Physics-Informed Neural Network (PINN) for 2D Cylinder Flow

1. Project Overview

This repository contains a Physics-Informed Neural Network (PINN) implementation designed to solve the steady-state, incompressible Navier-Stokes equations. The model simulates fluid flow around a circular cylinder within a 2D channel, a classic benchmark in computational fluid dynamics (CFD).

Unlike traditional mesh-based solvers (FVM/FEM), this approach uses a neural network as a continuous functional approximator, integrating physical laws directly into the loss function via automatic differentiation.

2. Technical Architecture

The model is built using a fully connected deep neural network that maps spatial coordinates (x, y) to the velocity field (u, v) and pressure p.

Physics Constraints

The total loss function is a weighted sum of several components:

• Navier-Stokes Residuals: Ensures momentum conservation (u \nabla u = -\nabla p + \nu \Delta u).
• Continuity Equation: Enforces mass conservation (\nabla \cdot \mathbf{u} = 0).
• Boundary Conditions (BCs):
  • Inlet: Parabolic velocity profile.
  • Outlet: Zero pressure boundary.
  • No-Slip: Zero velocity at channel walls and cylinder surface.

3. Implementation Highlights

• Dynamic Curriculum Learning: Implements a phased training schedule. The model first masters the geometry (Boundary Conditions) before gradually increasing the weight of the PDE residuals to handle lower viscosity (\nu) regimes.
• Advanced Visualization: Includes a custom plotting suite using matplotlib.streamplot to visualize flow topology and identify divergence errors or recirculation zones.
• Adaptive Weighting: Utilizes a custom weight curriculum to prevent the "gradient pathology" often found in PINNs where BC losses and PDE losses compete destructively.

4. Key Results (Epoch 15,000+)

The current state of the model demonstrates:

• Laminar Flow Reconstruction: Successfully captures the deflection of the flow field around the obstacle.
• Pressure Gradient: Realistic high-pressure zones at the stagnation point and low-pressure zones in the wake.
• Topology Analysis: Streamline visualization confirms that the model has learned the global structure of the flow field without manual labeling.

5. Getting Started

Prerequisites

• Python 3.x
• PyTorch
• NumPy
• Matplotlib
• Pandas
• Polars

Installation

git clone [https://github.com/fabianfrank-dev/NavierStokes.git]
cd NavierStokes
pip install -r requirements.txt

# RUnning the training
python3 main.py

6 Future Benchmarks

• Transition to transient (time-dependent) Navier-Stokes.

• Integration of Re-number as a parameterized input to the network.

• Benchmarking performance against OpenFOAM/SimScale data.