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

Francesco Della Santa, Sandra Pieraccini, Maria Strazzullo

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.

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

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.
Paper Structure (23 sections, 19 equations, 15 figures, 5 tables)

This paper contains 23 sections, 19 equations, 15 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Continuous problem
  4. Discrete formulation
  5. State of the art for efficiently solving parametric PDEs
  6. Surrogate Models based on ROMs
  7. Surrogate Models based on ML
  8. $\boldsymbol{\mu}$BC-NNs: NNs for Parametric PDEs with Parametric Boundary Conditions
  9. Graph-Instructed Layers and Graph-Instructed NNs
  10. Architecture Archetypes for $\boldsymbol{\mu}$BC-NNs
  11. $\boldsymbol{\mu}$BC-GINN: $\boldsymbol{\mu}$BC-NN Based on GI Layers
  12. $\boldsymbol{\mu}$BC-NN Based on Conv1D and FC Layers
  13. Preprocessing and Postprocessing Operations
  14. Training Loss for $\boldsymbol{\mu}$BC-NNs
  15. Numerical Results
  16. ...and 8 more sections

Figures (15)

  • Figure 1: Representation of a general domain $\Omega$ for two values of $\mu_b$, i.e., $\mu_b = \mu_b^*$ (left) and $\mu_b = \overline{\mu_b}$ (right). The solid, the dotted, and the dashed-dotted lines represent $\Gamma_D^{\boldsymbol{\mu}_b}, \Gamma_N^{\boldsymbol{\mu}_b}$, and $\Gamma_{\text{fix}}$, respectively.
  • Figure 2: Schematic representation of a mesh discretization on a square domain $\Omega$ with a portion of the boundary under the action of the parameters $\boldsymbol{\mu}_b$ and $\boldsymbol{\mu}_v$. This portion is $\Gamma^{\boldsymbol{\mu}_b}$, the continuous, grey, and thick line. The nodes that may change features are denoted with square and circle dots, while the nodes of the boundary with fixed features are denoted with black crosses. the color of the square and circle dotes denote the value of the BCs described by $\boldsymbol{\mu}_v$.
  • Figure 3: Visual representation of \ref{['eq:ginn_node_action']}. Example with $i=1$ and a non-directed graph of four nodes.
  • Figure 4: Visual representation of a GI layer as a "constrained" FC layer (subfigure ($A$)), with weight matrix defined by \ref{['eq:GI_weights_simple']} (subfigure ($B$)). This figure is based on the same graph illustrated in Figure \ref{['fig:ginnfilter']}.
  • Figure 5: Experiment 1. Spatial domain $\Omega$ for a specific parametric instance: schematic representation. (A) The homogeneous Dirichlet boundary, the non-homogeneous and homogeneous Neumann boundaries are applied to $\Gamma_D$ (solid line), $\Gamma_1^{\boldsymbol{\mu}_b}$ (dashed line), and $\Gamma_0^{\boldsymbol{\mu}_b} \cup \Gamma_N$ (dash-dotted and dotted lines), respectively. (B) Example of selection of circle intervals. The summation only represents the detection of the start and the end node of the intervals.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Remark 2.1
  • Remark 4.1
  • Remark 4.2
  • Remark 5.1: ROMs for Neumann varying boundary conditions