mlsanityData to 7 Oct 2026

Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver

arXiv 2105.11045 ↗First posted 23 May 2021cs.LG cs.NA math.NAcited by 15

Not on the map. Why?

Abstract (arXiv)

Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and image processing, and many analytic approaches or traditional numerical methods have been developed and widely used for their solutions. Inspired by rapidly growing impact of deep learning on scientific and engineering research, in this paper we propose a novel neural network, GF-Net, for learning the Green's functions of linear reaction-diffusion equations in an unsupervised fashion. The proposed method overcomes the challenges for finding the Green's functions of the equations on arbitrary domains by utilizing physics-informed approach and the symmetry of the Green's function. As a consequence, it particularly leads to an efficient way for solving the target equations under different boundary conditions and sources. We also demonstrate the effectiveness of the proposed approach by experiments in square, annular and L-shape domains.

Key facts. Cited by 15 papers in the map (data through 7 Oct 2026). Builds on 21 of its 49 references that are in the map. First posted on arXiv 23 May 2021.

Builds on

21 of 49 references.

References · 21

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  4. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  5. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  6. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  7. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  8. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  9. 2017PDE-Net: Learning PDEs from Datacited by 304
  10. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  11. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  12. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
9 more
  1. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  2. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  3. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 122
  4. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 63
  5. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 61
  6. 2021Data-driven discovery of Green's functions with human-understandable deep learningcited by 43
  7. 2018Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Datacited by 36
  8. 2017DR-RNN: A deep residual recurrent neural network for model reductioncited by 20
  9. 2019Reduced Order Modeling for Nonlinear PDE-constrained Optimization using Neural Networkscited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 4; 2025: 4; 2026: 4.

15 papers.

Sort:

  1. 2023TransNet: Transferable Neural Networks for Partial Differential Equationscited by 13
  2. 2024Neural Green's Operators for Parametric Partial Differential Equationscited by 10
  3. 2025Neural Green's Functionscited by 6
  4. 2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
  5. 2024Learning Domain-Independent Green's Function For Elliptic Partial Differential Equationscited by 5
  6. 2024An explainable operator approximation framework under the guideline of Green's functioncited by 3
  7. 2025Orthogonal greedy algorithm for linear operator learning with shallow neural networkcited by 3
  8. 2025Learning Singularity-Encoded Green's Functions with Application to Iterative Methodscited by 3
  9. 2023Learning Green's Function Efficiently Using Low-Rank Approximationscited by 1
  10. 2026Variational Green's Functions for Volumetric PDEscited by 1
  11. 2024Approximation of Solution Operators for High-dimensional PDEscited by 0
  12. 2025WellPINN: Accurate Well Representation for Transient Fluid Pressure Diffusion in Subsurface Reservoirs with Physics-Informed Neural Networkscited by 0
3 more
  1. 2026A transition-density-based operator learning method for Fokker-Planck equations with various initial conditionscited by 0
  2. 2026SDC-GON: Singular Decomposition and Consistency-Regularized Green's Operator Networks for Solving Partial Differential Equationscited by 0
  3. 2026Neural Harmonic Measure Operatorcited by 0