# Neural Green's Operators for Parametric Partial Differential Equations arXiv: https://arxiv.org/abs/2406.01857 ยท HTML: https://mlsanity.com/paper/2406.01857/ ## Key facts - Cited by 11 papers in the map (data through 14 Oct 2026). - Builds on 26 of its 68 references that are in the map. - First posted on arXiv 4 Jun 2024. ## Builds on (most cited first, 26 of 26) - [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015) - [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020) - [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019) - [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021) - [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022) - [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020) - [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023) - [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023) - [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020) - [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022) - [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023) - [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022) - [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023) - [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021) - [Multigrid-augmented deep learning preconditioners for the Helmholtz equation](https://mlsanity.com/paper/2203.11025/) (2022) - [Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems](https://mlsanity.com/paper/1906.06925/) (2019) - [Variationally Mimetic Operator Networks](https://mlsanity.com/paper/2209.12871/) (2022) - [Neural incomplete factorization: learning preconditioners for the conjugate gradient method](https://mlsanity.com/paper/2305.16368/) (2023) - [Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver](https://mlsanity.com/paper/2105.11045/) (2021) - [A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples](https://mlsanity.com/paper/2402.07156/) (2024) - [Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows](https://mlsanity.com/paper/2010.02866/) (2020) - [Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layers](https://mlsanity.com/paper/2306.17486/) (2023) - [Generating synthetic data for neural operators](https://mlsanity.com/paper/2401.02398/) (2024) - [Deep Generalized Green's Functions](https://mlsanity.com/paper/2306.02925/) (2023) - [Resolution invariant deep operator network for PDEs with complex geometries](https://mlsanity.com/paper/2402.00825/) (2024) - [AI-enhanced iterative solvers for accelerating the solution of large scale parametrized systems](https://mlsanity.com/paper/2207.02543/) (2022) ## Built on by (most cited first, 11 of 11) - [On the definition and importance of interpretability in scientific machine learning](https://mlsanity.com/paper/2505.13510/) (2025) - [Learning Singularity-Encoded Green's Functions with Application to Iterative Methods](https://mlsanity.com/paper/2509.11580/) (2025) - [Accelerating Conjugate Gradient Solvers for Homogenization Problems with Unitary Neural Operators](https://mlsanity.com/paper/2508.02681/) (2025) - [An optimal Petrov-Galerkin framework for operator networks](https://mlsanity.com/paper/2503.04024/) (2025) - [The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositions](https://mlsanity.com/paper/2602.21910/) (2026) - [PIKFNO: An Interpretable Neural Operator Based on Physics Informed Kernel Function](https://mlsanity.com/paper/2608.14619/) (2026) - [Nonlinear elliptic homogenization with the parametric Deep Ritz method](https://mlsanity.com/paper/2609.05778/) (2026) - [SDC-GON: Singular Decomposition and Consistency-Regularized Green's Operator Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2609.23206/) (2026) - [Why Directly Learning Periodic Trajectories Can Fail](https://mlsanity.com/paper/2609.32254/) (2026) - [Learning PDE Dynamics between Submanifolds Using Green's Observation Operators](https://mlsanity.com/paper/2610.01697/) (2026) - [Green-Routed Neural Operators: Physics Determines Where the Network Reads](https://mlsanity.com/paper/2610.05337/) (2026) ## Abstract (arXiv) 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.