#
APS March Meeting 2016

## Volume 61, Number 2

##
Monday–Friday, March 14–18, 2016;
Baltimore, Maryland

### Session E22: Predicting and Classifying Materials via High-Throughput Databases and Machine Learning I

8:00 AM–11:00 AM,
Tuesday, March 15, 2016

Room: 321

Sponsoring
Unit:
DCOMP

Chair: Gus Hart, Brigham Young University

Abstract ID: BAPS.2016.MAR.E22.4

### Abstract: E22.00004 : Deep Wavelet Scattering for Quantum Energy Regression*

8:36 AM–9:12 AM

Preview Abstract
Abstract

####
Author:

Matthew Hirn

(Michigan State University)

Physical functionals are usually computed as solutions of variational
problems or from solutions of partial differential equations, which
may require huge computations for complex systems. Quantum chemistry
calculations of ground state molecular energies is such an
example. Indeed, if $x$ is a quantum molecular state, then the ground state
energy $E_0(x)$ is the minimum eigenvalue solution of the
time independent Schr\"{o}dinger Equation, which is computationally intensive for large
systems. Machine learning algorithms do not simulate the physical system but estimate
solutions by interpolating values provided by a training set of known
examples $\{ (x_i, E_0(x_i) \}_{i \leq n}$. However, precise
interpolations may require a number of examples that is exponential in
the system dimension, and are thus intractable. This curse of
dimensionality may be circumvented by computing interpolations in
smaller approximation spaces, which take advantage of physical
invariants. Linear regressions of $E_0$ over a dictionary $\Phi = \{
\phi_k \}_k$ compute an approximation $\widetilde{E}_0$ as:
$
\widetilde{E}_0 (x) = \sum_k w_k \phi_k (x),
$
where the weights $\{ w_k \}_k$ are selected to minimize the error between $E_0$ and
$\widetilde{E}_0$ on the training set. The key to such a regression approach then lies in the design of the dictionary $\Phi$. It must be intricate enough to capture the
essential variability of $E_0(x)$ over the molecular states $x$ of interest, while
simple enough so that evaluation of $\Phi (x)$ is significantly less
intensive than a direct quantum mechanical computation (or
approximation) of $E_0 (x)$. In this talk we present a novel dictionary $\Phi$ for the regression of quantum mechanical energies based on the \textit{scattering transform} of an intermediate, approximate electron density representation $\rho_x$ of the state $x$. The scattering transform has the architecture of a deep convolutional network, composed of an alternating sequence of linear filters and nonlinear maps. Whereas in
many deep learning tasks the linear filters are learned from the
training data, here the physical properties of $E_0$ (invariance to
isometric transformations of the state $x$, stable to deformations of
$x$) are leveraged to design a collection of linear filters $\rho_x \ast \psi_{\lambda}$ for
an appropriate wavelet $\psi$. These linear filters are composed with
the nonlinear modulus operator, and the process is iterated upon so
that at each layer stable, invariant features are extracted:
$
\phi_k(x) = \| || \rho_x \ast
\psi_{\lambda_1} | \ast \psi_{\lambda_2} | \ast \cdots \ast
\psi_{\lambda_m} \|, \quad k = (\lambda_1, \ldots, \lambda_m),
\enspace m = 1, 2, \ldots
$
The scattering transform thus encodes not only interactions at
multiple scales (in the first layer, $m = 1$), but also features
that encode complex phenomena resulting from a cascade of interactions
across scales (in subsequent layers, $m \geq 2$). Numerical
experiments give state of the art accuracy over data bases of organic
molecules, while theoretical results guarantee performance for the
component of the ground state energy resulting from Coulombic
interactions.

*Supported by the ERC InvariantClass 320959 grant.

To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2016.MAR.E22.4