Solving Navier-Stokes Equations Using Data-free Physics-Informed Neural Networks With Hard Boundary Conditions

TL;DR

This work demonstrates a data-free, unsupervised PINN framework with hard boundary conditions for solving the 2D incompressible Navier–Stokes equations across four flow scenarios, including a transient case. By encoding the NSE residuals and boundary conditions directly into the loss and constructing outputs that satisfy boundary constraints exactly, the method achieves accurate velocity and pressure fields with ($L_2$) errors spanning $\mathcal{O}(10^{-4})$ to $\mathcal{O}(10^{-1})$ for moderate Reynolds numbers. Validation against COMSOL-based CFD confirms good agreement for the lid-driven cavity and flow past a circular obstacle, with drag coefficients closely matching CFD values; accuracy degrades at higher $Re$, highlighting current PINN limitations. A learning-rate scheduler improves convergence stability, and the study's extension to transient internal flows demonstrates the framework's potential for mesh-free, physics-consistent simulations of complex, time-dependent fluid dynamics and boundary geometries, while pointing to avenues for method enhancements such as CV-PINN, CAN-PINN, and LSFD-PINN.

Abstract

In recent years, Physics-Informed Neural Networks (PINNs) have emerged as a powerful and robust framework for solving nonlinear differential equations across a wide range of scientific and engineering disciplines, including biology, geophysics, astrophysics and fluid dynamics. In the PINN framework, the governing partial differential equations, along with initial and boundary conditions, are encoded directly into the loss function, enabling the network to learn solutions that are consistent with the underlying physics. In this work, we employ the PINN framework to solve the dimensionless Navier-Stokes equations for three two-dimensional incompressible, steady, laminar flow problems without using any labeled data. The boundary and initial conditions are enforced in a hard manner, ensuring they are satisfied exactly rather than penalized during training. We validate the PINN predicted velocity profiles, drag coefficients and pressure profiles against the conventional computational fluid dynamics (CFD) simulations for moderate to high values of Reynolds number ($Re$). It is observed that the PINN predictions show good agreement with the CFD results at lower $Re$. We also extend our analysis to a transient condition and find that our method is equally capable of simulating complex time-dependent flow dynamics. To quantitatively assess the accuracy, we compute the $L_2$ normalized error, which lies in the range $\mathcal{O}(10^{-4})$ - $\mathcal{O}(10^{-1})$ for our chosen case studies.

Figures (24)

• Figure 1: The architecture of a Physics-Informed Neural Network (PINN).
• Figure 2: Schematic of mesh generation in CFD for the three benchmark cases - the 2D lid-driven cavity flow (top left), flow passing a circular obstacle in a pipe (top right) and flow in a pipe with sinusoidal boundary (bottom).
• Figure 3: The geometry of the 2D square cavity with the top wall moving at a constant horizontal velocity ($u=1, v=0$). The bottom and side walls are considered to be no-slip walls.
• Figure 4: Velocity field for the 2D lid-driven cavity , with $Re = 100$, obtained from CFD (left) and PINN (right). Arrows represent velocity vectors, and color indicates velocity magnitude.
• Figure 5: Comparison of the results obtained from PINN (right) and CFD simulations (left) for the lid-driven cavity with Re = 100. The top and bottom panels correspond to the $x$- and $y$-components of velocity ($u, v$).