A Swarm Variant for the Schr\"odinger Solver

TL;DR

This paper presents EM-PSO, a swarm-based derivative-free optimizer, applied to neural network solvers for the Schr"odinger equation, demonstrating improved exploration and robustness over gradient methods.

Contribution

Introduces EM-PSO as a novel derivative-free optimizer for neural network-based Schr"odinger equation solutions, with mathematically supported hyper-parameters.

Findings

EM-PSO effectively approximates gradients for Schr"odinger equation solutions.

The method outperforms gradient-based optimizers in exploration and robustness.

Optimal hyper-parameters are mathematically derived and validated.

Abstract

This paper introduces application of the Exponentially Averaged Momentum Particle Swarm Optimization (EM-PSO) as a derivative-free optimizer for Neural Networks. It adopts PSO's major advantages such as search space exploration and higher robustness to local minima compared to gradient-descent optimizers such as Adam. Neural network based solvers endowed with gradient optimization are now being used to approximate solutions to Differential Equations. Here, we demonstrate the novelty of EM-PSO in approximating gradients and leveraging the property in solving the Schr\"odinger equation, for the Particle-in-a-Box problem. We also provide the optimal set of hyper-parameters supported by mathematical proofs, suited for our algorithm.