Introduction to Scientific Machine Learning (edX)

The course also reviews the state-of-the-art in physics-informed deep learning and ends with a discussion of automated Bayesian inference using probabilistic programming (Markov chain Monte Carlo, sequential Monte Carlo, and variational inference). Throughout the course, the instructor follows a probabilistic perspective that highlights the first principles behind the presented methods with the ultimate goal of teaching the student how to create and fit their own models.

What you'll learn

After completing this course, you will be able to:

- Represent uncertainty in parameters in engineering or scientific models using probability theory

- Propagate uncertainty through physical models to quantify the induced uncertainty in quantities of interest

- Solve basic supervised learning tasks, such as: regression, classification, and filtering

- Solve basic unsupervised learning tasks, such as: clustering, dimensionality reduction, and density estimation

- Create new models that encode physical information and other causal assumptions

- Calibrate arbitrary models using data

- Apply various Python coding skills

- Load and visualize data sets in Jupyter notebooks

- Visualize uncertainty in Jupyter notebooks

- Recognize basic Python software (e.g., Pandas, numpy, scipy, scikit-learn) and advanced Python software (e.g., pymc3, pytorch, pyrho, Tensorflow) commonly used in data analytics

Syllabus

Section 1: Introduction

- Introduction to Predictive Modeling

Section 2: Review of Probability Theory

- Basics of Probability Theory

- Discrete Random Variables

- Continuous Random Variables

- Collections of Random Variables

- Random Vectors

Section 3: Uncertainty Propagation

- Basic Sampling

- The Monte Carlo Method for Estimating Expectations

- Monte Carlo Estimates of Various Statistics

- Quantify Uncertainty in Monte Carlo Estimates

Section 4: Principles of Bayesian Inference

- Selecting Prior Information

- Analytical Examples of Bayesian Inference

Section 5: Supervised Learning: Linear Regression and Logistic Regression

- Linear Regression Via Least Squares

- Bayesian Linear Regression

- Advanced Topics in Bayesian Linear Regression

- Classification

Section 6: Unsupervised Learning

- Clustering and Density Estimation

- Dimensionality Reduction

Section 7: State-Space Models

- State-Space Models – Filtering Basics

- State-Space Models – Kalman Filters

Section 8: Gaussian Process Regression

- Gaussian Process Regression – Priors on Function Spaces

- Gaussian Process Regression – Conditioning on Data

- Bayesian Global Optimization

Section 9: Neural Networks

- Deep Neural Networks

- Deep Neural Networks Continued

- Physics-Informed Deep Neural Networks

- Section 10: Advanced Methods for Characterizing Posteriors

- Sampling Methods

- Variational Inference