, an incorrect number was used. The correct number is "2015C03Bd051.". The publisher apologizes for this mistake.The original version of this article has been updated.In the published article, there was a mistake in the Funding statement. The funding statement for the Key Development Project of the Department of Science and Technology was displayed as "2015CBd051". The correct statement is "Key Development Project of Department of Science and Technology (2015C03Bd051).''Template continues on the next page Last Name author1 et al. Frontiers Media SA PAGE \* Arabic \* MERGEFORMAT 3In the published article, there was an error in the Funding statement. [The author(s) declare that financial support was received for there search, authorship, and/or publication of this article. This study was supported by the Technology Project of State Grid Jiangsu Electric Power Company (Grant No. J2022096). The funder was not involved in the study design, collection, analysis, interpretation of data, the writing of this article, or the decision to submit it for publication.]. The correct Funding statement appears below.

• please read through all the templates before choosing • pick the most relevant text template(s) from the following page and delete all others.• edit the text as necessary, ensuring that the original incorrect text is included for the record, please see the below. • please do not use any extra formatting when editing the templates, and only modify the red text unless absolutely necessary • submit to Frontiers following the instructions on this page.When the original text contained incorrect information, to preserve the scientific record, please include that text when editing the below templates. For example:There was a mistake in the Funding statement, an incorrect number was used. The correct number is "2015C03Bd051.". The publisher apologizes for this mistake.The original version of this article has been updated.In the published article, there was a mistake in the Funding statement. In the published article, there was an error in the Funding statement. [Due to a different source of funding for the article, the funding statement should be changed]. The correct Funding statement appears below.

[This corrects the article DOI: 10.3389/fonc.2019.00224.].

[This corrects the article DOI: 10.3389/fnut.2022.1033265.].

• edit the text as necessary, ensuring that the original incorrect text is included for the record, please see the below.• please do not use any extra formatting when editing the templates, and only modify the red text unless absolutely necessary • submit to Frontiers following the instructions on this page.When the original text contained incorrect information, to preserve the scientific record, please include that text when editing the below templates. For example:There was a mistake in the Funding statement, an incorrect number was used. The correct number is "2015C03Bd051.". The publisher apologizes for this mistake.The original version of this article has been updated. In the published article, there was an error in the Funding statement. A source of funding was omitted from the original Funding statement: "The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This research was supported by the Southern California Earthquake Center with funding from National Science Foundation Cooperative Agreements EAR-1600087 and EAR-2225216 and United States Geological Survey Cooperative Agreements G17AC00047 and G22AC00070 (SCEC Contribution #13382). Additional support was provided by the Pacific Gas and Electric Company. This research used the Pegasus Workflow Management System funded by the National Science Foundation under the OAC SI2-SSI program, grant #1664162. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which was supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This computational work was performed using the DOE ALCC allocation, Improved Seismic Hazard Modeling Using Physics-based Simulations."The additional source of funding should appear after "Additional support was provided by the Pacific Gas and Electric Company". The correct Funding statement appears below.The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This research was supported by the Southern California Earthquake Center with funding from National Science Foundation Cooperative Agreements EAR-1600087Frontiers Media SA Template continues on the next page 

[This corrects the article DOI: 10.3389/fnut.2022.804046.].

Frontiers Corrigendum Template for AuthorsA corrigendum refers to a change to their article that the author wishes to publish after publication. The publication of this article is subject to Frontiers Editorial approval.Instructions:● please read through all the templates before choosing ● pick the most relevant text template(s) from the following page and delete all others.● edit the text as necessary, ensuring that the original incorrect text is included for the record, please see the below.● please do not use any extra formatting when editing the templates, and only modify the red text unless absolutely necessary● submit to Frontiers following the instructions on this page.When the original text contained incorrect information, to preserve the scientific record, please include that text when editing the below templates. For example:There was a mistake in the Funding statement, an incorrect number was used. The correct number is “2015C03Bd051.”. The publisher apologizes for this mistake. The original version of this article has been updated. In the published article, there was a mistake in the Funding statement. The funding statement for the Key Development Project of the Department of Science and Technology was displayed as “2015CBd051”. The correct statement is “Key Development Project of Department of Science and Technology (2015C03Bd051).’’ Corrigendum: Will the development of smart agriculture improve food safety in international markets? A theoretical approachAffiliations of all authors as they appear in the published original version of the article Lamia Meziani: LaMOS Research Unit, Faculty of Technology, University of Bejaia, Bejaia, Algeria, Mohammed Said Radjef: LaMOS Research Unit, Faculty of Exact Sciences, University of Bejaia, Bejaia, AlgeriaMaria Angela Perito: Department of Bioscience and Technology for Food, Agriculture and Environment, University of Teramo, Teramo, ItalyAbdelhakim Hammoudi: INRAE, AgroParisTech, Paris-Saclay Applied Economics, Paris-Saclay University, Palaiseau, France* Correspondence: abdelhakim.hammoudi@inrae.frKeywords: same as original article:food safety, producers’ digital capabilities, public policy, health control system, industrial organizationIncorrect AffiliationIn the published article, there was an error in affiliation(s) [1;2]. Instead of “[LaMOS Research Unit, Faculty of Exact Sciences, University of Bejaia, Bejaia, Algeria,]”, it should be “[1. LaMOS Research Unit, Faculty of Technology, University of Bejaia, Bejaia, Algeria; 2. LaMOS Research Unit, Faculty of Exact Sciences, University of Bejaia, Bejaia, Algeria]”. The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.

[This corrects the article DOI: 10.3389/fphar.2017.00224.].

[This corrects the article DOI: 10.3389/fmed.2022.876207.].

• please read through all the templates before choosing • pick the most relevant text template(s) from the following page and delete all others.• edit the text as necessary, ensuring that the original incorrect text is included for the record, please see the below. • please do not use any extra formatting when editing the templates, and only modify the red text unless absolutely necessary • submit to Frontiers following the instructions on this page.When the original text contained incorrect information, to preserve the scientific record, please include that text when editing the below templates. For example:There was a mistake in the Funding statement, an incorrect number was used. The correct number is "2015C03Bd051.". The publisher apologizes for this mistake.The original version of this article has been updated.In the published article, there was a mistake in the Funding statement. The funding statement for the Key Development Project of the Department of Science and Technology was displayed as "2015CBd051". The correct statement is "Key Development Project of Department of Science and Technology (2015C03Bd051).'' Template continues on the next page ê Corrigendum on: Rannankari, L., Burnham, R., and Duffus, D. (2024). Evidence of fin whale (Balaenoptera physalus velifera) recovery in the Canadian Pacific. Front. Conserv. Sci. 5:1392039. doi: 10.3389/fcosc.2024.1392039 In the published article, there was an error. In the published article, there was a mistake in the population abundance number for fin whales that was reported by Wright et al. 2021, as well as two other text mistakes within the same paragraph.A correction has been made to 4 Population abundance and structure, Paragraph Number 2. This sentence previously stated: "To date, an estimate of population abundance for fin whales in Canadian waters, especially for offshore regions is lacking where fin whales are presumed to be most numerous (COSEWIC, 2019). Dedicated, systematic surveys have estimated the population in BC to be approximately 400-500 individuals (2004-2005 survey, 496 individuals (95% CI: 202-1218) Williams and Thomas, 2007;2004-2008 survey, 446 individuals (95% CI: 263-759) Best et al., 2015). Nichol et al. ( 2018) confirmed this estimate from surveys conducted between 2009 and 2014 (405 individuals (95% CI: 363-469)), complemented using photo-identification to better estimate the number of individuals. These surveys highlighted whale 'hotspots' in Hecate Strait, and Queen Charlotte and Caamano Sounds (Harvey et al., 2017; Figure 1). Sightings interpolated using density surface modeling from the 2018 PRISM survey suggested a total count of 23,692 (95% CI: 19,121-29,356) fin whales for British Columbia from 29 sightings (Wright et al., 2021), far exceeding earlier estimates (see COSEWIC, 2019). Much more of these efforts were given to offshore survey. For the north-coast region, in an area comparable to the earlier work of Best et al. (2015) but ten years later, the model predicted 2,893 fin whales (95% CI: 2,171-3,855, Wright et al., 2021). Each of these dedicated surveys highlighted similar areas of increased whale density in BC."PAGE \* Arabic \* MERGEFORMAT 3The corrected sentence appears below: "To date, an estimate of population abundance for fin whales in Canadian waters, especially for offshore regions is lacking where fin whales are presumed to be most numerous (COSEWIC, 2019). Dedicated, systematic surveys have estimated the population in BC to be approximately 400-500 individuals (2004-2005 survey, 496 individuals (95% CI: 202-1218) Williams and Thomas, 2007;2004-2008survey, 446 individuals (95% CI: 263-759) Best et al., 2015). Nichol et al. (2018) confirmed this estimate from surveys conducted between 2009 and 2014 (405 individuals (95% CI: 363-469)), complemented using photo-identification to better estimate the number of individuals. These surveys highlighted whale 'hotspots' in Hecate Strait, and Queen Charlotte and Caamano Sounds (Harvey et al., 2017; Figure 1). Sightings interpolated using density surface modelling from the 2018 PRISMM survey suggested a total abundance of 2,893 (95% CI: 2171 -3855) fin whales in BC estimated from 235 sightings across two survey strata (Wright et al., 2021). This survey found over six times as many fin whale sightings in the offshore than the north coast stratum and, overall, exceeded earlier abundance estimates (see The Committee on the Status of Endangered Wildlife in Canada (COSEWIC), 2019)"The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.End of template, if you would like to request a correction for a reason not seen here, please contact the journal's Editorial Office

[This corrects the article DOI: 10.3389/fcimb.2022.1000291.].

• please read through all the templates before choosing • pick the most relevant text template(s) from the following page and delete all others.• edit the text as necessary, ensuring that the original incorrect text is included for the record, please see the below. • please do not use any extra formatting when editing the templates, and only modify the red text unless absolutely necessary • submit to Frontiers following the instructions on this page.When the original text contained incorrect information, to preserve the scientific record, please include that text when editing the below templates. For example:There was a mistake in the Funding statement, an incorrect number was used. The correct number is "2015C03Bd051.". The publisher apologizes for this mistake.The original version of this article has been updated.In the published article, there was a mistake in the Funding statement. The funding statement for the Key Development Project of the Department of Science and Technology was displayed as "2015CBd051". The correct statement is "Key Development Project of Department of Science and Technology (2015C03Bd051).'' Template continues on the next page 