Hermite Neural Network Simulation for Solving the 2D Schrodinger Equation

TL;DR

This work presents a Hermite-function-based neural network to solve the time-independent 2D Schrödinger equation on an infinite domain by employing Hermite roots as collocation points and Hermite activations. The method formulates a trial solution combining a boundary-satisfying component with a neural correction, optimized via gradient-based methods (including Adam) and autograd for derivatives. The network architecture uses Hermite-based hidden layers, with the output expressed as $y_{ ext{LNN}}(x)=\sum_{n=0}^N w_n \tilde{H}_n(x)$, trained to minimize a mean-squared error against the differential equation residual and boundary conditions, and implemented in MATLAB/Simulink. Numerical results show competitive accuracy and clear convergence patterns, with explicit comparisons to Physics-Informed Neural Networks (PINNs) and demonstrated visualization of energy levels and wave functions for 2D quantum states. The approach leverages spectral properties, enabling efficient handling of infinite domains and offering a practical framework for quantum-mechanical simulations with high fidelity and integration into standard simulation environments.

Abstract

The Schrodinger equation is a mathematical equation describing the wave function's behavior in a quantum-mechanical system. It is a partial differential equation that provides valuable insights into the fundamental principles of quantum mechanics. In this paper, the aim was to solve the Schrodinger equation with sufficient accuracy by using a mixture of neural networks with the collocation method base Hermite functions. Initially, the Hermite functions roots were employed as collocation points, enhancing the efficiency of the solution. The Schrodinger equation is defined in an infinite domain, the use of Hermite functions as activation functions resulted in excellent precision. Finally, the proposed method was simulated using MATLAB's Simulink tool. The results were then compared with those obtained using Physics-informed neural networks and the presented method.

Figures (14)

• Figure 1: Neural Network Structure, $x_{i}:(i=1,...,m)$ are the input vectors, $w_{i}:(i=1,...,m)$ are the weight vectors, $\sum$ is the sum operator, $f$ is the activation function and $y$ is output.
• Figure 2: Feedforward
• Figure 3: Hermite Neural Network Structure
• Figure 4: The present method flowchart
• Figure 5: Neural Network Structure

Theorems & Definitions (2)

• Example 1
• Example 2