Neural Green's Operators for Parametric Partial Differential Equations

[2406.01857] Neural Green's Operators for Parametric Partial Differential Equations Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Machine Learning arXiv:2406.01857 (cs) [Submitted on 4 Jun 2024 ( v1 ), last revised 22 Jan 2026 (this version, v5)] Title: Neural Green's Operators for Parametric Partial Differential Equations Authors: Hugo Melchers , Joost Prins , Michael Abdelmalik View a PDF of the paper titled Neural Green's Operators for Parametric Partial Differential Equations, by Hugo Melchers and 2 other authors View PDF HTML (experimental) Abstract: This work introduces a paradigm for constructing parametric neural operators that are derived from finite-dimensional representations of Green's operators for linear partial differential equations (PDEs). We refer to such neural operators as Neural Green's Operators (NGOs). Our construction of NGOs preserves the linear action of Green's operators on the inhomogeneity fields, while approximating the nonlinear dependence of the Green's function on the coefficients of the PDE using neural networks. This construction reduces the complexity of the problem from learning the entire solution operator and its dependence on all parameters to only learning the Green's function and its dependence on the PDE coefficients. Furthermore, we show that our explicit representation of Green's functions enables the embedding of desirable mathematical attributes in our NGO architectures, such as symmetry, spectral, and conservation properties. Through numerical benchmarks on canonical PDEs, we demonstrate that NGOs achieve comparable or superior accuracy to Deep Operator Networks, Variationally Mimetic Operator Networks, and Fourier Neural Operators with similar parameter counts, while generalizing significantly better when tested on out-of-distribution data. For parametric time-dependent PDEs, we show that NGOs that are trained on a single time step can produce pointwise-accurate dynamics in an auto-regressive manner over arbitrarily large numbers of time steps. For parametric nonlinear PDEs, we demonstrate that NGOs trained exclusively on solutions of corresponding linear problems can be embedded within iterative solvers to yield accurate solutions, provided a suitable initial guess is available. Finally, we show that we can leverage the explicit representation of Green's functions returned by NGOs to construct effective matrix preconditioners that accelerate iterative solvers for PDEs. Subjects: Machine Learning (cs.LG) ; Numerical Analysis (math.NA) MSC classes: 68T07 ACM classes: I.2.6; G.1.8 Cite as: arXiv:2406.01857 [cs.LG] (or arXiv:2406.01857v5 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2406.01857 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Computer Methods in Applied Mechanics and Engineering 455 (2026) 118893 Related DOI : https://doi.org/10.1016/j.cma.2026.118893 Focus to learn more DOI(s) linking to related resources Submission history From: Hugo Melchers [ view email ] [v1] Tue, 4 Jun 2024 00:02:52 UTC (7,725 KB) [v2] Wed, 19 Feb 2025 17:20:19 UTC (19,779 KB) [v3] Thu, 20 Feb 2025 09:28:22 UTC (19,779 KB) [v4] Mon, 20 Oct 2025 09:59:13 UTC (6,672 KB) [v5] Thu, 22 Jan 2026 09:00:01 UTC (7,324 KB) Full-text links: Access Paper: View a PDF of the paper titled Neural Green's Operators for Parametric Partial Differential Equations, by Hugo Melchers and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG < prev | next > new | recent | 2024-06 Change to browse by: cs cs.NA math math.NA References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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