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Michael Abdelmalik (Eindhoven University of Technology): Neural Green's Operators for Parametric Partial Differential Equations
We consider operator networks as promising machine learning tools for reduced order modeling of a wide range of physical systems described by partial differential equations (PDEs). This work introduces neural Green's operators (NGOs), a novel neural operator network architecture that learns the solution operator for a parametric family of linear partial differential equations (PDEs). Our construction of NGOs is derived directly from the Green's formulation of such a solution operator. Such a NGO acts as a surrogate for the PDE solution operator: it maps the PDE’s input functions (e.g. forcing, boundary conditions, PDE coefficients) to the solution. We apply NGOs to relevant canonical PDEs to demonstrate their efficacy and robustness as compared to a standard Deep Operator Networks [1], Variationally Mimetic Operator Networks [2] and Fourier Neural Operators [3]. Furthermore, we show that the explicit representation of the Green's function that is returned by NGOs enables the construction of effective preconditioners for numerical solvers for PDEs.
References
[1] Lu, Lu, et al. "Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.” In: Nat. Mach. Intell. 3.3 (2021).
[2] Dhruv Patel et al. “Variationally mimetic operator networks”. In: Comput. Methods Appl. Mech. Eng. 419 (2024).
[3] Zongyi Li et al. “Fourier neural operator for parametric partial differential equations”. In: arXiv preprint arXiv:2010.08895 (2020).
