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.

翻译：本研究提出了一种构建参数化神经算子的范式，该范式源于线性偏微分方程格林算子的有限维表示。我们将此类神经算子称为神经格林算子。NGO的构建保留了格林算子对非齐次场的线性作用，同时利用神经网络近似格林函数对偏微分方程系数的非线性依赖关系。这一构建将问题的复杂性从学习整个解算子及其对所有参数的依赖，降低为仅学习格林函数及其对偏微分方程系数的依赖。此外，我们证明了格林函数的这种显式表示能够将所需的数学特性嵌入到NGO架构中，例如对称性、谱特性和守恒特性。通过对典型偏微分方程的数值基准测试，我们证明在参数数量相近的情况下，NGO的精度与深度算子网络、变分模仿算子网络和傅里叶神经算子相当或更优，且在分布外数据测试中展现出显著更好的泛化能力。对于参数化时变偏微分方程，我们证明仅需单时间步训练的NGO能够以自回归方式在任意多时间步上产生逐点精确的动力学行为。对于参数化非线性偏微分方程，我们证明在可获得合适初始猜测的前提下，仅在线性问题解上训练的NGO可嵌入迭代求解器中以获得精确解。最后，我们展示了可利用NGO返回的格林函数显式表示来构建有效的矩阵预处理器，以加速偏微分方程的迭代求解过程。