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Error estimates for the generalization error for both state and control are derived from classical observability inequalities and energy estimates for the considered PDE. These error bounds, that apply to any exact controllable linear system of PDEs and in any dimension, provide a rigorous justification for the use of neural networks in this field. Preliminary numerical simulation results for three different types of PDEs are carried out to illustrate the performance of the proposed methodology.<\/jats:p>","DOI":"10.1007\/s10957-022-02100-4","type":"journal-article","created":{"date-parts":[[2022,9,17]],"date-time":"2022-09-17T06:02:36Z","timestamp":1663394556000},"page":"391-414","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Control of Partial Differential Equations via Physics-Informed Neural Networks"],"prefix":"10.1007","volume":"196","author":[{"given":"Carlos J.","family":"Garc\u00eda-Cervera","sequence":"first","affiliation":[]},{"given":"Mathieu","family":"Kessler","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7323-1809","authenticated-orcid":false,"given":"Francisco","family":"Periago","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,17]]},"reference":[{"key":"2100_CR1","unstructured":"B\u00e1rcenas-Petisco, J.A.: Optimal control for neural ode in a long time horizon and applications to the classification and simultaneous controllability problems. https:\/\/hal.archives-ouvertes.fr\/hal-03299270\/ (2022)"},{"key":"2100_CR2","first-page":"1","volume":"18","author":"AG Baydin","year":"2018","unstructured":"Baydin, A.G., Pearlmutter, B.A., Radul, A.A., Siskind, J.M.: Automatic differentiation in Machine Learning: a survey. 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