Solving nonlinear PDEs with Quantum Neural Networks: A variational approach to the Bratu Equation

Quantum Physics arXiv:2601.04372 (quant-ph) [Submitted on 7 Jan 2026] Title:Solving nonlinear PDEs with Quantum Neural Networks: A variational approach to the Bratu Equation Authors:Nikolaos Cheimarios View a PDF of the paper titled Solving nonlinear PDEs with Quantum Neural Networks: A variational approach to the Bratu Equation, by Nikolaos Cheimarios View PDF Abstract:We present a variational quantum algorithm (VQA) to solve the nonlinear one-dimensional Bratu equation. By formulating the boundary value problem within a variational framework and encoding the solution in a parameterized quantum neural network (QNN), the problem reduces to an optimization task over quantum circuit parameters. The trial solution incorporates both classical approximations and boundary-enforcing terms, allowing the circuit to focus on minimizing the residual of the differential operator. Using a noiseless quantum simulator, we demonstrate that the method accurately captures both solution branches of the Bratu equation and shows excellent agreement with classical pseudo arc-length continuation results. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2601.04372 [quant-ph] (or arXiv:2601.04372v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2601.04372 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Nikolaos Cheimarios [view email] [v1] Wed, 7 Jan 2026 20:29:51 UTC (881 KB) Full-text links: Access Paper: View a PDF of the paper titled Solving nonlinear PDEs with Quantum Neural Networks: A variational approach to the Bratu Equation, by Nikolaos CheimariosView PDF view license Current browse context: quant-ph new | recent | 2026-01 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark 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?) Links to Code Toggle Papers with Code (What is Papers with Code?) 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?) 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?)