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 ground state 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 the regularity of the physical system. We introduce deep multiscale learning architectures that compute such approximations via iterated wavelet transforms. The transforms are applied to an intermediate electron density representation, in relation to Density Functional Theory. Numerical results for computing the atomization energies of organic molecules indicate that a machine learning technique accurate enough to be competitive with expensive quantum chemical methods at a fraction of the computational cost is within reach.
Joint work with: Stéphane Mallat (École normale supérieure, France) and Nicolas Poilvert (Pennsylvania State University, USA)