Update README.md

README.md

@@ -1,6 +1,112 @@
 ---
-license: apache-2.0
+title: PINN for Navier-Stokes Equation
+emoji: 🌊
+pinned: false
+license: mit
 tags:
-- climate
-- physics
----
+- computational-fluid-dynamics
+- physics-informed-neural-networks
+- navier-stokes
+- machine-learning
+- fluid-flow
+datasets:
+- Allanatrix/CFD
+pipeline_tag: tabular-regression
+---
+
+# Physics-Informed Neural Network for Solving the Navier-Stokes Equation
+
+## Model Overview
+
+This repository contains a **Physics-Informed Neural Network (PINN)** designed to solve the **Navier-Stokes equation** for incompressible fluid flows, focusing initially on laminar flow regimes. The PINN embeds the governing partial differential equations (PDEs) into its loss function, enabling data-efficient solutions for velocity and pressure fields. The model takes time and spatial coordinates as inputs and predicts velocity components ($u$, $v$) and pressure ($p$). It was trained on a computational fluid dynamics (CFD) dataset and evaluated using machine learning metrics (training/validation loss) and problem-specific metrics (velocity/pressure field accuracy).
+
+Developed for researchers and CFD professionals, this model offers a promising alternative to traditional numerical solvers like direct numerical simulation (DNS), with potential for extension to turbulent flows through hybrid architectures and automated optimization.
+
+## Dataset
+
+- **Source**: Computational fluid dynamics simulations of laminar flows.
+- **Split**: 80% training, 20% validation.
+- **Description**: Includes velocity (u, v) and pressure (p) fields over a 2D domain with temporal evolution.
+
+## Evaluation Metrics
+
+| Metric Type         | Metric                     | Value  | Description                              |
+|---------------------|----------------------------|--------|------------------------------------------|
+| Machine Learning    | Final Training Loss        | 25.0   | Combined data and physics loss           |
+|                     | Final Validation Loss      | 4.9    | Generalization to unseen data            |
+| Problem-Specific    | Predicted u Range        | [-0.4, 0.8] | Velocity component $u$ prediction range  |
+|                     | True u Range             | [-0.4, 1.2] | True velocity $u$ range                  |
+|                     | Predicted v Range        | [-0.4, 0.2] | Velocity component $v$ prediction range  |
+|                     | True v Range             | [-0.4, 0.4] | True velocity $v$ range                  |
+|                     | Predicted p Range        | [-3.5, 0]   | Pressure prediction range                |
+|                     | True p Range             | [-7, 0]     | True pressure range                      |
+
+## Results
+
+- **Machine Learning**: Achieved a final training loss of 25.0 and validation loss of 4.9, indicating effective learning of physics constraints and good generalization.
+- **Problem-Specific**: Captured general trends in velocity (u, v) and pressure (p) fields, but smoothed fine-scale variations and underestimated the dynamic range, especially for pressure.
+- **Visualizations**: Scatter plots of true vs. predicted u, v, and p at a specific time step, and loss curves over 30 epochs, highlight the model’s strengths and limitations.
+
+## Installation
+
+1. Clone the repository:
+   ```bash
+   git clone https://github.com/your-repo/pinn-navier-stokes.git
+   cd pinn-navier-stokes
+
+
+Install dependencies:
+pip install -r requirements.txt
+
+Required packages (example requirements.txt):
+torch
+numpy
+pandas
+matplotlib
+
+
+Prepare the dataset:
+
+Obtain a CFD dataset for laminar flows (e.g., simulated velocity/pressure fields).
+Place the data in a data/ directory or update the dataset path in the code.
+
+
+# Usage
+
+Prepare the Data:
+
+Ensure the CFD dataset is formatted as numpy arrays of time ($t$), spatial coordinates ($x$, $y$), and fields ($u$, $v$, $p$).
+
+
+Train the Model:
+python main.py --data-path /path/to/data --mode train
+
+
+Options:
+--mode: train (train the PINN), predict (generate predictions).
+--data-path: Path to the dataset.
+
+Example
+To train the PINN and predict fluid fields:
+python main.py --data-path data/cfd_laminar --mode train
+
+This will:
+
+Train the PINN for 50 epochs with early stopping.
+Predict velocity ($u$, $v$) and pressure ($p$) fields.
+Save results (loss curves, field comparisons) to the results/ directory.
+
+# Limitations
+
+Fine-Scale Dynamics: The feedforward architecture smooths fine-scale variations, limiting accuracy for complex flows.
+Pressure Prediction: Underestimates the dynamic range of pressure, affecting Navier-Stokes compliance.
+Turbulent Flows: Current model is limited to laminar flows; turbulent regimes require more expressive architectures.
+
+Future Enhancements
+
+Develop hybrid models with recurrent (RNN), convolutional (CNN), and attention-based architectures for turbulent flows.
+Implement a meta-learner to aggregate model predictions based on physical fidelity.
+Automate hyperparameter tuning using Optuna for efficient convergence.
+Visualize the optimization landscape to understand training dynamics.
+
+For issues or contributions, contact the maintainers.