Adapting reservoir computing to solve the Schr\"odinger equation

TL;DR

This paper extends reservoir computing to solve the time-dependent Schr"odinger equation by handling complex data and introducing a multi-step learning strategy, demonstrated on molecular vibrational dynamics problems.

Contribution

It adapts reservoir computing for complex-valued wavefunctions and proposes a multi-step training method to improve accuracy in quantum dynamics simulations.

Findings

Successfully applied to four molecular vibrational problems

Achieved accurate wavefunction propagation over time

Enhanced reservoir computing with complex data handling

Abstract

Reservoir computing is a machine learning algorithm that excels at predicting the evolution of time series, in particular, dynamical systems. Moreover, it has also shown superb performance at solving partial differential equations. In this work, we adapt this methodology to integrate the time-dependent Schr\"odinger equation, propagating an initial wavefunction in time. Since such wavefunctions are complex-valued high-dimensional arrays the reservoir computing formalism needs to be extended to cope with complex-valued data. Furthermore, we propose a multi-step learning strategy that avoids overfitting the training data. We illustrate the performance of our adapted reservoir computing method by application to four standard problems in molecular vibrational dynamics.