Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes

[2603.27936] Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes Skip to main content Search Submit Donate Log in Mathematics > Numerical Analysis arXiv:2603.27936 (math) [Submitted on 30 Mar 2026 ( v1 ), last revised 28 Aug 2026 (this version, v3)] Title: Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes Authors: Sean Disarò , Ruma Rani Maity , Aras Bacho View PDF HTML (experimental) Abstract: Nonlinear Partial Differential Equations (PDEs) are ubiquitous in mathematical physics and engineering. Although Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving PDE problems, they typically struggle to identify multiple distinct solutions, since they are designed to find one solution at a time. To address this limitation, we introduce Deflation-PINNs, a novel framework that integrates a deflation loss with an architecture based on PINNs and Deep Operator Networks (DeepONets). By incorporating a deflation term into the loss function, our method systematically forces the Deflation-PINN to seek and converge upon distinct finitely many solution branches. We provide theoretical results on the approximation capabilities of our model and demonstrate the efficacy of Deflation-PINNs through numerical experiments on the Landau-de Gennes model of liquid crystals, a system renowned for its complex energy landscape and multiple equilibrium states, and on an Allen--Cahn benchmark whose solution set is provably known. Our results show that Deflation-PINNs can successfully identify and characterize multiple distinct crystal structures: a single unsupervised run recovers all six stable states of the benchmark, each branch certified to lie in the basin of attraction of a different equilibrium, and the discovered branches are refined to percent-level accuracy by a purely neural Deflation--Deep-Ritz stage and to the accuracy of a mesh-converged reference by a classical solver that they initialize. Subjects: Numerical Analysis (math.NA) ; Machine Learning (cs.LG) Cite as: arXiv:2603.27936 [math.NA] (or arXiv:2603.27936v3 [math.NA] for this version) https://doi.org/10.48550/arXiv.2603.27936 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Aras Bacho [ view email ] [v1] Mon, 30 Mar 2026 01:09:00 UTC (1,660 KB) [v2] Tue, 31 Mar 2026 21:49:34 UTC (1,660 KB) [v3] Fri, 28 Aug 2026 11:02:18 UTC (4,922 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2026-03 Change to browse by: cs cs.LG cs.NA math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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? ) 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? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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