Graph-Instructed Neural Networks for parametric problems with varying boundary conditions

TL;DR

This paper introduces Graph-Instructed Neural Networks (GINNs) to efficiently simulate parametric PDEs with varying boundary conditions, overcoming limitations of classical methods for real-time applications.

Contribution

The paper presents a novel GINN framework that learns the mapping between parametric domain descriptions and PDE solutions, enabling scalable and accurate simulations.

Findings

GINN models effectively handle complex parametric PDEs.

Compared to fully connected architectures, GINNs are more robust and scalable.

The approach is suitable for real-time simulation scenarios.

Abstract

This work addresses the accurate and efficient simulation of physical phenomena governed by parametric Partial Differential Equations (PDEs) characterized by varying boundary conditions, where parametric instances modify not only the physics of the problem but also the imposition of boundary constraints on the computational domain. In such scenarios, classical Galerkin projection-based reduced order techniques encounter a fundamental bottleneck. Parametric boundaries typically necessitate a re-formulation of the discrete problem for each new configuration, and often, these approaches are unsuitable for real-time applications. To overcome these limitations, we propose a novel methodology based on Graph-Instructed Neural Networks (GINNs). The GINN framework effectively learns the mapping between the parametric description of the computational domain and the corresponding PDE solution. Our results demonstrate that the proposed GINN-based models, can efficiently represent highly complex parametric PDEs, serving as a robust and scalable asset for several applied-oriented settings when compared with fully connected architectures.