Neural Green's Operators for Parametric Partial Differential Equations

TL;DR

This work presents Neural Green's Operators (NGOs), a parametric neural operator framework built from finite-dimensional Green's operators with learnable Green's functions for linear PDEs. NGOs preserve the linear action of Green's operators on inhomogeneities while learning the nonlinear PDE-coefficient dependence through weighted-average inputs, reducing operator-learning complexity and enabling multi-scale resolution. Empirically, NGOs match or exceed the accuracy of DeepONets, VarMiONs, and Fourier neural operators at similar parameter counts, while generalizing significantly better to out-of-distribution data and enabling pointwise autoregressive dynamics for time-dependent problems. The explicit Green's-function representation also facilitates effective preconditioning for iterative solvers and the embedding of inductive biases such as symmetry, spectral properties, and conservation laws, with demonstrated benefits in stability and long-time integration.

Abstract

This work introduces a paradigm for constructing parametric neural operators that are derived from finite-dimensional representations of Green's operators, with learnable Green's functions, 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 that take weighted averages of such coefficients as input. 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. Moreover, taking weighted averages, rather than point samples, of input functions decouples the network size from the number of sampling points, enabling efficient resolution of multiple scales in the input fields. 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 time-dependent PDEs, we show that NGOs can produce pointwise-accurate dynamics in an auto-regressive manner when trained on a single time step. 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.

Figures (28)

• Figure 1: Architecture of the neural Green's operator (NGO), that maps the material parameter $\theta(\vb{x})$, forcing $f(\vb{x})$ and boundary conditions $g_i(\vb{x})$ onto the solution $u(\vb{x})$. For a model NGO, $\vb{F}$ is given by \ref{['E MNGO']}, whereas for a data NGO, $\vb{F}$ is given by \ref{['E DNGO']}. The system network, shown in blue, is the only trainable component in the NGO.
• Figure 2: The use cases and characteristics of the model NGO, data-free NGO and data NGO.
• Figure 3: Left: the exact Green's function for the 1D advection-diffusion equation $-u" + pu' = f$, for two different values of the advection speed $p$. Middle: the corresponding approximations to these Green's functions, approximated by an NGO. Right: the true Green's functions projected onto the NGO's basis, showing the best possible approximation of the Green's function by an NGO.
• Figure 4: The square domain $\Omega$, with Neumann boundaries $\Gamma_{\mathrm{N}}$ on the top and bottom, and Dirichlet boundaries $\Gamma_{\mathrm{D}}$ on the left and right.
• Figure 5: (a) Relative $L^2$ 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 \ref{['E Neumann series ansatz']}. (b) The relative $L^2$ test error versus the spectral radius $\rho(-\delta \vb{F} \vb{F}_0^{-1})$ for a vanilla model NGO, and a Neumann model NGO. the models have been trained on dataset C (Appendix \ref{['A data generation']}). Points and error bars are, respectively, averages and 95% confidence intervals on 1000 manufactured steady diffusion solutions, generated in the same way as dataset C.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4: Relation to CNOs