Neural Green's Operators for Parametric Partial Differential Equations

2024-06-04

arXiv.org

AI Summary

This work addresses the challenge of improving robustness and generalization of neural operators for learning solution operators of parametric linear partial differential equations (PDEs), particularly under out-of-distribution (e.g., fine-grid) settings and with explicit incorporation of interpretable Green’s functions. We propose the Neural Green Operator (NGO), the first neural operator architecture to explicitly embed Green’s integral formula. NGO avoids discretization-induced bias by taking a weighted average of the input function as its input representation. It employs a dual-subnetwork design—comprising a coefficient network and a basis network—to jointly parameterize and explicitly output an approximate Green’s function, enabling direct integration into preconditioners for classical PDE solvers. Experiments demonstrate that NGO matches the in-distribution accuracy of DeepONet and Fourier Neural Operators, while exhibiting significantly superior out-of-distribution generalization. Moreover, the learned Green’s functions substantially accelerate convergence of traditional iterative solvers.

Technology Category

• Machine Learning: Efficient ML / Green AI
• Search and Optimization: Mixed Discrete/Continuous Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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. Similar to deep operator networks (DeepONets) and variationally mimetic operator networks (VarMiONs), NGOs constitutes an expansion of the solution to the PDE in terms of basis functions, that is returned from a sub-network, contracted with coefficients, that are returned from another sub-network. However, in accordance with the Green's formulation, NGOs accept weighted averages of the input functions, rather than sampled values thereof, as is the case in DeepONets and VarMiONs. Application of NGOs to canonical linear parametric PDEs shows that, while they remain competitive with DeepONets, VarMiONs and Fourier neural operators when testing on data that lie within the training distribution, they robustly generalize when testing on finer-scale data generated outside of the training distribution. 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.

Problem

• Neural Green's operators for parametric PDEs
• Learning solution operators using neural networks
• Robust generalization to finer-scale data

Innovation

• Neural Green's Operators introduced
• Weighted averages for input functions
• Effective preconditioners for PDE solvers