Structure-Preserving and Pressure-Robust PINNs for Incompressible Oseen Problems
For researchers in scientific machine learning and computational fluid dynamics, this work provides a principled PINN formulation with rigorous error guarantees, addressing the lack of stability and pressure robustness in standard PINNs.
The paper develops a consistent PINN (CPINN) framework for stationary Oseen equations, achieving pressure-robust velocity approximations that eliminate the influence of gradient forces. The method yields optimal error bounds for velocity in H1 and pressure in L2, with numerical experiments confirming theoretical results.
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.