Machine Learning on Dynamical Systems

I’ve started taking an off-cycle course 12.S592 on machine learning taught by Sai Ravela, with a focus on (non-Gaussian) dynamical systems. My desired final project is on extratropical cyclone downscaling, using a reduced stochastic model. Here are some PSet notes and schematic solutions, because they are very interesting problems to think about.

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L0 vs L1 Problem: Particle Tracking in Non-Gaussian Flow. Hungarian Algo (L0 Method), and Sinkhorn Algo (Doubly Stochastic)

Subset Selection Problem: What to do when nonlinear functional mapping is expensive? (Resampling and mapping approximation, to identify most damaging cyclones from massive input features)

Sparse Recovery: How to identify particles with spike in wavelength spectra, convoluted and added by noises, as siganls received in telescopes? Retrieving 4 particles from a 1000by1000 space and identify their weight!

Earthquake Sourcing: How to identify source location and onset time for multiple earthqukes (with spurious noises like local ppl stamping on the groud!)

Simplest Regression: Can we get accurate polynomial coefficients? Vandermonde matrix, multicolinearity problem

Quadratic Programming and Gaussian Belief Propagation: revisit nodes correspondence

Gaussian Belief Propagation under a Bayesian viewpoint

[Regularization: Ridge, Lasso, Tikhonov, Ivanov, Morozov, Mixed-Integer Quadratic Programming (MIQP), Mixed-Integer Linear Programming (MILP), GUROBI]

Proof that each MLQP corresponds to a unique MILP (vice versa). Fortet (compact) linearization

Generative Adversarial Networks, & Conditional Gaussian Process Notes

How to do Sparsity L0 Problem

Sequential Projection, QR Decomposition

Mutual Coherence, L1 problem

Meanwhile, Sai has some preferred naming of the above psets, which I’ve also included here:

• L0 vs L1 Problem: Correspodent Problem
• Subset Selection Problem: Where’s Waldo
• Sparse Recovery: All mixed up
• Earthquake Sourcing: Boom!
• MIQP ~ MILP, Paper by Zhu et al., 2020 PNAS
• Downscaling, Paper by Saha and Ravela, 2024