Abstract:
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 molecular energies is such an example. Machine learning algorithms do not simulate the physical system but estimate solutions by interpolating values provided by a training set of known examples. 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 avoided by computing interpolations in smaller approximation spaces, which take advantage of physical invariants. We present a novel approach for the regression of quantum mechanical energies based on the scattering transform of an intermediate electron density representation. The scattering transform has the structure of a deep convolutional network, composed of iterated wavelet transforms and modulus operators, and possesses the appropriate invariant and stability properties for molecular energy regression. Numerical experiments give state of the art accuracy over data bases of planar organic molecules.

Biography:
Matthew Hirn is an assistant professor in the Department of Computational Mathematics, Science and Engineering and the Department of Mathematics at Michigan State University, where he has been a faculty member since 2015.

Hirn received his B.A. in mathematics from Cornell University and his Ph.D. in mathematics from the University of Maryland, College Park. Before arriving at Michigan State, he held postdoctoral appointments in the Department of Mathematics at Yale University and in the Département 'Informatique at the Ecole normale supérieure, Paris.

Hirn's research interests are at the interface of harmonic analysis and machine learning. Broadly speaking, he develops mathematically provable machine learning algorithms for the analysis of high dimensional data and to circumvent prohibitively costly computations in scientific computing.

Specific research interests include:

• Applied harmonic analysis
• Manifold learning
• Smooth extensions and interpolations
• Quantum chemistry and N-body problems
• Deep learning
• Computer vision

Host:
Dr. Metin Aktulga