Physics-Informed Neural Networks as Solvers for the Time-Dependent Schrödinger Equation Karan Shah

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Physics-Informed Neural Networks as Solvers for the

Time-Dependent Schrödinger Equation

Karan Shah

Center for Advanced Systems Understanding

Helmholtz-Zentrum Dresden-Rossendorf

Görlitz, Germany

k.shah@hzdr.de

Patrick Stiller

Institute of Radiation Physics

Helmholtz-Zentrum Dresden-Rossendorf

Dresden, Germany

p.stiller@hzdr.de

Nico Hoffmann

Institute of Radiation Physics

Helmholtz-Zentrum Dresden-Rossendorf

Dresden, Germany

n.hoffmann@hzdr.de

Attila Cangi

Center for Advanced Systems Understanding

Helmholtz-Zentrum Dresden-Rossendorf

Görlitz, Germany

a.cangi@hzdr.de

Abstract

We demonstrate the utility of physics-informed neural networks (PINNs) as solvers

for the non-relativistic, time-dependent Schrödinger equation. We study the perfor-

mance and generalisability of PINN solvers on the time evolution of a quantum

harmonic oscillator across varying system parameters, domains, and energy states.

1 Introduction

In recent years, there was a surge in the use of machine learning (ML) techniques in the physical

sciences [

], referred to as scientiﬁc machine learning. This has led to the rise of physics-informed

machine learning [

], where physics-based constraints are used to guide ML models, achieved by

incorporating structured prior information derived from physical laws into the learning algorithm.

Given the importance of numerical simulations across scientiﬁc disciplines, many ML surrogates for

solving differential equations on a large scale [

] have been developed. These approaches include the

deep Galerkin method [4], PINNs [5], neural operators [6], and DeepONets [7].

PINNs are a class of ML algorithms for solving forward and inverse problems that are represented

by partial differential equations. As opposed to numerical solvers where a new solution must be

computed whenever there is a change in the domain or system parameters, PINNs offer a mesh-

free alternative where the solver can be used for systems on arbitrary grid resolutions and system

parameters at a ﬁxed inference cost [

]. While training the PINN models can be a substantial

computational cost initially, it can be amortized over time due to rapid inference over a wider set

of system parameters. However, PINNs also have demonstrated shortcomings such as spectral bias

against high-frequency solutions [

], overﬁtting to trivial solutions [

], and poor performance for

systems with shocks [

] and for larger time domains [

]. The theory and applications of PINNs

are active areas of research, with developments such as the RNN-DCT-PINN [

], the gated-PINN

[

], and the variational-PINN [

] which increase their performance and scalability. Popular ML

frameworks for PINN models include Neural Solvers [13], Modulus [15], and DeepXDE [16].

The non-relativistic Schrödinger equation is the fundamental equation for describing quantum systems.

The wavefunction

, from which all observables of a system can be calculated, is obtained by solving

it. We distinguish two classes of problems: the time-independent Schrödinger equation (TISE) yields

the wave function

and the associated energies for a static system in terms of an eigenvalue problem,

Preprint. Under review.

arXiv:2210.12522v1 [quant-ph] 22 Oct 2022

while the time-dependent Schrödinger equation (TDSE) describes the dynamics of a quantum system,

i.e., the time evolution of

. Prior works have dealt with using ML to solve the SE. These include

models such as fully-connected networks (FCN) [

], reservoir computing [

], PINNs [

] for

the eigenvalue problem deﬁned by the TISE, and residual networks [

] and LSTMs [

] for the

TDSE. The TDSE has also been used as benchmark for PINNs [

]. However, in this work, we

go beyond prior investigations by examining the utility of PINN solvers for the TDSE across varying

system parameters, domains, and energy states.

2 Methods

2.1 PINNs

PINNs are constructed by encoding the constraints posed by a given differential equation and

its boundary conditions into the loss function of a deep learning model, usually, an FCN. These

constraints guide the network to ﬁnding a solution to the differential equation.

A partial differential equation (PDE) is deﬁned by fwith solution u(x, t)governed by

f(u) := ut+N[u;λ],x∈Ω, t ∈[0, T ], f(u) = 0 ,(1)

where N[u;λ]is a differential operator parameterised by λ,Ω∈RD, and x= (x1, x2, ..., xd)

with boundary conditions B(u, x, t)=0on ∂Ωand initial conditions T(u, x, t)=0at t= 0 .

A neural network

unet :RD+1 7→ R1

is constructed as a surrogate model

fnet =f(unet)

for the

true solution u. Constraints are encoded in the loss term Lfor neural network optimization

L=λfLf+λBC LBC +λIC LIC ,(2)

with

λf, λBC , λIC

being the regularization parameters. The PDE loss

Lf=

(1/Nf)PNf

i=1 



fnet xi

f, ti

f



denotes the error in the solution within the interior

points of the system and is calculated for

collocation points. The boundary loss

LBC = (1/NBC )PNBC

i=1 

unet xi

BC , ti

BC −uxi

BC , ti

BC 



is the constraint imposed by

the boundary conditions, whereas

LIC = (1/NIC )PNIC

i=1 

unet xi

IC , ti

IC −uxi

IC , ti

IC 



imposes the initial conditions. Both of them are calculated on a set of

NBC

boundary points and

NIC

initial points, respectively, where

unet

refers to the approximate solution predicted by the PINN

and

denotes the true value according to the boundary and initial conditions at that point. The

distribution of collocation points is visualized in Fig. 5b.

Once trained, the neural network is used to solve the PDE, potentially for a range of parameters

[

2.2 Time Dependent Schrödinger Equation

A PINN is constructed for solving the TDSE

i∂φ(r, t)

∂t −ˆ

Hφ(r, t) = 0 ,(3)

where

denotes the Hamiltonian of the problem and

φ(r, t)

the solution which is commonly referred

to as a wave function that depends on a spatial coordinate

in three-dimensional space and time

Note that we adopt Hartree atomic units, i.e.,

~=kB=me= 1

, so energies are measured in Hartree

and lengths in Bohr radii. The Hamiltonian represents the speciﬁc problem, such as the kinetic and

potential energies of particle species and their interactions among each other and with external ﬁelds.

In this conceptual work, we resort to a simple but archetypical model problem, namely non-interacting

and spinless particles in a quantum harmonic oscillator in one spatial dimension with Hamiltonian

Hx=−1

∂2

∂x2+ω2

2x2,(4)

where

denotes the frequency of the harmonic oscillator. The most fundamental kind of quantum

dynamics is achieved by a superposition of eigenstates which are solutions of the corresponding

TISE. The analytical eigenstates of the harmonic oscillator are

φn(x) = φ0(x)(2nn!)−1/2Hn(√ωx)

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Physics-InformedNeuralNetworksasSolversfortheTime-DependentSchrödingerEquationKaranShahCenterforAdvancedSystemsUnderstandingHelmholtz-ZentrumDresden-RossendorfGörlitz,Germanyk.shah@hzdr.dePatrickStillerInstituteofRadiationPhysicsHelmholtz-ZentrumDresden-RossendorfDresden,Germanyp.stiller@hzdr.deNico...

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Physics-Informed Neural Networks as Solvers for the Time-Dependent Schrödinger Equation Karan Shah

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