This course is a graduate-level course in numerical methods for solving ordinary and partial differential equations (ODEs and PDEs, respectively). The curriculum is designed to cover both the rigorous theoretical foundations of traditional numerical schemes and the latest machine-learning-based methodologies that are driving innovation in modern computational science and engineering.

The course is structured into two main parts:
1. Part 1 (Classical Methods): The first half focuses on traditional numerical methods, prominently featuring the Finite Difference Method (FDM). The emphasis is on understanding the essential ideas that underlie the development, analysis, and practical use of finite difference methods to solve ODEs and PDEs.
2. Part 2 (Modern AI Methods): The second half shifts focus to cutting-edge machine learning techniques that integrate data and physical laws, called scientific machine learning. We will explore new paradigms for solving differential equations, e.g., Physics-Informed Neural Networks (PINN), and also cover other novel architectures such as DeepONet and Fourier Neural Operators (FNO).

Through theoretical analysis, algorithmic understanding, and implementation, students will gain a comprehensive perspective on how modern scientific computing bridges mathematics and machine learning.

A good background in analysis and linear algebra, and experience in writing computer programs (in Python).

The required readings and literature (e.g., research papers) will be provided in class.

It covers the following materials: 

- Finite difference approximations for boundary value problems, initial value problems and parabolic problems.
- Machine learning foundations for scientific machine learning, physics-informed neural networks and operator learning.

No late homeworks will be accepted without an approved excuse.