Research
Dynamics under randomness.
My research connects stochastic analysis and dynamical systems. I am interested in the mechanisms behind stability, chaos, and mixing, and in how these ideas illuminate learning algorithms.
Stability, chaos, and stochastic flows.
I study the long-term behavior of systems driven by noise. Lyapunov exponents provide a way to distinguish stability from chaotic behavior and to understand how local dynamics influence the evolution of a random system.
My work includes shear-induced chaos in noisy oscillators, conditioned Lyapunov spectra for systems with absorption, and questions about stochastic flows, strong completeness, and synchronization.
Lyapunov exponents · Shear-induced chaos · Stochastic flows · Synchronization
Stochastic Flows with Strong Shear - Part I: Strong Completeness and Set Attractors Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise The conditioned Lyapunov spectrum for random dynamical systems Stability of Random Dynamical Systems in Noisy Oscillators with Shear and in Stochastic Gradient DescentMixing and the scales of dissipation.
I am interested in the transport and mixing of passive scalars by random velocity fields. A central question is how small-scale structures and the rate of dissipation behave as molecular diffusion becomes weak.
In joint work with Robin Chemnitz, we study explicit white-in-time flows on the torus and establish Batchelor-scale behavior through bounds on the exponential dissipation rate. The work also examines mixing in the absence of diffusion.
Passive scalar transport · Mixing · Enhanced dissipation · Batchelor scale
Mixing at the Batchelor Scale for White-In-Time FlowsLearning algorithms as dynamical systems.
I study gradient descent and stochastic gradient descent through the lens of dynamical systems. In overparameterized models, many parameters can interpolate the training data, so the dynamics of the algorithm matter for which solutions can be reached.
My work characterizes dynamical stability near interpolation solutions using Lyapunov exponents. I also contribute to a broader dynamical systems perspective on neural networks, including their architecture, training, and mean-field limits.
Stochastic gradient descent · Overparameterization · Dynamical stability · Neural networks
Characterizing Dynamical Stability of Stochastic Gradient Descent in Overparameterized Learning A Dynamical Systems Perspective on the Analysis of Neural NetworksFor paper details and citations, see the full publication list.
Publications