Shiv Mishra

Arbaz Khan, Kent-Andre Mardal, Shiv Mishra

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.

Shiv Mishra, Arbaz Khan

We develop a new class of physics-informed neural network approximations for the stationary Oseen equations based on stability-consistent loss constructions. In contrast to standard PINN formulations, which are typically heuristic, the proposed consistent PINN (CPINN) framework is systematically derived from the stability structure of the continuous problem. Within this setting, we introduce two fundamentally new approaches. First, we design standard CPINN formulations that exhibit clear improvements over conventional PINNs. Second, we propose pressure-robust CPINN formulations that provably eliminate the influence of gradient forces on the velocity approximation, yielding velocity errors that depend solely on the divergence-free component of the forcing and are independent of the pressure. The framework accommodates both exactly divergence-free architectures and unconstrained velocity approximations, providing a unified treatment of these two paradigms. Using techniques from optimal recovery theory, we establish, for the first time in the PINN setting for Oseen-type problems, quantitative recovery estimates and optimal error bounds for both velocity and pressure under suitable Besov regularity assumptions. In particular, we obtain optimal rates for the velocity in $\boldsymbol{H}^1(Ω)$ and for the pressure in $L^2(Ω)$. The proposed methodology introduces a pressure-robust CPINN paradigm for incompressible flows, combining structural consistency, robustness with respect to irrotational forces, and rigorous accuracy guarantees. Numerical experiments corroborate the theoretical findings and demonstrate the effectiveness of the approach.