Introduction to Partial Differential Equations

Partial Differential Equations (PDEs) are fundamental mathematical tools that describe physical phenomena across space and time. From fluid dynamics to quantum mechanics, PDEs model complex systems that are essential in scientific and engineering applications.

What are PDEs?

A Partial Differential Equation relates a function of multiple variables to its partial derivatives. Unlike Ordinary Differential Equations (ODEs) that depend on a single variable, PDEs describe how quantities change across multiple dimensions.

Example: The heat equation

∂u/∂t = α∇²u

describes how temperature u changes over time t and space.

Traditional Numerical Methods

Finite Difference Method

• Discretizes the domain into a grid
• Approximates derivatives using Taylor series
• Simple implementation but limited to regular geometries

Challenges with Traditional Methods

• Computational Cost: High resolution requires significant computational resources
• Curse of Dimensionality: Performance degrades exponentially with dimensions
• Adaptivity: Difficult to adapt to solution features automatically
• Uncertainty Quantification: Limited capabilities for handling uncertainties

The AI Revolution

Modern AI techniques are transforming PDE solving by:

• Learning solution operators directly from data
• Incorporating physical constraints into neural networks
• Enabling efficient high-dimensional solutions
• Providing uncertainty quantification capabilities