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Mixed Consistent PINNs for Elliptic Obstacle Problems with Stability Analysis

Arbaz Khan, Kent-Andre Mardal, Shiv Mishra

Abstract

We propose a consistent physics-informed neural networks (CPINNs) framework for elliptic obstacle problems formulated as variational inequalities. The method is based on a mixed loss functional that is rigorously aligned with the stability structure of the underlying problem and incorporates obstacle constraints through a consistent treatment of the associated Lagrange multiplier. Relying on optimal recovery theory under Besov regularity assumptions, we establish near-optimal convergence rates for the simultaneous reconstruction of the solution and the multiplier from pointwise interior and boundary data. To enable practical implementation, we construct discrete counterparts of the continuous stability norms and duality pairings, leading to fully computable and theoretically justified training losses. Numerical experiments on benchmark obstacle problems demonstrate the accuracy, stability, and robustness of the proposed approach, and highlight its clear advantages over standard PINNs.

Mixed Consistent PINNs for Elliptic Obstacle Problems with Stability Analysis

Abstract

We propose a consistent physics-informed neural networks (CPINNs) framework for elliptic obstacle problems formulated as variational inequalities. The method is based on a mixed loss functional that is rigorously aligned with the stability structure of the underlying problem and incorporates obstacle constraints through a consistent treatment of the associated Lagrange multiplier. Relying on optimal recovery theory under Besov regularity assumptions, we establish near-optimal convergence rates for the simultaneous reconstruction of the solution and the multiplier from pointwise interior and boundary data. To enable practical implementation, we construct discrete counterparts of the continuous stability norms and duality pairings, leading to fully computable and theoretically justified training losses. Numerical experiments on benchmark obstacle problems demonstrate the accuracy, stability, and robustness of the proposed approach, and highlight its clear advantages over standard PINNs.

Paper Structure

This paper contains 27 sections, 5 theorems, 94 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Outline of the paper
  4. Functional setting and optimization formulation
  5. Function spaces
  6. Minimization problem
  7. Loss formulation
  8. Discretization method of PINNs
  9. CPINNs framework using mixed loss
  10. Optimal recovery
  11. Optimal recovery of $u$
  12. Optimal recovery of the primal solution $u$
  13. Optimal recovery of the mixed pair $(u,\lambda)$
  14. Norm discretization
  15. $L^\tau(\Omega)$ norm discretization
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $f\in H^{-1}(\Omega)$, $\psi\in H^1(\Omega)$ and $g\in H^{1/2}(\partial\Omega)$. Then the problems 1 and minprob are equivalent. Moreover, the variational problem minprob admits a unique solution $\boldsymbol{u}\in K^s$, and the following estimate holds where the constant $C_J$ depends only on domain $\Omega$. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Visualization of the true solution (left), standard PINNs prediction using $\mathcal{L}^p_{M}$ (middle) and CPINNs prediction using $\mathcal{L}_{M}^{c}$ (right), evaluated on $15 \times 15$ grid for $u_1$ (top row) and $u_2$ (bottom row).
  • Figure 2: Visualization of the true solution (left), standard PINNs prediction using $\mathcal{L}^p_{M}$ (middle) and CPINNs prediction using $\mathcal{L}_{M}^{c}$ (right), evaluated on $15 \times 15$ grid for $u_3$ (top row) and $u_4$ (bottom row).
  • Figure 3: Visualization of the true solution (left), standard PINNs prediction using $\mathcal{L}^p_{M}$ (middle) and CPINNs prediction using $\mathcal{L}_{M}^{c}$ (right), evaluated on $15 \times 15$ grid for $\boldsymbol{u_5}$ on $\nu = 0.4$(top row) and $\nu = 0.49$(bottom row).
  • Figure 4: Plots of the true solution (left), standard PINNs prediction using $\mathcal{L}^p_{M}$ (middle) and CPINNs prediction using $\mathcal{L}_{M}^{c}$ (right), evaluated on $15 \times 15$ grid for for $u_6$ (top row) and $u_7$ (bottom row).

Theorems & Definitions (10)

  • Theorem 2.1
  • Proof 1
  • Theorem 4.1
  • Proof 2
  • Theorem 4.2
  • Proof 3
  • Remark 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Proof 4