Solving Multi-Dimensional Stationary Schr odinger Equations Using Extended Physics-Informed Neural Networks Jinde Liu Xilong Dou Chen Yang Gang Jiang

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Solving Multi-Dimensional Stationary Schr¨odinger Equations

Using Extended Physics-Informed Neural Networks

Jinde Liu, Xilong Dou, Chen Yang∗

, Gang Jiang

February 17, 2023

Abstract

Due to the good performance of neural networks in high-dimensional and nonlinear problems,

machine learning is replacing traditional methods and becoming a better approach for eigenvalue

and wave function solutions of multi-dimensional Schr¨odinger equations. This paper proposes a

numerical method based on neural networks to solve multiple excited states of multi-dimensional

stationary Schr¨odinger equation. We introduce the orthogonal normalization condition into the loss

function, use the frequency principle of neural networks to automatically obtain multiple excited

state eigenfunctions and eigenvalues of the equation from low to high energy levels, and propose a

degenerate level processing method. The use of equation residuals and energy uncertainty makes the

error of each energy level converge to 0, which eﬀectively avoids the order of magnitude interference of

error convergence, improves the accuracy of wave functions, and improves the accuracy of eigenvalues

as well. Comparing our results to the previous work, the accuracy of the harmonic oscillator problem is

at least an order of magnitude higher with fewer training epochs. We complete numerical experiments

on typical analytically solvable Schr¨odinger equations, e.g., harmonic oscillators and hydrogen-like

atoms, and propose calculation and evaluation methods for each physical quantity, which prove the

eﬀectiveness of our method on eigenvalue problems. Our successful solution of the excited states of

the hydrogen atom problem provides a potential idea for solving the stationary Schr¨odinger equation

for multi-electron atomic molecules.

1 Introduction

Solving the eigenvalue problem of the multi-dimensional stationary Schr¨odinger equation (SSE) has

been an issue since quantum mechanics established that many physicists pay attention to. The solution

to this problem is of great signiﬁcance to the development of quantum chemistry and computational

materials science. Except for a few models that can be solved analytically, e.g., inﬁnitely deep potential

wells, harmonic oscillators, the hydrogen atom, Morse potentials, etc., the SSE can only oﬀer approximate

solutions.

As a speciﬁc eigenvalue problem, the solution of the SSE has great versatility with other types of

eigenvalue problems. The key process of ﬁnding an approximate solution consists of the representation

method of the eigen wave function, the model transformation, and the solving procedure. The model

transformation and the solving procedure are generally needed to transform the equation-solving prob-

lem into an optimization problem, and then use the optimization algorithm to solve it. More generally,

for the eigenvalue problem of partial diﬀerential equations (PDE), the solving process has the follow-

ing key points: problem transformation, eigenvalue solving, normalization condition (non-zero solution

construct), orthogonal condition (excited state treatment), boundary conditions, and sampling method.

In view of the above key points, diﬀerent numerical calculation methods based on neural networks are

proposed [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

Lagaris et al.[1] proved for the ﬁrst time the feasibility of using neural networks to solve eigenvalue

problems. Nakanishi et al.[2, 3] was trained to solve the eigenvalues as parameters for the ﬁrst time, and

the excited state at high energy levels was calculated. Shirvany et al.[4] adds boundary conditions to the

loss function to obtain the solution of multiple energy levels of an one-dimensional harmonic oscillator.

After the extensive development of neural networks, E et al.[5] for the ﬁrst time used the variational

form of the residual network and eigenvalue problem to solve the ground state of the high-dimensional

Schr¨odinger equation. Han et al.[6] for the ﬁrst time used the backward stochastic diﬀerential equations

∗yangchen@scu.edu.cn

arXiv:2210.00454v2 [physics.comp-ph] 16 Feb 2023

form of diﬀusion Monte Carlo to solve the ground and excited states of the high-dimensional Schr¨odinger

equation in combination with the neural network, and the automatic constraint of periodic boundary

conditions is realized by adding a layer of trigonometric functions to the network. Li et al.[8] constructs a

multi-output network to represent multiple eigenstates, solves the multi-dimensional harmonic oscillator

problem, proposes an algorithm that obtains multiple states in a single training, and discusses the resource

consumption problem when solving high-dimensional problems. Zhang et al.[9, 10, 11, 12, 13] also made

some explorations on the issue of eigenvalues.

There are some limitations of previous methods, e.g., only low-dimensional cases and a small number

of excited states were solved, degeneracy problems were rarely discussed, and the solution accuracy

obtained was poor. At present, there is still a lack of research on a large number of excited states and

degenerate problems for multi-dimensional complex eigenvalue problems. We use the multi-output neural

network as multiple eigenstate ansatz functions, embed the boundary conditions into the ansatz functions,

calculate the sum of squares of the equation residuals of each energy level on the spatial sampling point,

calculate the orthogonal normalization error and energy uncertainty by Monte Carlo integral, minimize

the weighted sum of these three items by gradient descent, and ﬁnally acquire the solution of multiple

energy levels in a single training. At the same time, a method of obtaining multiple states from multiple

training is given. Energy uncertainty has been added to the loss function to improve convergence accuracy.

And we discussed the situations and approaches to degenerate levels in multi-dimensional problems. This

method does not directly adopt the variational Monte Carlo (VMC) method, thus the sampling accuracy

requirements are low, which can further save computing resources. And because the solution occurs at the

zero point of each item, it is possible to solve the excited state without having to consider the weighting

coeﬃcients of each energy level. For simple boundary conditions, we generally use terms that multiply

the solution function to meet the boundary conditions so that the solution function automatically satisﬁes

the boundary conditions. For complex boundary conditions, points are generally taken at the boundary

according to the idea of supervised learning so that the solution function meets the boundary conditions

at these points. In order to verify the eﬀectiveness of the algorithm, we ﬁrst solved the problems of

1, 2, 3, 5, and 10 dimensions on the simple system of harmonic oscillators. For the one-, two-, and

ﬁve-dimensional harmonic oscillator problems, our energy average absolute error is 0.0070%, 0.0385%,

and 0.0306%, respectively, and the reference [8] energy average absolute error is 0.3850%, 0.5927%, and

1.6598%, respectively. Especially for one-dimensional problems, we got this result after more than 30,000

iterations, and the reference[8] got this result after more than 70,000 iterations. Fewer iterations shows

that our method can achieve higher accuracy convergence with less resource consumption. Subsequently,

we generalized the method to the hydrogen atom problem in Cartesian coordinate system, and successfully

obtained the solution of the lowest ﬁve energy levels, which indicates that our method has the ability

to solve singularity-containing potentials, and further shows that our algorithm has the potential to

be extended to calculate the multi-excited states of the Schr¨odinger equation for multi-electron atomic

molecules.

2 Results

We ﬁrst verify the feasibility of our algorithm in a simple harmonic oscillator system and determine

the numerical ranges of some key hyperparameters. Then it is extended to the actual system like hydrogen

atom and the wave function and energy calculation of the ﬁrst few energy levels are completed. All the

problems are solved directly by using Cartesian coordinate system. According to the analysis in Section 4,

for any potential ﬁeld, we need to ﬁrst determine the form of potential function and boundary conditions,

then set reasonable boundary terms according to the potential ﬁeld and boundary conditions, and design

reasonable eigenvalue terms according to the possible forms of eigenvalues to speed up the convergence

rate. It is also necessary to select a suitable sampling scheme according to the characteristics of the

potential ﬁeld. These steps are also illustrated in the following examples.

2.1 Multi-dimensional uncoupled harmonic oscillator potential

We ﬁrst calculate the classical problem of harmonic oscillator. To make simpler, let m=¯

h=ω= 1,

the harmonic oscillator potential function is V(x) = 1

2PD

i=1 x2

i, the wave function is Ψ( ~

R). It is easy to

know that the term satisfying the boundary condition is Bn(x) = exp(−1

2PD

iνnix2

i). And the real wave

function of D-dimensional harmonic oscillator is ψn=QD

d=1 ψni(xd), the real energy is En=PD

d=1 n+D

where ψnd =NndHnd(xd)exp(−1

2x2

d) is the real wave function of the energy level of the dth dimension,

is Hermitian polynomial, Nnd is the normalization coeﬃcient[14, 15]. We calculate the wave functions

and energy levels of 1, 2, 3, 5, and 10 dimension, respectively, and give the relationship between the key

parameters and the convergence error.

The spatial sampling of this algorithm uses Gaussian sampling, and the density function is P(X) =

exp(−PD

d=1(xd

µ)2), thus the weights of each sampling point are ω(Xs) = 1

P(Xs). Through the change

of each loss function with the number of iterations, we know what happens in the iterative convergence

process, which reﬂects the performance characteristics of the solver and the reliability of the solution. The

key parameter settings are shown in Table 1. In pre-training, we generally adopted larger lr and smaller

Sto facilitate the fast convergence of the model, and larger sums to ensure that the wave functions of

diﬀerent energy levels can be obtained. During the transfer training, we reduced lr2,α2

2and α2

3, improved

S(2), reduced α2

2and α2

3in order to improve the stability and accuracy of the convergence of each energy

level in the pre-training.

In the calculation of the one-dimensional harmonic oscillator, the wave function and energy conver-

gence curves of 16 energy levels are given, as shown in Figures 1 and 2. From Figure 1(a), the convergence

process will initially undergo a disorderly convergence, and then ψθ

n(Xs) will continue to appear at each

energy level, and the overall trend will gradually transition from low energy level to high energy level.

Since it is not speciﬁed which energy level the output wave function corresponds to, there is also a sit-

uation of energy level exchange, which is similar to the situation shown in the reference[8]. Note that

our convergence does not stipulate that it must converge to a minimum of Nenergy levels, but in the

one-dimensional case, it still occupies the lowest energy level as possible. This is consistent with the

assumption we introduced in Section 4.1. Finally, the Eθ

nare sorted to determine the corresponding

analytic wave function. At the same time, it is also necessary to determine the subspace of each energy

level in the case of degeneracy, the speciﬁc method is shown in the Supplementary Information. lr and

αjointly control the energy level transition speed, while lr,α1, and α2control convergence stability

and convergence accuracy. Since our method uses Monte Carlo integrals, it is clear that the number of

sampling points and sampling methods directly aﬀect the convergence accuracy. Figure 1(b) shows that

there is a mutation in the total loss function Lduring transfer training, which is caused by the decrease

of the weight coeﬃcient of the loss term, but it can be seen that the attenuation trend of Lalso has

a mutation, which is because after reducing the loss weight coeﬃcient of the orthogonal normalization

condition, Figure 1(c) and (d) Fand ∆Ebecome the main loss terms, which are rapidly reduced, so that

the accuracy of the wave function is rapidly improved, and at the same time, the orthogonal condition

Figure 1(e) is naturally easy to meet.

−2

−1

0 2 4 6 8

−4

−2

0 2 4 6 8

−9

−6

−3

0 2 4 6 8

−9

−6

−3

(a) (b) (c)

(d) (e) (f)

Figure 1: Convergence curve of one-dimensional harmonic oscillator, (a) is the average energy convergence

curve, the black dotted line is the real energy level, (b) is the total loss function convergence curve, (c) is

the energy uncertainty convergence curve, (d) is the residual equation convergence curve, (e) orthogonal

condition convergence curve, (f) is the normalization condition convergence curve. The red vertical lines

in ﬁgures represent the iteration starting points.

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SolvingMulti-DimensionalStationarySchrodingerEquationsUsingExtendedPhysics-InformedNeuralNetworksJindeLiu,XilongDou,ChenYang*,GangJiangFebruary17,2023AbstractDuetothegoodperformanceofneuralnetworksinhigh-dimensionalandnonlinearproblems,machinelearningisreplacingtraditionalmethodsandbecomingabettera...

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Solving Multi-Dimensional Stationary Schr odinger Equations Using Extended Physics-Informed Neural Networks Jinde Liu Xilong Dou Chen Yang Gang Jiang

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