Pattern Formation and Systems Identification
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I am currently working with Shankar Venkataramani and Alan Newell on systems identification methods for natural stripe patterns. We propose the Swift-Hohenberg PDE as an interesting test case for systems identification methods that can learn order parameter dynamics from the underlying data. This is in contrast to many latent space methods, where the latent variable is not a priori interpreted as an order parameter. While Swift-Hohenberg fields exhibit a large variety of striped patterns and defect configurations, they all possess a local wave vector that evolves slowly in time and space relative to the microscopic coordinates. This observation has motivated a macroscopic model known as the Regularized Cross-Newell equation, whose solutions exhibit many common defect structures seen in experiments and in simulations. The main goal of this work is to find a suitable regularization to the joint optimization problem of coordinate transformation and dynamics discovery that yields order parameter dynamics.
Sample simulation of Swift-Hohenberg on the circle, where the initial condition is chosen to have a small spatial wavenumber. The left column shows the full field, middle column is zoomed in, right column is microscopic energy density of the middle column.
A heuristic description of the macroscopic model. To derive the Regularized Cross-Newell equations, one averages the microscopic density over the local periodicity of the pattern. The top figure is a sample microscopic energy density from a Swift-Hohenberg simulation, the bottom figure is an approximation of its macroscopic energy density.
A simple feed forward neural network with task of performing phase extraction on perturbed versions of a commonly observed defect known as the phase grain boundary. This experiment demonstrated the ability to map the pattern to a phase (at least for data that doesn't contain point defects.) A major goal of this project is to achieve a mapping like this in the form of an encoder, where the loss function does not include the phase surface.
Sparse Sensing of Tsunami Waves
In collaboration with Arvind Mohan, Darren Engwirda, and Javier Santos at Los Alamos National Lab, I have also worked on a transformer model known as the Senseiver which solves sparse sensing problems in a variety of physical contexts. My work has been to modify the model to reconstruct full field tsunami waves (~150,000 pixels), from only ~60 input pixels. The inputs correspond to real buoys that measure wave heights in real time for the purpose of tsunami detection. The goal of this work is to have a mapping from the sparse set of wave height readings to a full field estimate of the tsunami wave. A publication is in the works for this project.
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