AMULTI -CATEGORY INVERSE DESIGN NEURAL NETWORK AND ITS APPLICATION TO DIBLOCK COPOLYMERS Dan Wei1z Tiejun Zhou1z Yunqing Huang1and Kai Jiang1

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AMULTI-CATEGORY INVERSE DESIGN NEURAL NETWORK AND

ITS APPLICATION TO DIBLOCK COPOLYMERS

Dan Wei1,‡, Tiejun Zhou1,‡, Yunqing Huang1and Kai Jiang∗1

1Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent

Computing and Information Processing of Ministry of Education, School of Mathematics and Computational Science,

Xiangtan University, Xiangtan, Hunan, China, 411105.

‡These authors contributed equally to this work.

October 26, 2022

ABSTRACT

In this work, we design a multi-category inverse design neural network to map ordered periodic

structure to physical parameters. The neural network model consists of two parts, a classiﬁer and

Structure-Parameter-Mapping (SPM) subnets. The classiﬁer is used to identify structure, and the SPM

subnets are used to predict physical parameters for desired structures. We also present an extensible

reciprocal-space data augmentation method to guarantee the rotation and translation invariant of

periodic structures. We apply the proposed network model and data augmentation method to two-

dimensional diblock copolymers based on the Landau-Brazovskii model. Results show that the

multi-category inverse design neural network is high accuracy in predicting physical parameters for

desired structures. Moreover, the idea of multi-categorization can also be extended to other inverse

design problems.

Keywords

Inverse design; Multi-category network; Reciprocal-space data augmentation method; Landau-Brazovskii

model; Diblock copolymers; Periodic structure.

1 Introduction

Material properties are mainly determined by microscopic structures. Therefore, to obtain satisfactory properties,

how to ﬁnd desired structures is very important in material design. The formation of ordered structures directly relies

on physical conditions, such as temperature, pressure, molecular components, geometry conﬁnement. However, the

relationship between ordered structures and physical conditions is extremely complicated and diversiﬁed. A traditional

approach is a trial-and-error manner, i.e., passively ﬁnding ordered structures for given physical conditions. This

approach, in terms of solving direct problem, is time-consuming and expensive. A wise way is inverse design that turns

to ﬁnd physical conditions for desired structures.

In this work, we are concerned about the theoretical development of inverse design method for block copolymers. Block

copolymer systems are important materials in industrial applications since they can self-assemble into innumerous

ordered structures. There are many solving direct problem approaches of block copolymer systems, such as the ﬁrst

principle calculation [

], Monte Carlo simulation [

], molecular dynamic [

], dissipative particle dynamics [

], self

consistent ﬁeld simulation [

], and density functional theory [

]. In the past decades, a directed self-assembly (DSA)

method has been developed to inverse design block copolymers. Liu et al. [

] presented an integration scheme of block

copolymers directed assembly with 193 nm immersion lithography, and provided a pattern quality that was comparable

with existing double patterning techniques. Suh et al. [

] obtained nanopatterns via DSA of block copolymer ﬁlms

with a vapour-phase deposited topcoat. Many DSA strategies have been also developed for the fabrication of ordered

square patterns to satisfy the demand for lithography in semiconductors [11, 12, 13, 14].

∗kaijiang@xtu.edu.cn

arXiv:2210.13453v1 [cond-mat.soft] 12 Oct 2022

A multi-category inverse design neural network and its application to diblock copolymers

With the rise of data science and machine learning, many deep-learning inverse design methods have been developed to

learn the mapping between structures and physical parameters [

]. These new techniques and methods are

beginning to study block copolymers. Yao et al. combined machine learning with self consistent ﬁeld theory (SCFT) to

accelerate the exploration of parameter space for block copolymers [

]. Lin and Yu designed a deep learning solver

inspired by physical-informed neural networks to tackle the inverse discovery of the interaction parameter and the

embedded chemical potential ﬁelds for an observed structure [

]. Based on the idea of classifying ﬁrst and ﬁtting later,

Katsumi et al. estimated Flory-Huggins interaction parameters of diblock copolymers from cross-sectional images of

phase-separated structures [

]. The phase diagrams of block copolymers can be predicted by combining deep learning

technique and SCFT [22, 23].

In this work, we propose a new neural network to address inverse design problem based on the idea of multi-

categorization. We take AB diblock copolymer system as an example to demonstrate the performance of our network.

The training and test data sets are generated from the Landau-Brazovskii (LB) model [

]. LB model is an effective

tool to describe the phases and phase transition of diblock copolymers [

]. Let

φ(r)

be the order

parameter, a function of spatial position

, which represents the density distribution of diblock copolymers. The free

energy functional of LB model is

E(φ(r)) = 1

|Ω|ZΩξ2

2[(∆ + 1)φ]2+τ

2!φ2−γ

3!φ3+1

4!φ4dr,(1.1)

φ(r)

satisﬁes the mass conservation

|Ω|RΩφ(r)dr= 0

Ω

is the system volume. The model parameters in (1.1) are

associated to physical conditions of diblock coplymers. Concretely,

is a temperature-like parameter related to the

Flory-Huggins interaction parameter, the degree of polymerization

, and the A monomer fraction

of each diblock

copolymer chain.

can control the onset of the order-disorder spinodal decomposition. The disordered phase becomes

unstable at

τ= 0

is associated with

and

, it’s nonzero only if AB diblock copolymers chain is asymmetric.

the bare correlation length. Further relationship can be found in [

]. The stationary states of LB free

energy functional correspond to ordered structures.

The rest of the paper is organized as follows. In Section 2, we solve the LB model

(1.1)

to obtain data set. In Section 3,

we present the multi-category inverse design neural network, and reciprocal-space data augmentation (RSDA) method

for periodic structures. In Section 4, we take the diblock copolymer system conﬁned in two dimension as an example to

test the performance of our proposed inverse design neural network model. In Section 5, we draw a brief summary of

this work.

2 Direct problem

Solving direct problem is optimizing LB free energy functional (1.1) to obtain stationary states corresponding to ordered

structures

min

φ(r)E(φ(r)),s.t. 1

|Ω|ZΩ

φ(r)dr= 0.(2.1)

Here we only consider periodic structures. Therefore, we can apply Fourier pseudospectral method to discretize the

above optimization problem.

2.1 Fourier pseudospectral method

For a periodic order parameter φ(r),r∈Ω := Td=Rd/AZd, where A= (a1,a2,...,ad)∈Rd×dis the primitive

Bravis lattice. The primitive reciprocal lattice B= (b1,b2,...,bd)∈Rd×d, satisfying the dual relationship

ABT= 2πI.(2.2)

The order parameter φ(r)can be expanded as

φ(r) = X

k∈Zd

φ(k)ei(Bk)Tr,r∈Td,(2.3)

where the Fourier coefﬁcient

φ(k) = 1

|Td|ZTd

φ(r)e−i(Bk)Trdr,(2.4)

|Td|is the volume of Td.

A multi-category inverse design neural network and its application to diblock copolymers

We deﬁne the discrete grid set as

N=n(r1,j1, r2,j2, ..., rd,jd) = Aj1/N, j2/N, ..., jd/NT,0≤ji< N, ji∈Z, i = 1,2, ..., do,(2.5)

where the number of elements of

M=Nd

. Denote grid periodic function space

f:Td

N7→ C

periodic . For any periodic grid functions F, G ∈ GN, the `2-inner product is deﬁned as

hF, GiN=1

rj∈Td

F(rj)¯

G(rj).(2.6)

The discrete reciprocal space is

N=nk= (kj)d

j=1 ∈Zd:−N/2≤kj< N/2o,(2.7)

the discrete Fourier coefﬁcients of φ(r)in Td

Ncan be represented as

φ(k) = hφ(rj), ei(Bk)TrjiN=1

rj∈Td

φ(rj)e−i(Bk)Trj,k∈Kd

N.(2.8)

For k∈Zd, and l∈Zd, we have the discretize orthogonality,

hei(Bk)Trj, ei(Bl)TrjiN=1,k=l+Nm,m∈Zd,

0, otherwise. (2.9)

Therefore, the discrete Fourier transform of φ(rj)is,

φ(rj) = X

k∈Kd

φ(k)ei(Bk)Trj,rj∈Td

N.(2.10)

The Nd-order trigonometric polynomial is

INφ(r) = X

k∈Kd

φ(k)ei(Bk)Tr,r∈Td.(2.11)

Then for rj∈Td

N, we have φ(rj)≈INφ(rj).

Due to the orthogonality (2.9), the LB energy functional E(φ)can be discretized as

Eh[ˆ

Φ] = ξ2

h1+h2=0h1−(Bh1)T(Bh2)i2ˆ

φ(h1)ˆ

φ(h2) + τ

2! X

h1+h2=0

φ(h1)ˆ

φ(h2)

−γ

3! X

h1+h2+h3=0

φ(h1)ˆ

φ(h2)ˆ

φ(h3)

4! X

h1+h2+h3+h4=0

φ(h1)ˆ

φ(h2)ˆ

φ(h3)ˆ

φ(h4),

(2.12)

where

hi∈Kd

i= 1,2,3,4

, and

Φ = ˆ

φ1,ˆ

φ2,..., ˆ

φMT

∈CM

. The convolutions in the above expression

can be efﬁciently calculated through the fast Fourier transform (FFT). Moreover, the mass conservation constraint

|Ω|RΩφ(r)dr= 0 is discretized as

eTˆ

Φ=0,(2.13)

where e= (1,0, ..., 0)T∈RM. Therefore, (2.1) reduces to a ﬁnite dimensional optimization problem

min

Φ∈CM

Eh[ˆ

Φ] = Gh[ˆ

Φ] + Fh[ˆ

Φ], s.t. eTˆ

Φ=0,(2.14)

where Ghand Fhare discretize interaction and bulk energies

Gh(ˆ

Φ) = ξ2

h1+h2=0h1−(Bh1)T(Bh2)i2ˆ

φ(h1)ˆ

φ(h2),

Fh(ˆ

Φ) = τ

2! X

h1+h2=0

φ(h1)ˆ

φ(h2)−γ

3! X

h1+h2+h3=0

φ(h1)ˆ

φ(h2)ˆ

φ(h3)

4! X

h1+h2+h3+h4=0

φ(h1)ˆ

φ(h2)ˆ

φ(h3)ˆ

φ(h4).

(2.15)

In the work, we employ the adaptive APG method to solve (2.14).

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AMULTI-CATEGORYINVERSEDESIGNNEURALNETWORKANDITSAPPLICATIONTODIBLOCKCOPOLYMERSDanWei1,z,TiejunZhou1,z,YunqingHuang1andKaiJiang11HunanKeyLaboratoryforComputationandSimulationinScienceandEngineering,KeyLaboratoryofIntelligentComputingandInformationProcessingofMinistryofEducation,SchoolofMathematicsand...

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