Physics-Informed Neural Networks as Solvers for the Time-Dependent Schrödinger Equation Karan Shah
2025-05-02
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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 [
1
], referred to as scientific machine learning. This has led to the rise of physics-informed
machine learning [
2
], 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 [
3
] 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 [
8
]. 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 [
9
], overfitting to trivial solutions [
10
], and poor performance for
systems with shocks [
11
] and for larger time domains [
12
]. The theory and applications of PINNs
are active areas of research, with developments such as the RNN-DCT-PINN [
12
], the gated-PINN
[
13
], and the variational-PINN [
14
] 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
φ
, 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
φ
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
φ
. Prior works have dealt with using ML to solve the SE. These include
models such as fully-connected networks (FCN) [
17
,
18
], reservoir computing [
19
], PINNs [
20
] for
the eigenvalue problem defined by the TISE, and residual networks [
21
] and LSTMs [
22
] for the
TDSE. The TDSE has also been used as benchmark for PINNs [
5
,
2
,
13
]. 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
2
denotes the error in the solution within the interior
points of the system and is calculated for
Nf
collocation points. The boundary loss
LBC = (1/NBC )PNBC
i=1
unet xi
BC , ti
BC −uxi
BC , ti
BC
2
is the constraint imposed by
the boundary conditions, whereas
LIC = (1/NIC )PNIC
i=1
unet xi
IC , ti
IC −uxi
IC , ti
IC
2
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
u
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
λ
[
8
].
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
ˆ
H
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
r
in three-dimensional space and time
t
.
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
ˆ
Hx=−1
2
∂2
∂x2+ω2
2x2,(4)
where
ω
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)
2
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Physics-InformedNeuralNetworksasSolversfortheTime-DependentSchrödingerEquationKaranShahCenterforAdvancedSystemsUnderstandingHelmholtz-ZentrumDresden-RossendorfGörlitz,Germanyk.shah@hzdr.dePatrickStillerInstituteofRadiationPhysicsHelmholtz-ZentrumDresden-RossendorfDresden,Germanyp.stiller@hzdr.deNico...
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时间：2025-05-02
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