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[Submitted on 15 Sep 2026] Title:Lecture notes on Physics Informed Neural Networks, Neural Operators, and their applications View a PDF of the paper titled Lecture notes on Physics Informed Neural Networks, Neural Operators, and their applications, by Alessandro Bombini View PDF HTML (experimental) Abstract:This is the set of lecture notes for the PhD course \href{this https URL}{\textit{Physics Informed Neural Network}, held at the University of Bozen/Bolzano} in the academic year 2025/2026. The goal of the course was to introduce the concept of Physics Informed Deep Neural Networks (PINN) and Neural Operators (NOs), discuss their implementation from scratch in PyTorch and using advanced ad-hoc developed open-source libraries such as NVIDia PhysicsNeMo to address real-world problems in various fields (engineering, physics, petroleum reservoir). We discuss recent topics such as Mixture-of-Models, Fourier Neural Operators, Physics-Informed Kolmogorov-Arnold Networks (PIKANs) and Fourier Neural Operators. Comments: 264 pages; Lecture notes for the PhD course Physics Informed Neural Network held at the University of Bozen/Bolzano Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computational Engineering, Finance, and Science (cs.CE); Computational Physics (physics.comp-ph) Cite as: arXiv:2609.17638 [cs.LG] (or arXiv:2609.17638v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2609.17638 arXiv-issued DOI via DataCite (pending registration) Submission history From: Alessandro Bombini [view email] [v1] Tue, 15 Sep 2026 12:57:03 UTC (37,047 KB) Full-text links: Access Paper: View a PDF of the paper titled Lecture notes on Physics Informed Neural Networks, Neural Operators, and their applications, by Alessandro Bombini View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-09 Change to browse by: cs cs.AI cs.CE physics physics.comp-ph References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)