Neural network solution of the Schrödinger equation for a two-dimensional harmonic oscillator

Abstract

We present computer simulations of a neural network capable of learning to perform transformations generated by the Schrödinger equation required to find eigenenergies of a two-dimensional harmonic oscillator. We show that this task can be achieved by a not fully connected back-propagation neural network containing 49 input neurons, 3 hidden layer neurons and 1 output neuron. The investigated neural network turns out to be capable of predicting eigenenergies with an average error of less than one percent. We demonstrate that the CPU time required to teach a neural network of performing the transformation produced by the Schrödinger equation is about 2 min to reach 41000 learning iterations, thus making foreseeable a direct application of a neural network in this and other more complex physical and chemical problems. A discussion of the errors due to the generalization of acquired knowledge is presented and related to a limited number of examples in learning mode and the number of neurons in the hidden layer. Decreasing the number of neurons in the hidden layer increases the apparent ability of the neural network for generalization.