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arxiv: 2406.01857 · v5 · submitted 2024-06-04 · 💻 cs.LG · cs.NA· math.NA

Neural Green's Operators for Parametric Partial Differential Equations

Hugo Melchers , Joost Prins , Michael Abdelmalik This is my paper

Pith reviewed 2026-05-24 00:23 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA
keywords neural operatorsGreen's functionsparametric PDEsoperator learningdeep learninggeneralizationpreconditionersPDE solvers
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The pith

Neural Green's Operators learn the coefficient dependence of Green's functions while exactly preserving their linear action on source terms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents Neural Green's Operators as a way to build parametric neural operators for PDEs by starting from finite-dimensional Green's operator representations. Instead of learning the full solution map and all its parameter dependencies at once, the construction uses neural networks only to approximate how the Green's function varies with PDE coefficients. The linear mapping from inhomogeneities to solutions remains exact by design in the finite representation. This setup allows natural inclusion of properties like symmetry and conservation laws. Tests on standard PDEs show the approach matches or exceeds the accuracy of other neural operator methods with comparable size while generalizing much better outside the training data distribution.

Core claim

The central construction defines Neural Green's Operators by approximating the nonlinear dependence of the Green's function on PDE coefficients via neural networks, while the linear action of the Green's operator on inhomogeneity fields is preserved exactly within the finite-dimensional representation. This reduces the learning problem from the full solution operator to only the Green's function and its coefficient dependence. The explicit form permits embedding of symmetry, spectral, and conservation properties into the architecture. Benchmarks demonstrate comparable or superior accuracy to Deep Operator Networks, Variationally Mimetic Operator Networks, and Fourier Neural Operators at same

What carries the argument

Finite-dimensional Green's operator representations in which neural networks approximate only the nonlinear coefficient dependence of the Green's function.

If this is right

  • Comparable or better accuracy than DeepONet, VMON, and FNO with similar parameter counts on canonical PDEs.
  • Significantly better generalization when tested on out-of-distribution data.
  • Single time-step training produces pointwise-accurate dynamics in autoregressive manner over arbitrarily large numbers of time steps for time-dependent parametric PDEs.
  • Training exclusively on linear problem solutions allows accurate solutions for nonlinear PDEs when embedded in iterative solvers with suitable initial guess.
  • Explicit Green's function representations enable construction of effective matrix preconditioners that accelerate iterative PDE solvers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Separating the linear and nonlinear parts in this way may make it easier to incorporate physical constraints or hybrid numerical-neural schemes.
  • The preconditioner application suggests NGOs could speed up traditional solvers even without full replacement.
  • Improved OOD performance might be due to the lower complexity of learning only the Green's function map.
  • Extension to more complex geometries or higher dimensions would test whether the finite-representation assumption scales.

Load-bearing premise

The nonlinear dependence of the Green's function on PDE coefficients can be approximated by neural networks in a way that keeps the exact linear action on inhomogeneity fields for the finite-dimensional representations used.

What would settle it

Run an NGO on a fixed set of coefficients and check whether solutions for linear combinations of source terms exactly match the linear combination of individual solutions, or measure accuracy collapse on coefficients outside the training distribution range.

Figures

Figures reproduced from arXiv: 2406.01857 by Hugo Melchers, Joost Prins, Michael Abdelmalik.

Figure 1
Figure 1. Figure 1: Architecture of the neural Green’s operator (NGO), that maps the material parameter θ(x), forcing f(x) and boundary conditions gi(x) onto the solution u(x). For a model-NGO, F is given by (3.2), whereas for a data-NGO, F is given by (3.5). The system￾network, shown in blue, is the only machine learning based component in the NGO. 3.1.2 Data-free NGO The data-free NGO can be used in the case where the targe… view at source ↗
Figure 2
Figure 2. Figure 2: The use cases and characteristics of the model NGO, data-free NGO and data NGO. data is available (for example, when dealing with experimental data). The input to the system net of a data NGO is given by Fn[θ] = Z Ω ψnθdx ′ , (3.5) and it is trained on the solution loss as given by Equation 3.3. Just like for a model NGO, the data NGO error lower bound in the norm of 3.3 is given by ∥u p − u∥. The use case… view at source ↗
Figure 3
Figure 3. Figure 3: Left: the exact Green’s function for the 1D advection-diffusion equation [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An overview of the filtering process as done in CNOs. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: An overview of the interpolation process as done in CNOs. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: a single Dirichlet kernel basis function. Right: a full function basis, [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relative test L2 error of model NGOs with and without embedded scale equivariance, versus the (a) offset bθ and (b) amplitude cθ of the material parameter θ(x) = bθ + cθGRFλθ (x). The scale equivariance makes the NGO more robust against scale variations in the material parameter θ(x). Points and error bars are, respectively, averages and 95% confidence intervals on 1000 steady diffusion manufactured soluti… view at source ↗
Figure 8
Figure 8. Figure 8: (a) Relative L2 test error versus the order K of the truncated Neumann series of a FEM that uses an approximate matrix inversion using a Neumann series, and a Neumann model NGO as defined in Equation (3.19)). The model NGO effectively learns about two additional terms of the Neumann series, and converges to the projection error at K ≈ 5. (b) The relative L2 test error versus the spectral radius of −δFF−1 0… view at source ↗
Figure 9
Figure 9. Figure 9: The unit square domain Ω, with Neumann boundaries ΓN on the top and bottom, and Dirichlet boundaries ΓD on the left and right. expressed in terms of the Green’s function as u(x) = Z Ω G[θ](x, x ′ )f(x ′ )dx ′ + Z ΓN G[θ](x, x ′ )η(x ′ )dx ′ − Z ΓD gˆn · ∇x′G[θ](x, x ′ )dx ′ . (4.2) All neural operators were trained to learn the solution operator (θ, f, η, g) 7→ u. The way the training data is generated is … view at source ↗
Figure 10
Figure 10. Figure 10: Five random samples of solutions from both the finite element data set [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Error box plots of the different models on the heat equation problem. [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: True solution and relative solution error of four different models, tested on [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: True solution and predicted solutions of four different models, tested on [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Relative L2 test error versus test data length scale for a bare NN (here a U￾Net), and the same NN, when used in a DeepONet, VarMiON, data NGO, data-free NGO and model NGO. Only the NGOs manage to keep the errors small inside and in particular outside of the training data distribution. This is a result of the NGOs’ embedded Green’s operator structure, and their more favorable scaling of model size with qu… view at source ↗
Figure 15
Figure 15. Figure 15: The relation between the input function quadrature points xq and the NN input for bare NNs, DeepONets, VarMiONs, data NGOs and model NGOs. In contrast to NNs, DeepONets and VarMiONs, the input vector size (and thereby often the model size) is independent of the quadrature density Q, which makes them more scalable to multiscale problems where finer quadrature is required. In [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 16
Figure 16. Figure 16: (a) Comparison of the error of a few canonical NN architectures when used naively in isolation, versus the same NN when used as system net in a 10x10 cubic B-spline model NGO. Whereas the FNO is most accurate of the bare NNs, the U-Net performs best when used as system net in this particular B-spline model NGO. (b) Comparison of the L2 projection errors of solutions projected on a few canonical bases havi… view at source ↗
Figure 17
Figure 17. Figure 17: The relative L2 test error of a data NGO, data-free NGO, model NGO, in comparison to FEM and projection errors, versus the (a) number of assembly quadrature points per dimension Q1/d, (b) number of loss quadrature points per dimension Q 1/d L , (c) number of basis functions N, (d) number of trainable parameters Nw, (e) number of training samples Ns, and (f) number of training epochs Ne. For all tests, the… view at source ↗
Figure 18
Figure 18. Figure 18: Convergence behaviour of GMRES for 100 × 100 finite-difference dis￾cretisations of the diffusion equation, without preconditioner (left) and with the NGO-based preconditioner (right). Each line corresponds to one linear system being solved [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Convergence behaviour of GMRES for 100 ×100 finite-difference discreti￾sations of the diffusion equation, with only a block-Jacobi preconditioner (left) and with both a block-Jacobi and NGO-based preconditioner (right). Each line corre￾sponds to one linear system being solved. 5 Extension to Other Problems 5.1 Time-Dependent Diffusion The last test problem we consider is a time-dependent diffusion problem… view at source ↗
Figure 20
Figure 20. Figure 20: The space-time domain Ω × [0, T], with Neumann boundaries ΓN on the top and bottom, Dirichlet boundaries ΓD on the sides, and the initial condition boundary t = 0 on the left side. solution to problem (5.1) can be expressed in terms of a Green’s operator as u(x, t) = Z Ω,T G[θ]f dx ′ dt′ − Z ΓN,T G[θ]ηdx ′ dt′ + Z ΓD,T gˆn · ∇G[θ]dx ′ dt′ − Z Ω G[θ](t = 0)u0dx ′ . (5.2) The derivation of Equation (5.2) is… view at source ↗
Figure 21
Figure 21. Figure 21: Relative L2 test error versus test data (a) time scale τ /T and (b) length scale λ/L for a bare NN (here a U-Net), and the same NN, when used in a DeepONet, VarMiON, data NGO, data-free NGO and model NGO. Trends are similar to those in [PITH_FULL_IMAGE:figures/full_fig_p034_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Example of a manufactured solution u with length and time scales λ/L = 0.1 and τ /T = 0.1, together with the projection u p onto the 4x10x10 B-spline basis, and the solution ˆu predicted by the model NGO. The manufactured solution is based on θ and u, defined as in Equation 3.13, where bu ∼ U(−1, 1), cu ∼ U(0, 1), bθ = 1, cθ ∼ U(0, 0.2). • Model NGOs construct the input matrix using the weak form from Baz… view at source ↗
Figure 23
Figure 23. Figure 23: A plot showing the mean relative error of FNO, CNO, data NGO, and [PITH_FULL_IMAGE:figures/full_fig_p037_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: From top to bottom: architectures of the DeepONet, FNO, and VarMiON. [PITH_FULL_IMAGE:figures/full_fig_p046_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Architecture of the model-based NGO (top) and data-based NGO (bot [PITH_FULL_IMAGE:figures/full_fig_p047_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: UNet (system) architectures of the VarMiON (top left), Data-NGO (top [PITH_FULL_IMAGE:figures/full_fig_p048_26.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces Neural Green's Operators (NGOs) as parametric neural operators derived from finite-dimensional representations of Green's operators for linear PDEs. The construction exactly preserves the linear action of the Green's operator on inhomogeneity fields while using neural networks to approximate the nonlinear dependence of the Green's function on PDE coefficients. This reduces the learning task and enables embedding of mathematical properties such as symmetry, spectral, and conservation attributes. Numerical benchmarks on canonical PDEs show NGOs achieving comparable or superior accuracy to DeepONet, VMON, and FNO with similar parameter counts and significantly better out-of-distribution generalization. Additional results cover auto-regressive time-stepping for time-dependent PDEs trained on single steps, embedding in iterative solvers for nonlinear PDEs, and construction of effective preconditioners from the explicit Green's function representations.

Significance. If the results hold, the work offers a principled way to incorporate classical Green's function theory into neural operator learning, separating linear action (preserved exactly) from nonlinear coefficient dependence (learned). This structure could improve generalization and enable embedding of desirable properties, with practical extensions to time-dependent problems, nonlinear solvers, and preconditioning. The approach stands out for reducing the learning problem while maintaining mathematical fidelity, potentially influencing future operator architectures for parametric PDEs.

minor comments (2)
  1. [Abstract] The abstract refers to 'canonical PDEs' without naming them; adding the specific equations (e.g., Poisson, heat, wave) would improve immediate readability.
  2. Notation for the finite-dimensional Green's operator representation and its discretization could be introduced with a brief example in the introduction or early methods section to aid readers unfamiliar with the construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for recommending acceptance. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs NGOs from finite-dimensional Green's operator representations for linear PDEs, preserving the exact linear action on inhomogeneity fields while using NNs only to approximate nonlinear coefficient dependence. This separation follows directly from classical Green's function theory without reduction to fitted inputs or self-referential definitions. Benchmarks compare against external baselines (DeepONet, FNO, etc.) on out-of-distribution data, and extensions (auto-regressive stepping, iterative solvers) are presented with explicit caveats. No load-bearing step equates a prediction to its own inputs by construction, and no self-citation chain is invoked for uniqueness or ansatz. The derivation remains self-contained against external mathematical and numerical evidence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full details on free parameters, axioms, and invented entities are unavailable. The abstract invokes standard Green's operator theory for linear PDEs and neural network approximation of coefficient dependence.

axioms (1)
  • domain assumption Green's operators exist and admit finite-dimensional representations for the linear PDEs under consideration
    Invoked implicitly when stating that NGOs are derived from such representations.

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Forward citations

Cited by 1 Pith paper

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