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An Iterative Deep Ritz Method for Monotone Elliptic Problems

Tianhao Hu, Bangti Jin, Fengru Wang

TL;DR

This work develops the Iterative Deep Ritz Method (IDRM) to solve a broad class of monotone elliptic problems with weak regularity. By formulating a sequence of convex surrogate losses and introducing a locally convex potential together with a dual objective, IDRM achieves stable, preconditioned updates and provable convergence rates, balancing optimization and learning errors. The authors provide convergence and learning-error analyses grounded in Banach-space geometry and Rademacher complexity, and validate the method through diverse high-dimensional numerical experiments where IDRM outperforms existing solvers like PINN, DRM, and WAN in nonsmooth settings. The approach offers a practical, theory-backed pathway for robust neural PDE solvers applicable to nonlinear and nonvariational elliptic problems, with potential extensions to interface problems and discontinuous coefficients.

Abstract

In this work, we present a novel iterative deep Ritz method (IDRM) for solving a general class of elliptic problems. It is inspired by the iterative procedure for minimizing the loss during the training of the neural network, but at each step encodes the geometry of the underlying function space and incorporates a convex penalty to enhance the performance of the algorithm. The algorithm is applicable to elliptic problems involving a monotone operator (not necessarily of variational form) and does not impose any stringent regularity assumption on the solution. It improves several existing neural PDE solvers, e.g., physics informed neural network and deep Ritz method, in terms of the accuracy for the concerned class of elliptic problems. Further, we establish a convergence rate for the method using tools from geometry of Banach spaces and theory of monotone operators, and also analyze the learning error. To illustrate the effectiveness of the method, we present several challenging examples, including a comparative study with existing techniques.

An Iterative Deep Ritz Method for Monotone Elliptic Problems

TL;DR

This work develops the Iterative Deep Ritz Method (IDRM) to solve a broad class of monotone elliptic problems with weak regularity. By formulating a sequence of convex surrogate losses and introducing a locally convex potential together with a dual objective, IDRM achieves stable, preconditioned updates and provable convergence rates, balancing optimization and learning errors. The authors provide convergence and learning-error analyses grounded in Banach-space geometry and Rademacher complexity, and validate the method through diverse high-dimensional numerical experiments where IDRM outperforms existing solvers like PINN, DRM, and WAN in nonsmooth settings. The approach offers a practical, theory-backed pathway for robust neural PDE solvers applicable to nonlinear and nonvariational elliptic problems, with potential extensions to interface problems and discontinuous coefficients.

Abstract

In this work, we present a novel iterative deep Ritz method (IDRM) for solving a general class of elliptic problems. It is inspired by the iterative procedure for minimizing the loss during the training of the neural network, but at each step encodes the geometry of the underlying function space and incorporates a convex penalty to enhance the performance of the algorithm. The algorithm is applicable to elliptic problems involving a monotone operator (not necessarily of variational form) and does not impose any stringent regularity assumption on the solution. It improves several existing neural PDE solvers, e.g., physics informed neural network and deep Ritz method, in terms of the accuracy for the concerned class of elliptic problems. Further, we establish a convergence rate for the method using tools from geometry of Banach spaces and theory of monotone operators, and also analyze the learning error. To illustrate the effectiveness of the method, we present several challenging examples, including a comparative study with existing techniques.
Paper Structure (13 sections, 11 theorems, 150 equations, 9 figures, 3 tables, 1 algorithm)

This paper contains 13 sections, 11 theorems, 150 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mathematical formulation
  4. Challenges with existing neural PDE solvers
  5. Iterative deep Ritz method (IDRM)
  6. The derivation of IDRM loss
  7. Practical aspects of the IDRM solver
  8. Convergence results for IDRM
  9. Numerical experiments and discussions
  10. Proofs of Theorems \ref{['thm:modelconv']} and \ref{['thm:modelerr']}
  11. Proof of Theorem \ref{['thm:modelconv']}
  12. Proof of Theorem \ref{['thm:modelerr']}
  13. Conclusion

Key Result

Theorem 4.1

Let $u^*$ be the solution to problem maineq, let $c_{\rm p}$, $c_{\rm r}$ and $c_{\rm s}$ be three constants that depend on $p,\rho,R$ and $\mu$ (given in the proof) and let the sequence $\{u_{k}\}_{k=0}^\infty$ be generated by Algorithm alg:alg1 with the following step size schedule Then, under Assumptions eqassp and assump:bound, the following statements hold with $c_{\rm t}=c_{\rm p}^{-1}c_{\r

Figures (9)

  • Figure 1: The DNN approximations for Example \ref{['exam1:elli']}, slices at $x_i=\frac{1}{2}\ (i=3,4,\cdots,10)$. (top: IDRM, middle: WAN, bottom: PINN)
  • Figure 2: The training dynamics of IDRM for Example \ref{['exam1:elli']}: (a) the loss $L_\sigma^k$ versus the iteration index $i$, (b) the error $e$ versus the iteration index $i$.
  • Figure 3: The DNN approximation for Example \ref{['exam2:pLap']}(i), slices at $x_i=\frac{1}{2}(i=3,4,\cdots,10)$.
  • Figure 4: The training dynamics of IDRM for Example \ref{['exam2:pLap']}(i): (a) the loss $L_\sigma^k$ versus the iteration index $i$ and (b) the error $e$ versus the iteration index $i$.
  • Figure 5: The DNN approximations for Example \ref{['exam2:pLap']}(ii), slices at $x_i=\frac{1}{2}(i=3,4,\cdots,10)$ (top: IDRM, middle: WAN, bottom: PINN).
  • ...and 4 more figures

Theorems & Definitions (32)

  • Remark 2.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 4.1
  • Theorem 4.1
  • Remark 4.1
  • Theorem 4.2
  • Example 5.1
  • Example 5.2
  • ...and 22 more