Michael Abdelmalik

Hugo Melchers, Michael Abdelmalik, Victorita Dolean

Neural operators are increasingly used as drop-in accelerators inside classical numerical methods, but it is rarely clear which architectural ingredients matter for which role. We answer this question for one important role: the coarse-space correction inside a two-level preconditioner for discretised linear partial differential equations. By systematically varying four DeepONet-like architectures along two design axes - input discretisation (sampling versus integration against a basis) and source-term linearity - we show that the favourable corner of this 2$\times$2 design is occupied by a single architecture, the Neural Green's Operator (NGO), and that moving away from it produces predictable failure modes: structurally non-symmetric preconditioned spectra, breakdown of preconditioned conjugate gradients on self-adjoint problems, and stagnation on non-self-adjoint ones. Used as a coarse-space correction, the NGO matches the iteration count of an exact coarse solve on diffusion and advection-diffusion problems. We also characterise the failure of fixed-size learned coarse spaces at high Helmholtz wave numbers, isolating it as a property of the basis rather than of the architecture. The principle generalises: integrating inputs against the basis used for the output is what allows a neural operator to serve as a Galerkin-type coarse-space correction.

Hugo Melchers, Joost Prins, Michael Abdelmalik

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.