# Research

Author: Zhenzhao Tu

I am a Master’s student in Applied Mathematics at the University of Washington. The work that currently occupies me sits between numerical analysis, dynamical systems, and the theory of machine learning.

The short version: neural networks can train like kernel machines, or they can actually learn features. Those two poles are usually called the [lazy and rich regimes](/research/network-richness/). I care about the scale of that difference — what I have been calling network richness — and about how width, depth, and parameterization pull on it.

Before that I spent time on more classical applied math: ODE models of turbulence, iterative methods, and a little image reconstruction posed as optimization. The [papers](/papers/) page is a reading list, not a publication list. I have not published much yet.

## Current

- [Network richness](/research/network-richness/) — notes on the lazy–rich transition, and on what “richness” should mean once depth is allowed to move.

<h2 id="ftle">Finite-Time Lyapunov Exponents</h2>

Earlier work used ODE models of turbulence and compared Runge–Kutta integration with Kalman filtering. Finite-Time Lyapunov Exponents are still on the desk when I think about how trajectories separate.

<h2 id="dynamical-systems">Dynamical systems</h2>

I like systems that actually move. Neural-network training is one of them, if you squint; fluid models are another, if you do not.

<h2 id="numerical-methods">Numerical methods</h2>

Proximal gradient, iterative linear solvers, SVD as a compression theorem rather than a button. The [note](/notes/numerical-methods/) is shorter than a methods course and longer than a tweet.