Prerequisites: linear algebra, multivariable calculus, elementary probability, some programming experience.
In recent years, machine learning techniques have penetrated a tremendous variety of scientific fields. In particular, they have truly opened new horizons in solving numerically partial differential equations (PDEs) by enabling us to address problems that used to be intractable due to the curse of dimensionality. Notoriously high-dimensional PDEs ubiquitously arise in chemical physics applications. Many problems of interest in this field are to identify transition mechanism between metastable states of a biomolecule or a cluster of interacting particles, and estimate the transition rate. Often these transitions are rare in the sense that direct Monte Carlo simulations take prohibitively long time even if enhanced sampling techniques are used.
An alternative approach is deterministic. It consists in solving a boundary-value problem for an elliptic PDE, the backward Kolmogorov equation, to find the so-called committor function. The committor allows one to readily calculate the reactive current and find the reaction rate which give a complete quantitative description of the transition process . It worth noting that molecular configurations are typically described by means of physically motivated collective variables such as dihedral angles, distances between particular atoms, etc. While the dimension of the space of collective variable is much lower than the dimension of the space of atomic coordinates, it is often still too high to apply traditional finite difference (FD) or finite element (FE) methods to solve the committor PDE. Recent successes in solving the committor PDEs by means of neural networks (NNs) and in harnessing diffusion maps for quantifying rare events are really inspiring. However, contrary to FD and FE methods, there is no convergence theory for DM-based and NN-based PDE solvers. Currently, this is an area of active research. Many questions remain unanswered: How to choose the scaling parameter for DMs to minimize the error in the committor? How to preprocess the set of data points for DMs? How do various features of the architecture of NNs affect the accuracy of the solution? How does the number and arrangement of training points affect the accuracy of the solution? In this REU, we will explore these questions by means numerical experiments and analytical tools.