Academics

  /  

Courses

  /  

Descriptions

ELEC_ENG/COMP_ENG 395, 495: Scientific Machine Learning

VIEW ALL COURSE TIMES AND SESSIONS

Description

Course coordinator: Aggelos Katsaggelos
Instructor: Manuel Ballester

Description

This course introduces students to Scientific Machine Learning, a modern approach that combines machine learning with physical modeling. The goal is to help students see how the classical physical laws and modern machine learning complement each other. This course provides the tools to build hybrid models for engineering, scientific, and data-driven applications.

Students will learn how deep learning models can incorporate physically meaningful information to improve prediction, stability, and interpretability. The emphasis is on learning concepts carefully from first principles (depth versus breadth), focusing on understanding rather than covering large amounts of material. Students will program basic models for physics-informed neural networks, neural operators, and Hamiltonian or Lagrangian networks, among others.

Instruction is supported by material from the book “Physics Based Machine Learning” by Manuel Ballester, Christoph Wuersch and Aggelos Katsaggelos, Springer, 2026 (forthcoming), together with lecture slides and reference papers when needed.

Prerequisites

No specific prerequisites are required. Essential material in optimization, deep learning, and differential equations will be reviewed during the first weeks so students from different backgrounds can follow. However, it is recommended that students have some familiarity with:
- Machine learning and optimization theory
- Basic physics (from general physics courses or an engineering physics sequence)
- Undergraduate mathematics (linear algebra, differential calculus, and vector calculus)
The course is intended for advanced undergraduates, MS, and PhD students. Students without the mentioned background are still welcome but should expect to spend additional time on the early review material.

Course Outline (10 Weeks)

• Course overview, introduction to Scientific Machine Learning, optimization (essential results, dual problems, algorithms).

• Deep Learning. Fundamentals (regression vs classification, supervised vs unsupervised, essential results). Perceptron and fully connected networks. Quick review of advanced architectures (convolutional, recurrent, transformers, and graph neural networks).

• Ordinary Differential Equations (ODE). Fundamental definitions, classifications, interpretation, existence and uniqueness. Classical numerical solvers and stability. Dynamical behavior.

• Neural ODE and Normalizing Flows. Neural ODEs, the adjoint method, normalizing flows, limitations, and hybrid approaches.

• Lagrangian and Hamiltonian Neural Networks (LNN and HNN). Euler-Lagrange equations and the principle of least action. Lagrangian and Hamiltonian formulations to describe physical systems. Modern neural architectures that incorporate these physical structures.

• Partial Differential Equations (PDEs). PDE types and classic equations (e.g., continuity, wave, diffusion), simple analytical solutions, numerical methods (such as FDM and FVM), stability analysis and CFL conditions.

• Geometric Deep Learning. Spectral bias, Fourier features, geometric representations, and MeshGraphNets.

• Physics-Informed Neural Networks (PINNs). PINN formulation, several examples solving classical PDEs, known limitations and extensions.

• Neural Operators. DeepONets. Fourier Neural Operators. Physics-informed neural operator (PINO).

• Differentiable Physics. Develop differentiable simulators. Automatic differentiation for inverse physical problems. Adjoint methods.

• Applications of physics-informed approaches for advanced dynamical problems (star evolution, weather prediction, medical imaging).

Grading:

This course has no exams.

60% weekly exercises (theoretical and coding), individual or in pairs.

40% final project: Students select a scientific or engineering system and develop a physics-informed machine learning model (e.g., PINN, Neural Operator, HNN). Deliverables include code (GitHub repository), short 2-page report (in Latex), and a pre-recorded short presentation.