arXiv subjects

Michael Abdelmalik

Publications and source records attributed to Michael Abdelmalik.

4 recordsLinked to original sources

arXiv2026

When can a neural operator replace a coarse solve? Architectural principles for two-level preconditioning

Neural operators are increasingly used as accelerators inside classical numerical methods, but it is rarely clear which architectural ingredients matter for which application. We answer this question for one important use case: the coarse-space correction inside a two-level preconditioner for discretised linear partial differential equations. We systematically vary four DeepONet-like architectures along two design axes: input discretisation (sampling versus integration against a basis) and source-term linearity. In doing this, 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. 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.

math.NA↗

arXiv2024

Neural Green's Operators for Parametric Partial Differential Equations

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.

cs.LG↗

arXiv2023

A conservative Galerkin solver for the quasilinear diffusion model in magnetized plasmas

The quasilinear theory describes the resonant interaction between particles and waves with two coupled equations: one for the evolution of the particle probability density function(\textit{pdf}), the other for the wave spectral energy density(\textit{sed}). In this paper, we propose a conservative Galerkin scheme for the quasilinear model in three-dimensional momentum space and three-dimensional spectral space, with cylindrical symmetry. We construct an unconditionally conservative weak form, and propose a discretization that preserves the unconditional conservation property, by "unconditional" we mean that conservation is independent of the singular transition probability. The discrete operators, combined with a consistent quadrature rule, will preserve all the conservation laws rigorously. The technique we propose is quite general: it works for both relativistic and non-relativistic systems, for both magnetized and unmagnetized plasmas, and even for problems with time-dependent dispersion relations. We represent the particle \textit{pdf} by continuous basis functions, and use discontinuous basis functions for the wave \textit{sed}, thus enabling the application of a positivity-preserving technique. The marching simplex algorithm, which was initially designed for computer graphics, is adopted for numerical integration on the resonance manifold. We introduce a semi-implicit time discretization, and discuss the stability condition. In addition, we present numerical examples with a "bump on tail" initial configuration, showing that the particle-wave interaction results in a strong anisotropic diffusion effect on the particle \textit{pdf}.

physics.comp-ph↗

arXiv2021

Goal-Oriented Adaptive THB-Spline Schemes for PDE-Based Planar Parameterization

This paper presents a PDE-based planar parameterization framework with support for Truncated Hierarchical B-Splines (THB-splines). For this, we adopt the a posteriori refinement strategy of Dual Weighted Residual and present several adaptive numerical schemes for the purpose of approximating an inversely harmonic geometry parameterization. Hereby, the combination of goal-oriented a posteriori strategies and THB-enabled local refinement avoids over-refinement, in particular in geometries with complex boundaries. To control the parametric properties of the outcome, we introduce the concept of domain refinement. Hereby, the properties of the domain into which the mapping maps inversely harmonically, are optimized in order to fine-tune the parametric properties of the recomputed geometry parameterization.

math.NA↗