AMATH 563: Inferring Structure of Complex Systems

Setting Up CVX

• Open MATLAB
• In the MATLAB console/terminal type:

	cd cvx-w64/cvx
	cvx_setup

Lecture Topics

Regularization and Promoting Sparsity

• Solving overdetermined & underdetermined systems
• L1 Norm vs L2 Norm
• Gradient descent, LASSO, and QR Decomposition
• Regularization and the benefits of sparsity

Model Discovery

• SINDy: Discovering models for ODEs & PDEs using data
• Time-Embeddings: Discovering hidden (latent) variables
• Pareto Frontiers
• Cross-validation
• k fold cross validation
• Information Criteria (Bayesian,KL, AIC)

Data Assimilation

• Kalman Filters & Model auto-correction

Dynamic Mode Decomposition

• Model discovery in low-data settings

Clustering and Classification

• Unsupervised Methods
  • K-Means
  • Hierarchical Clustering
• Supervised Methods
  • Support Vector Machines
  • Classification/Decision Trees

Neural Networks

• Feedforward neural networks
• Convolutional neural networks
• Regularization in Neural Networks

Randomized Linear Algebra

• Extracting low-rank structure in Big-data

Homework Topics

HW1: Regression and Sparsity

• MNIST database (hand-drawn digits)
• Digit recognition using linear regression
• Promoting Sparsity with L1 Norm
• MATLAB optimization with CVX

HW2: Model Discovery

• Nonlinear sparse regression
• Discovery of dynamical systems through data
• Model Assessment: Information Criterion and KL Divergence

HW3: Clustering and Classification with Neural Networks

• Lorenz System, Reaction-Diffusion Equations, kuramoto sivashinsky equations
• Future State Predictions