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A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems

Tianhao Hu, Bangti Jin, Fengru Wang, Yifeng Xu

arXiv:2609.27325v1math.NAcs.LGcs.NA

TL;DR

Elliptic interface problems challenge neural solvers because solutions can be globally low-regularity and interfaces geometrically complex. The H-IDRM combines a mixed formulation with a level-set neural network, and numerical examples report better accuracy than DD-PINN and DRM in tested settings.

  • Problem

    Interface problems combine complex geometries with solutions that are regular within subdomains but may have singularities or internal layers across interfaces, challenging numerical resolution.

    Source: §1 · p. 1

  • Method

    H-IDRM uses a mixed formulation with iterative convex minimizations and a level-set neural network, converting interface coupling into volumetric integrals.

    Source: §5 · p. 16

  • Results

    The numerical results report that H-IDRM predicts the physical quantity of interest better than DD-PINN and DRM for the tested lower-regularity problem.

    Source: §4.2 · p. 9

  • Takeaways & Limitations

    The analysis establishes well-posedness of the mixed formulation, linear convergence of the iterative scheme, and approximation rates for the LSNN architecture.

    Source: §5 · p. 16

    The cited test setups cover a 3D unit ball with a level-set interface and a 5D unit ball with a concentric radius-0.5 interface.

    Source: §4.2 · pp. 11, 13

Abstract

from arXiv · show

In this work, we propose a hybrid iterative deep Ritz method (H-IDRM) for a class of interface problems for second-order elliptic operators. It is based on a new mixed formulation of the problem and involves solving a sequence of convex minimization problems. We employ a level-set neural network architecture, featuring a level-set representation of the interface, to accommodate the piecewise smoothness of the solution and the flux. The approach involves only volumetric representations instead of duality pairing on the interface and avoids explicit interface sampling that is inconvenient for complex interface geometries. Further, we present an analysis of the method, including the errors arising from the neural network approximation, Monte Carlo approximation, iterative scheme, and penalty parameters. Numerical experiments indicate that the H-IDRM outperforms existing neural solvers on problems with high-dimensional domains, intricate interface geometries, and lower subdomain regularity.

1 Introduction

Elliptic interface problems combine complex geometries with solutions that are smooth within subdomains but have lower global regularity, challenging numerical resolution. H-IDRM combines a mixed formulation, level-set networks, and iterative convex minimization, with analysis and experiments supporting its use in challenging settings.

Motivation

  • Solutions are typically highly regular within each subdomain but exhibit singularities or internal layers across the interface.This global low regularity and complex interface geometry challenge effective numerical resolution.Source: §1 · p. 1

Numerical challenges

  • Standard numerical methods can incur expensive mesh construction for irregular geometries, with computational cost escalating in high dimensions.PINNs are not directly applicable when the PDE does not hold strongly across the interface, and DRM can converge slowly under limited global regularity.Source: §1 · p. 2

Prior neural methods

  • Existing neural approaches face limitations in interface-condition enforcement, complex-interface sampling, single-network representation of jumps, and rigorous convergence analysis.Domain-decomposition methods commonly impose interface conditions using L2(Γ) penalties, while uniform sampling of complex interfaces remains challenging.Source: §1 · p. 2

Contributions

  • H-IDRM combines a mixed formulation, level-set neural networks, and iterative convex minimization, and the experiments report better performance than existing neural solvers.The paper also presents convergence analysis for this approach.Source: §1 · p. 2

2 Hybrid iterative deep Ritz method

H-IDRM combines a mixed formulation, an interface-aware level-set network, and an iterative optimization scheme. These components represent the solution and flux while reducing training to convex minimization subproblems.

2.1 Mixed formulation

  • The mixed formulation introduces flux p = a∇u and adds a regularized term to obtain a coercive formulation suited to iterative training.Its well-posedness is established under piecewise solution regularity and a positivity condition on c in each subdomain.Source: §2.1 · pp. 3–4

2.2 Level-set neural network

  • The LSNN encodes interface geometry through a smoothed level-set transformation, enabling distinct subdomain representations while maintaining solution continuity.Its transition steepness is controlled by α, and the implicit representation accommodates complex geometries without explicit meshing.Source: §2.2 · p. 5

2.3 Iterative deep Ritz method

  • The IDRM treats the nonsymmetric part of the mixed operator iteratively, producing a sequence of convex minimization subproblems.The practical surrogate loss adds a step-size-controlled update and a stabilization term enforcing p = a∇u.Source: §2.3 · pp. 5–6
  • The method discretizes loss integrals by Monte Carlo sampling and additionally penalizes boundary mismatch to enforce the Dirichlet condition in practice.The boundary penalty supplements the weak imposition in the mixed formulation.Source: §2.3 · p. 6

3 Convergence analysis

The analysis establishes local linear convergence of the iterative scheme and bounds learning error through approximation, statistical, and optimization components. It also derives approximation and statistical error bounds for the neural representation.

Iterative convergence

  • The iterative scheme converges locally at a linear rate during an initial phase, under a sufficiently small step size and controlled learning error.The operator's monotonicity allows the IDRM convergence analysis to carry over to the mixed formulation.Source: §3 · p. 7

Learning error

  • The learning error comprises approximation, statistical, and optimization errors, with the optimization component controlled by the stopping criterion.The analysis separately bounds approximation error and statistical error.Source: §3 · p. 7

Approximation error

  • The LSNN approximation result bounds solution error under the stated piecewise-regularity and level-set assumptions.The proof also yields a corresponding estimate in the Hdiv(Ω) norm for the flux.Source: §3 · p. 7

4 Numerical experiments and discussions

The experiments compare H-IDRM with neural PDE solvers across interface problems with varied geometry, regularity, degeneracy, and dimension. H-IDRM achieves accurate solution and flux approximations in these tests, including a five-dimensional problem with nonzero jumps.

4.1 Experimental setup

  • The experiments use 10,000 domain samples and 5,000 boundary samples, with Adam training for 50,000 epochs; baseline network capacities and training budgets are comparable.The setup compares H-IDRM with DRM, DD-PINN, and H-IDRM without LSNN using relative solution and flux errors.Source: §4.1 · p. 8, Table 1 · p. 8

4.2 Numerical results and discussions

  • H-IDRM attains better accuracy than DD-PINN and DRM for Example 4.1, where the solution's limited global regularity degrades both baselines.The DRM used a reduced-capacity network because matching H-IDRM capacity caused severe training instability.Source: §4.2 · pp. 8–10, Table 2 · p. 9, Figure 1 · p. 9
  • For the checkerboard problem, the crossing interface induces low solution regularity, and the reported results confirm H-IDRM's advantages on this case.The comparison is summarized in Table 3.Source: §4.2 · p. 9, Table 3 · p. 9
  • For Example 4.3, Case (a), H-IDRM outperforms the baselines; its convex-surrogate training shows monotonic loss decay and low eventual errors for u and p.The smooth-network variant has larger approximation error at a steep gradient.Source: §4.2 · p. 10, Figure 3 · p. 11, Figure 4 · p. 12
  • For degenerate Example 4.3, Case (b), regularized H-IDRM reaches relative errors eu = 1.44 × 10^-3 and ep = 8.56 × 10^-3.The mixed formulation lacks the desired coercivity in this case, so the method uses regularization.Source: §4.2 · p. 11, Figure 6 · p. 13
  • For the implicitly defined interface in Example 4.4, Case (a), H-IDRM reaches eu = 6.57 × 10^-3 and ep = 8.61 × 10^-3; omitting LSNN worsens accuracy by a factor of two to three.Direct uniform sampling on the interface is infeasible, and H-IDRM requires no interface quadrature.Source: §4.2 · pp. 11–12, Table 5 · p. 12, Figure 8 · p. 14
  • For degenerate Example 4.4, Case (b), H-IDRM achieves eu = 6.87 × 10^-3 and ep = 9.78 × 10^-3 despite the geometric complexity.The example uses a regularized formulation.Source: §4.2 · pp. 12–13, Figure 10 · p. 15
  • In the five-dimensional example, H-IDRM achieves relative errors eu = 2.71 × 10^-3 and ep = 4.84 × 10^-3 while handling inhomogeneous interface conditions.A two-stage procedure first matches the prescribed jumps, then applies H-IDRM to the zero-jump correction.Source: §4.2 · pp. 14–15, Figure 11 · p. 16

5 Conclusions

The H-IDRM combines a mixed formulation, level-set neural network, and iterative convex minimization to handle elliptic interface problems; analysis and experiments support its convergence properties and performance on challenging cases.

5 Conclusions

  • The H-IDRM combines a mixed formulation and level-set neural network with iterative convex minimization, avoiding explicit interface sampling through volumetric integrals.The work also proves well-posedness of the mixed formulation, linear convergence of the iterative scheme, and LSNN approximation rates.Source: §5 · p. 16
  • Numerical experiments show that H-IDRM consistently outperforms existing neural PDE solvers, particularly for intricate interfaces or lower global regularity.Source: §5 · p. 16

A Proof of Theorem 3.2

The proof constructs and approximates an auxiliary function from extended subdomain solutions, relates it to the LSNN output, controls errors near and away from the interface, and balances parameters to obtain the approximation rate.

A Proof of Theorem 3.2

  • The proof first constructs an auxiliary function from extended subdomain solutions and approximates it with a neural network.The auxiliary function interpolates the two extensions affinely in the added coordinate.Source: A Proof of Theorem 3.2 · pp. 16–17
  • The level-set transformation links the neural network approximation to the LSNN output, which is then compared with the target solution.Source: A Proof of Theorem 3.2 · pp. 16–17
  • The error estimate separates regions away from the interface from the interface-near region and bounds their contributions separately.The regions are defined using the level-set function and a small threshold.Source: A Proof of Theorem 3.2 · pp. 16–18
  • Balancing the error terms by choosing parameters of the same order yields the desired approximation rate.Source: A Proof of Theorem 3.2 · p. 18
  • A mollification argument extends the approximation bound beyond r ≥ 1 to the full range r > 0.Source: A Proof of Theorem 3.2 · p. 18
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