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Inexact Uzawa-Double Deep Ritz Method for Weak Adversarial Neural Networks

Emin Benny-Chacko, Ignacio Brevis, Luis Espath, Kristoffer G. van der Zee

TL;DR

The paper develops the Uzawa Deep Double Ritz Method, a mesh-free PDE solver that couples dual-norm residual minimization with Uzawa iterations, using neural networks to approximate both trial and test variables. It proves continuous-level convergence for an inexact (approximate) inner Ritz solves, provided updates move in the correct descent direction and the Uzawa step size is appropriately small. The method is instantiated as a Deep Ritz-energy-based framework with two neural networks per iteration and trained via a three-level, block-gradient scheme. Numerical experiments on a 1D transport problem validate the theoretical results, showing stable contraction and monotone energy decay even when inner problems are not fully solved. The approach offers a robust, scalable alternative to traditional mesh-based methods and adversarial neural PDE solvers, with strong theoretical guarantees and practical performance evidence.

Abstract

The emergence of deep learning has stimulated a new class of PDE solvers in which the unknown solution is represented by a neural network. Within this framework, residual minimization in dual norms -- central to weak adversarial neural network approaches -- naturally leads to saddle-point problems whose stability depends on the underlying iterative scheme. Motivated by this structure, we develop an inexact Uzawa methodology in which both trial and test functions are represented by neural networks and updated only approximately. We introduce the Uzawa Deep Double Ritz method, a mesh-free deep PDE solver equipped with a continuous level convergence showing that the overall iteration remains stable and convergent provided the inexact inner updates move in the correct descent direction. Numerical experiments validate the theoretical findings and demonstrate the practical robustness and accuracy of the proposed approach.

Inexact Uzawa-Double Deep Ritz Method for Weak Adversarial Neural Networks

TL;DR

The paper develops the Uzawa Deep Double Ritz Method, a mesh-free PDE solver that couples dual-norm residual minimization with Uzawa iterations, using neural networks to approximate both trial and test variables. It proves continuous-level convergence for an inexact (approximate) inner Ritz solves, provided updates move in the correct descent direction and the Uzawa step size is appropriately small. The method is instantiated as a Deep Ritz-energy-based framework with two neural networks per iteration and trained via a three-level, block-gradient scheme. Numerical experiments on a 1D transport problem validate the theoretical results, showing stable contraction and monotone energy decay even when inner problems are not fully solved. The approach offers a robust, scalable alternative to traditional mesh-based methods and adversarial neural PDE solvers, with strong theoretical guarantees and practical performance evidence.

Abstract

The emergence of deep learning has stimulated a new class of PDE solvers in which the unknown solution is represented by a neural network. Within this framework, residual minimization in dual norms -- central to weak adversarial neural network approaches -- naturally leads to saddle-point problems whose stability depends on the underlying iterative scheme. Motivated by this structure, we develop an inexact Uzawa methodology in which both trial and test functions are represented by neural networks and updated only approximately. We introduce the Uzawa Deep Double Ritz method, a mesh-free deep PDE solver equipped with a continuous level convergence showing that the overall iteration remains stable and convergent provided the inexact inner updates move in the correct descent direction. Numerical experiments validate the theoretical findings and demonstrate the practical robustness and accuracy of the proposed approach.

Paper Structure

This paper contains 15 sections, 3 theorems, 84 equations, 6 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Abstract Setting
  4. Residual Minimization and Saddle point problem
  5. Uzawa Method
  6. Convergence analysis
  7. Uzawa Double Ritz Method
  8. Inexact Uzawa Method
  9. Convergence analysis:
  10. Approximation with Neural Networks: Uzawa Deep Double Ritz Method
  11. Uzawa Deep Double Ritz Method
  12. Uzawa Deep Double Ritz Training
  13. Numerical Experiments
  14. Conclusion
  15. Acknowledgments

Key Result

Theorem 2.1

Classical Result: Convergence of Exact Uzawa Method Let $u^k \in U \text{ and } r^k \in V \text{ for } k= 0,1,2,\cdots$ be generated through the Uzawa iterative method uzawa_iterate. Suppose $u^* \in U \text{ and } r^* \in V$ be the saddle points of the saddle point problem mixed problem_cont. Then

Figures (6)

  • Figure 1: Residue r(x) plotted at Uzawa outer iteration: The shapes confirm the decay of the residual and illustrate the contraction properties of the Uzawa update
  • Figure 2: Evolution of inner-loop deep Ritz energy of r during the Uzawa outer iteration: shows the consistent descent behavior in each outer iteration.
  • Figure 3: Convergence of approximated $u$ towards the exact solution over successive outer Uzawa iterations.
  • Figure 4: Evolution of inner-loop deep Ritz energy of $u$ during the Uzawa outer iteration: shows the consistent descent behavior in each outer iteration.
  • Figure 5: Convergence of Uzawa Energy over successive Uzawa outer iteration
  • ...and 1 more figures

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Remark
  • proof
  • Remark