Applied Mathematics Seminar | Kirill Neklyudov, Applications of Deep Learning and Otto Calculus in Natural Sciences

MC 5501 

Speaker

Kirill Neklyudov, Vector Institute 

Title

Applications of Deep Learning and Otto Calculus in Natural Sciences

Abstract

Recently, Deep Learning has demonstrated success in many applications such as generative modeling, molecular dynamics, single-cell trajectory inference, and solving the Schrödinger equation. In this talk, I will demonstrate how these seemingly unrelated problems can be approached from a single perspective. In particular, this can be done through an extension of the optimal transport problem by introducing the Lagrangian formulation on the space of distributions. This provides a natural way to incorporate background knowledge to the considered dynamics such as stochastic transitions, laws of Physics, and birth and death of samples. Moreover, Action Matching, a subroutine in the Lagrangian perspective, has multiple applications in generative modeling and natural sciences. When compared to other approaches, the distinguishing part of Action Matching is its sampling-agnostic training procedure. Namely, unlike diffusion models, it does not rely on the prior knowledge of the image corruption process, significantly extending its scope of applications. Furthermore, optimal transport methods bring another perspective on the Schrödinger equation, which allows for efficient simulations. Namely, the Wasserstein Variational Monte Carlo algorithm is a novel way to estimate the ground state of a quantum system. It restricts the updates of the variational approximation to "local changes", which helps the sampling step to mix faster and results in better convergence. In my talk, I aim to convey the main abstract concepts behind my developments and demonstrate their utility in practice. Finally, we will discuss and envision future directions of this line of research including inference of a Hamiltonian, prediction of population dynamics, and optimal control problems.