Structure-Preserving and Pressure-Robust PINNs for Incompressible Oseen Problems

TL;DR

This paper introduces a new class of physics-informed neural networks for incompressible flow problems that are pressure-robust, structurally consistent, and provide optimal error bounds, improving accuracy and robustness over standard PINNs.

Contribution

The paper develops a systematic, stability-based framework for PINNs applied to Oseen equations, including pressure-robust formulations with rigorous error analysis and optimal convergence rates.

Findings

Pressure-robust CPINNs eliminate pressure influence on velocity errors.

The framework achieves optimal error rates in velocity and pressure approximations.

Numerical experiments confirm theoretical accuracy and robustness improvements.

Abstract

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 $\boldsymbol{H}^1(\Omega)$ and for the pressure in $L^2(\Omega)$. 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.