mlsanityData to 7 Oct 2026

DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems

arXiv 2101.07206 ↗First posted 31 Dec 2020math.NA cs.LG cs.NAcited by 61

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Abstract (arXiv)

Boundary value problems (BVPs) play a central role in the mathematical analysis of constrained physical systems subjected to external forces. Consequently, BVPs frequently emerge in nearly every engineering discipline and span problem domains including fluid mechanics, electromagnetics, quantum mechanics, and elasticity. The fundamental solution, or Green's function, is a leading method for solving linear BVPs that enables facile computation of new solutions to systems under any external forcing. However, fundamental Green's function solutions for nonlinear BVPs are not feasible since linear superposition no longer holds. In this work, we propose a flexible deep learning approach to solve nonlinear BVPs using a dual-autoencoder architecture. The autoencoders discover an invertible coordinate transform that linearizes the nonlinear BVP and identifies both a linear operator L and Green's function G which can be used to solve new nonlinear BVPs. We find that the method succeeds on a variety of nonlinear systems including nonlinear Helmholtz and Sturm--Liouville problems, nonlinear elasticity, and a 2D nonlinear Poisson equation. The method merges the strengths of the universal approximation capabilities of deep learning with the physics knowledge of Green's functions to yield a flexible tool for identifying fundamental solutions to a variety of nonlinear systems.

Key facts. Cited by 61 papers in the map (data through 7 Oct 2026). First posted on arXiv 31 Dec 2020.

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Citing papers per year: 2021: 8; 2022: 9; 2023: 13; 2024: 11; 2025: 10; 2026: 10.

61 papers.

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  13. 2023BINN: A deep learning approach for computational mechanics problems based on boundary integral equationscited by 10
  14. 2024Neural Green's Operators for Parametric Partial Differential Equationscited by 10
  15. 2023Interpreting and generalizing deep learning in physics-based problems with functional linear modelscited by 9
  16. 2022Data adaptive RKHS Tikhonov regularization for learning kernels in operatorscited by 8
  17. 2022Approximation of Functionals by Neural Network without Curse of Dimensionalitycited by 8
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  27. 2025Learning Singularity-Encoded Green's Functions with Application to Iterative Methodscited by 3
  28. 2026Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flowscited by 2
  29. 2021Operator Autoencoders: Learning Physical Operations on Encoded Molecular Graphscited by 1
  30. 2023Learning Green's Function Efficiently Using Low-Rank Approximationscited by 1
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  32. 2025chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Datacited by 1
  33. 2025Rates and architectures for learning geometrically non-trivial operatorscited by 1
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  35. 2026Learning Mesh-Free Discrete Differential Operators with Self-Supervised Graph Neural Networkscited by 1
  36. 2026Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networkscited by 1
  37. 2021Learning the structure of wind: A data-driven nonlocal turbulence model for the atmospheric boundary layercited by 0
  38. 2024A Hybrid Kernel-Free Boundary Integral Method with Operator Learning for Solving Parametric Partial Differential Equations In Complex Domainscited by 0

50 most cited of 61 All 61 (JSON)