Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm

[2602.08515] Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm Skip to main content Search Submit Donate Log in Mathematics > Numerical Analysis arXiv:2602.08515 (math) [Submitted on 9 Feb 2026 ( v1 ), last revised 25 Aug 2026 (this version, v3)] Title: Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm Authors: Muhammad Luthfi Shahab , Imam Mukhlash , Hadi Susanto View PDF HTML (experimental) Abstract: This work investigates shallow physics-informed neural networks (PINNs) for solving forward and inverse problems governed by nonlinear partial differential equations (PDEs). By formulating PINN training as a nonlinear least-squares problem, the Levenberg-Marquardt (LM) algorithm is used to efficiently optimize the network parameters. Exact analytical expressions for neural-network derivatives with respect to the input variables are derived, revealing the relationships between the network output and its spatial and temporal derivatives and providing a clearer interpretation of the PINN architecture. These expressions are then used to derive explicit formulas for the Jacobian matrix required by LM. The proposed approach is evaluated on the Burgers, Schrödinger, Allen-Cahn, and three-dimensional Bratu equations. Numerical results show that LM substantially outperforms BFGS, L-BFGS, and Adam in convergence speed, accuracy, and final loss values. Comparisons with deeper networks further demonstrate that shallow LM-PINNs can achieve higher accuracy with substantially fewer parameters, emphasizing the importance of considering network architecture and optimization strategy jointly. The explicit analytical Jacobian also provides computational and memory advantages that are particularly relevant to large-scale PINNs. Overall, these results suggest that, for a broad class of PDEs, shallow PINNs combined with effective second-order optimization can provide accurate and computationally efficient solutions to both forward and inverse problems. Subjects: Numerical Analysis (math.NA) ; Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE); Optimization and Control (math.OC) Cite as: arXiv:2602.08515 [math.NA] (or arXiv:2602.08515v3 [math.NA] for this version) https://doi.org/10.48550/arXiv.2602.08515 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Muhammad Luthfi Shahab [ view email ] [v1] Mon, 9 Feb 2026 11:05:57 UTC (905 KB) [v2] Wed, 11 Feb 2026 10:16:12 UTC (861 KB) [v3] Tue, 25 Aug 2026 02:08:41 UTC (1,125 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2026-02 Change to browse by: cs cs.LG cs.NA cs.NE math math.OC 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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