NEURAL-NETWORK SOLUTION OF THE SCHRODINGER-EQUATION FOR A 2-DIMENSIONAL HARMONIC-OSCILLATOR

We present computer simulations of a neural network capable of learnin g to perform transformations generated by the Schrodinger equation req uired to find eigenenergies of a two-dimensional harmonic oscillator. We show that this task can be achieved by a not fully connected back-p ropagation 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 le ss than one percent. We demonstrate that the CPU time required to teac h a neural network of performing the transformation produced by the Sc hrodinger equation is about 2 min to reach 41000 learning iterations, thus making foreseeable a direct application of a neural network in th is and other more complex physical and chemical problems. A discussion of the errors due to the generalization of acquired knowledge is pres ented and related to a limited number of examples in learning mode and the number of neurons in the hidden layer. Decreasing the number of n eurons in the hidden layer increases the apparent ability of the neura l network for generalization.