Online Program

Statistical Learning for Predicting Electron Dynamics (309718)

Harish Subrahmanya Bhat, University of California, Merced 
*Prachi Gupta, University of California, Merced 
Christine Isborn, University of California, Merced 
Karnamohit Ranka, University of California, Merced 

Keywords: statistical learning, Hamiltonian, electron dynamics, electron density

We develop a statistical method to learn a molecular Hamiltonian matrix from a time-series of electron density matrices. In this problem, the density matrices satisfy the time-dependent Hartree-Fock (TDHF) equations, a system of nonlinear differential equations, determined by the Hamiltonian. The Hamiltonian is itself a function of the density matrices; our method's goal is to learn this functional relationship.

We proceed by taking a linear model for the Hamiltonian together with a squared loss function derived from a time-discretization of the Hartree-Fock equations. Assuming a naive approach, for large molecular systems, this results in least squares problems with hundreds of thousands of regression coefficients. These larger systems feature Hessians with large numbers of zero eigenvalues, leading to excessive training times and inaccurate propagation results.

To solve these problems, we employ two main techniques, the first of which is dimensionality reduction, accounting for physical properties of the true Hamiltonian (namely, a real/imaginary splitting). This results in least squares problems with, on average, an order of magnitude fewer coefficients. We also apply ridge regression, which regularizes and therefore removes the zero eigenvalue issue. We use a validation set approach to choose the ridge regularization parameter.

These techniques allow us to learn Hamiltonians that can be used in place of the ground truth Hamiltonian in the TDHF equations. Learned Hamiltonians yield propagation results in close quantitative agreement with ground truth trajectories for both field-off and field-on problems. To compute the field-on trajectories, we use an augmented Hamiltonian (the learned Hamiltonian plus a time-dependent perturbation) in the TDHF equations. The close agreement between field-on trajectories (for the learned and ground truth systems) indicates the capacity of the learning approach to extrapolate to regimes outside that of the training data.