Mixed Consistent PINNs for Elliptic Obstacle Problems with Stability Analysis
This provides a theoretically justified method for solving elliptic obstacle problems, which are incremental improvements over existing PINN approaches.
The authors tackled elliptic obstacle problems by developing a consistent physics-informed neural networks framework with a mixed loss functional aligned with stability structure, achieving near-optimal convergence rates for solution and multiplier reconstruction. Numerical experiments demonstrated accuracy, stability, and robustness with clear advantages over standard PINNs.
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.