**Urban Chemical Safety: A Scientific Machine Learning Challenge**

🚨 What if you could predict and prevent a potential environmental disaster before it happens?

**The Critical Scenario**

In a densely populated urban area, a precision manufacturing facility uses trichloroethylene (TCE), a hazardous industrial solvent with severe environmental and health implications. Imagine a 1 km × 1 km urban monitoring zone (perfectly matching our unit square domain \([0,1] \\(\times\\) [0,1]\) where environmental scientists and safety experts are tracking a potential chemical tracer dispersion:

- TCE: A known carcinogen with extreme persistence in urban environments
- Square area represents a critical zone surrounding an industrial facility
- Potential accidental release poses significant public health risks
- Tracking how a chemical contaminant might spread through an urban landscape

**Tracking the Invisible Threat**
- Tracer released from some specific source
- Wind patterns causing directional spread:
--  Eastern flow (\\(b_{x} = 2\\))
-- Northern component (\\(b_y = 3\\))
- Extremely low atmospheric mixing (\\(\epsilon = 10^{-4}\\))

## Goal : 
**Predict precise concentration of chemical tracer across this compact urban region, potentially saving lives and preventing environmental catastrophe.**

# The Deep Learning Frontier: Physics-Informed Neural Networks 

Traditional simulation methods struggle with intricate transport phenomena. Enter Physics-Informed Neural Networks (PINNs)—a revolutionary approach that combines the power of deep learning with fundamental physical principles.

## The Technical Challenge

Participants will develop a Physics-Informed Neural Network to solve a Convection-Diffusion (CD) equation i.e., a mathematical model that captures intricate transport phenomena. The CD equation is given as:

$$-\varepsilon\Big(\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2}\Big) + b_x\dfrac{\partial u}{\partial x} + b_y\dfrac{\partial u}{\partial y} = f(x,y)$$

where:  
- \\(\varepsilon\\) : diffusion coefficient 
- \\(\vec{b} = (b_x, b_y)\\) : Convective field in \\(x-y\\) directions  
- \\(f(x,y)\\) : Forcing function (guides the solution of PDE)  

## Problem Specifics 

For our urban chemical safety challenge:
- Diffusion coefficient: \\(\varepsilon = 10^{-4}\\) (Extremely low mixing)
- Convective field: \\(\vec{b} = (2,3)\\) (Complex wind patterns)

The forcing function \\(f(x,y)\\) is a complex expression that captures the intricate dynamics of tracer dispersion.
$$f(x,y) = 2\epsilon \left( -x + \exp\left( \frac{2(x - 1)}{\epsilon} \right) \right) + x y^2 + 6 x y - x \exp\left( \frac{3(y - 1)}{\epsilon} \right) - y^2 \exp\left( \frac{2(x - 1)}{\epsilon} \right) + 2 y^2 - 6 y \exp\left( \frac{2(x - 1)}{\epsilon} \right) - 2 \exp\left( \frac{3(y - 1)}{\epsilon} \right) + \exp\left( \frac{2x + 3y - 5}{\epsilon} \right)$$

## Boundary Conditions

The problem is defined on a unit square domain \([0,1] \\(\times\\) [0,1]\) with Dirichlet boundary conditions:

$$u(x,y) = 0 \quad \text{for } (x,y) \in \partial \Omega$$

where, \\( \partial \Omega\\) represents domain boundary. This means the solution is constrained to zero along all boundaries of the domain, adding an extra layer of complexity to the physics-informed neural network's challenge.

# Your Mission

🚀 Develop a PINN that can:
- Accurately model chemical tracer dispersion
- Capture complex transport dynamics
- Perform precise predictions within a 1 km² urban environment [Hint : The dimensions are normalized and hence the domain is a unit square
- Outperform traditional numerical techniques
- Potentially save lives through advanced predictive modeling

## Beyond the Competition

This challenge is more than a hackathon—it's an opportunity to develop cutting-edge techniques that could revolutionize:
- Urban environmental monitoring
- Industrial safety protocols
- Emergency response planning
- Advanced scientific computing

**Are you ready to push the boundaries of scientific machine learning and make a real-world impact?**