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Best approximation results and essential boundary conditions for novel types of weak adversarial network discretizations for PDEs

Silvia Bertoluzza, Erik Burman, Cuiyu He

TL;DR

This work provides a rigorous analysis of weak adversarial networks (WAN) for high-dimensional PDEs, establishing the existence of weakly convergent discrete solutions and Céa-type quasi-optimality bounds under coercivity or discrete inf-sup frameworks. It introduces two stabilized WAN loss formulations that avoid direct normalization, along with boundary-imposing strategies (CutWAN) that effectively handle Dirichlet conditions via boundary-weighted test spaces and Hardy inequalities. The paper also develops and advocates XNODE and pseudo-time XNODE architectures to accelerate convergence for parabolic and static PDEs, with numerical experiments showing faster convergence and improved stability compared to traditional DNN/PINN approaches. Overall, the results provide both theoretical guarantees and practical tools to improve WAN-based PDE solvers in high dimensions, including stable boundary treatments and efficient neural-network structures.

Abstract

In this paper, we provide a theoretical analysis of the recently introduced weakly adversarial networks (WAN) method, used to approximate partial differential equations in high dimensions. We address the existence and stability of the solution, as well as approximation bounds. More precisely, we prove the existence of discrete solutions, intended in a suitable weak sense, for which we prove a quasi-best approximation estimate similar to Cea's lemma, a result commonly found in finite element methods. We also propose two new stabilized WAN-based formulas that avoid the need for direct normalization. Furthermore, we analyze the method's effectiveness for the Dirichlet boundary problem that employs the implicit representation of the geometry. The key requirement for achieving the best approximation outcome is to ensure that the space for the test network satisfies a specific condition, known as the inf-sup condition, essentially requiring that the test network set is sufficiently large when compared to the trial space. The method's accuracy, however, is only determined by the space of the trial network. We also devise a pseudo-time XNODE neural network class for static PDE problems, yielding significantly faster convergence results than the classical DNN network.

Best approximation results and essential boundary conditions for novel types of weak adversarial network discretizations for PDEs

TL;DR

This work provides a rigorous analysis of weak adversarial networks (WAN) for high-dimensional PDEs, establishing the existence of weakly convergent discrete solutions and Céa-type quasi-optimality bounds under coercivity or discrete inf-sup frameworks. It introduces two stabilized WAN loss formulations that avoid direct normalization, along with boundary-imposing strategies (CutWAN) that effectively handle Dirichlet conditions via boundary-weighted test spaces and Hardy inequalities. The paper also develops and advocates XNODE and pseudo-time XNODE architectures to accelerate convergence for parabolic and static PDEs, with numerical experiments showing faster convergence and improved stability compared to traditional DNN/PINN approaches. Overall, the results provide both theoretical guarantees and practical tools to improve WAN-based PDE solvers in high dimensions, including stable boundary treatments and efficient neural-network structures.

Abstract

In this paper, we provide a theoretical analysis of the recently introduced weakly adversarial networks (WAN) method, used to approximate partial differential equations in high dimensions. We address the existence and stability of the solution, as well as approximation bounds. More precisely, we prove the existence of discrete solutions, intended in a suitable weak sense, for which we prove a quasi-best approximation estimate similar to Cea's lemma, a result commonly found in finite element methods. We also propose two new stabilized WAN-based formulas that avoid the need for direct normalization. Furthermore, we analyze the method's effectiveness for the Dirichlet boundary problem that employs the implicit representation of the geometry. The key requirement for achieving the best approximation outcome is to ensure that the space for the test network satisfies a specific condition, known as the inf-sup condition, essentially requiring that the test network set is sufficiently large when compared to the trial space. The method's accuracy, however, is only determined by the space of the trial network. We also devise a pseudo-time XNODE neural network class for static PDE problems, yielding significantly faster convergence results than the classical DNN network.
Paper Structure (21 sections, 7 theorems, 97 equations, 9 figures, 3 tables)

This paper contains 21 sections, 7 theorems, 97 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The abstract setting
  3. The WAN method
  4. Analysis of the WAN method
  5. Coercive problems
  6. PDE without coercivity
  7. Two novel stabilized loss functions
  8. Stabilized WAN method
  9. A further stabilized WAN method
  10. Imposition of Dirichlet boundary conditions
  11. Neural Network Structures
  12. Deep Neural Network (DNN) Structure
  13. The recursive DNN model
  14. Comments about DNN
  15. XNODE model for parabolic PDE.
  16. ...and 6 more sections

Key Result

Lemma 1

\newlabellem:existence0 Let $\{w^n_{\boldsymbol{\theta}} \}$ be a minimizing sequence for eq:neteqred. Then, under assumption assumptionZT-1, there exists a subsequence weakly converging to an element $u^*_{\boldsymbol{\theta}} \in \mathrm{cl}^{seq}_w( {W_{\boldsymbol{\theta}}})$ satisfying

Figures (9)

  • Figure 1: A DNN network structure with $5$ hidden layers
  • Figure 1: Example \ref{['ex1']}: Relative $L^{2}$ Error versus Step for models using $L_{\text{pinn}}$
  • Figure 2: Example \ref{['ex1']}: Relative $L^{2}$ Error versus Step for models using $L_{\text{wan}}$
  • Figure 3: Example \ref{['ex1']}: Relative $L^{2}$ Error for XNODE models on $L_{\text{cwan}}$ and $L_{\text{scwan}}$
  • Figure 4: Example \ref{['ex1']}: The effect using $\tilde{L}_\text{bdry}({\boldsymbol{\theta}})$
  • ...and 4 more figures

Theorems & Definitions (24)

  • Remark 1.1
  • Remark 1.2
  • Lemma 1
  • Proof 1
  • Lemma 2
  • Proof 2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3
  • ...and 14 more