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Meshfree Variational Physics Informed Neural Networks (MF-VPINN): an adaptive training strategy

Stefano Berrone, Moreno Pintore

TL;DR

A Meshfree Variational-Physics-Informed Neural Network that does not require the generation of the triangulation of the entire domain and that can be trained with an adaptive set of test functions is introduced.

Abstract

In this paper, we introduce a Meshfree Variational-Physics-Informed Neural Network. It is a Variational-Physics-Informed Neural Network that does not require the generation of the triangulation of the entire domain and that can be trained with an adaptive set of test functions. In order to generate the test space, we exploit an a posteriori error indicator and add test functions only where the error is higher. Four training strategies are proposed and compared. Numerical results show that the accuracy is higher than the one of a Variational-Physics-Informed Neural Network trained with the same number of test functions but defined on a quasi-uniform mesh.

Meshfree Variational Physics Informed Neural Networks (MF-VPINN): an adaptive training strategy

TL;DR

A Meshfree Variational-Physics-Informed Neural Network that does not require the generation of the triangulation of the entire domain and that can be trained with an adaptive set of test functions is introduced.

Abstract

In this paper, we introduce a Meshfree Variational-Physics-Informed Neural Network. It is a Variational-Physics-Informed Neural Network that does not require the generation of the triangulation of the entire domain and that can be trained with an adaptive set of test functions. In order to generate the test space, we exploit an a posteriori error indicator and add test functions only where the error is higher. Four training strategies are proposed and compared. Numerical results show that the accuracy is higher than the one of a Variational-Physics-Informed Neural Network trained with the same number of test functions but defined on a quasi-uniform mesh.
Paper Structure (12 sections, 48 equations, 18 figures, 1 table)

This paper contains 12 sections, 48 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Problem Discretization
  4. Loss Function
  5. The a Posteriori Error Estimator
  6. The Choice of $\cal M$ and $\cal P$
  7. Numerical Results
  8. Implementation Details
  9. Adaptive Training Strategies
  10. The Importance of the Error Indicator
  11. Extension to More a Complex Domain
  12. Conclusions and Discussion

Figures (18)

  • Figure 1: Graphical representation of a set $\{P_i\}_{i=1}^{n_\text{patches}}$ obtained from a squared reference patch $\hat{P}$ with $\bm{c_{\hat{P}}}$ in its center covering the domain $\Omega=(0,1)^2$.
  • Figure 2: Graphical representation of the solution $u$ in \ref{['eq:sol4']}.
  • Figure 3: Strategy #1: Relative $H^1$ errors obtained at the end of each training iteration for $C_M=4$ (blue circles) and $C_M=9$ (red triangles).
  • Figure 4: Strategy #1: Patches used to train the MF-VPINN with $C_M=4$. Each dot represents a patch $P_i$, its position is the center $\bm{c_{P_i}}$ of the patch, its size is proportional to the patch size $h_i^2$, and its color is associated with the quantity $\eta_i^\gamma$. (a) Representation of ${\cal P}_2$; (b) Representation of ${\cal P}_3$; (c) Representation of ${\cal P}_4$; (d) Representation of ${\cal P}_6$; (e) Representation of ${\cal P}_8$; (f) Representation of ${\cal P}_9$.
  • Figure 5: Strategy #1: Patches used to train the MF-VPINN with $C_M=9$. Each dot represents a patch $P_i$, its position is the center $\bm{c_{P_i}}$ of the patch, its size is proportional to the patch size $h_i^2$, and its color is associated with the quantity $\eta_i^\gamma$. (a) Representation of ${\cal P}_1$; (b) Representation of ${\cal P}_2$; (c) Representation of ${\cal P}_3$; (d) Representation of ${\cal P}_4$; (e) Representation of ${\cal P}_5$; (f) Representation of ${\cal P}_6$.
  • ...and 13 more figures