In this project, denoising score matching (DSM) models are implemented to estimate the score function of a multi-dimensional stochastic dynamical system from simulated trajectory data. Samples are generated by numerically integrating a stochastic differential equation using the Euler Maruyama method, and a neural network was trained to recover the score from Gaussian perturbed data. The learned score will be validated by comparison with an analytical score function derived from the governing dynamics to demonstrate that the network successfully captured the underlying score structure. This implementation establishes a framework for estimating score functions in systems where analytical solutions are unavailable and provides the foundation for future applications of the Green’s Fluctuation Dissipation Theorem using learned scores.

I am participating in SuperUROP because it provides an opportunity to go deeper into my research while working on a challenging problem at the intersection of mathematics, machine learning, and statistics. Through this project, I have gained experience developing denoising score matching models, implementing and evaluating neural network based algorithms, and connecting theoretical concepts to computational methods. SuperUROP allows me to extend this work beyond an introductory research experience by exploring more advanced techniques, validating new methods, and contributing to the broader goal of estimating Green’s functions from stochastic data. I hope to strengthen both my mathematical foundation and research skills while preparing for a career involving computational modeling and machine learning.