Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach

Many fundamental problems in scientific computing can be reduced to the computation of eigenvalues and eigenfunctions of an operator. One primary example is the electronic structure calculations, namely, computing the leading eigenvalue and eigenfunction of the Schrödinger operator. If the dimension of the state variable is low, one can use classical approaches, such as the finite difference method or spectral method, to discretize the operator and to solve the eigenvalue problem. However, these conventional, deterministic approaches suffer from the so-called curse of dimensionality, when the underlying dimension becomes high, since the degree of freedom grows exponentially as the dimension increases. For high-dimensional problems, commonly arising from quantum mechanics, statistical mechanics, and finance applications, stochastic methods become more attractive and in many situations the only viable option. In the context of quantum mechanics, two widely used approaches for high-dimensional eigenvalue problems are the variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC) methods [1, 2, 3, 4, 5, 6]. These two approaches deal with the high dimensionality via different strategies. VMC relies on leveraging chemical knowledge to propose an ansatz of eigenfunction (wavefunction in the context of quantum mechanics) with parameters to be optimized under the variational formulation of the eigenvalue problem.