Mixed Consistent PINNs for Elliptic Obstacle Problems with Stability Analysis

TL;DR

This paper introduces a novel mixed PINNs framework for elliptic obstacle problems, ensuring stability and obstacle constraint handling, with proven near-optimal convergence and demonstrated numerical effectiveness.

Contribution

The paper develops a consistent mixed PINNs approach with stability-based loss functionals, providing theoretical convergence guarantees and practical implementation strategies.

Findings

Achieves near-optimal convergence rates for solution and multiplier reconstruction.

Demonstrates improved accuracy and robustness over standard PINNs in numerical tests.

Provides a fully computable loss functional aligned with stability norms.

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.