Optimal Control of Material Micro-Structures Aayushman Sharma Zirui Mao Haiying Yang Suman Chakravorty Michael J Demkowicz Dileep Kalathil Abstract In this paper we consider the optimal control

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Optimal Control of Material Micro-Structures

Aayushman Sharma, Zirui Mao, Haiying Yang, Suman Chakravorty, Michael J Demkowicz, Dileep Kalathil

Abstract— In this paper, we consider the optimal control

of material micro-structures. Such material micro-structures

are modeled by the so-called phase ﬁeld model. We study

the underlying physical structure of the model and propose

a data based approach for its optimal control, along with a

comparison to the control using a state of the art Reinforcement

Learning (RL) algorithm. Simulation results show the feasibility

of optimally controlling such micro-structures to attain desired

material properties and complex target micro-structures.

I. INTRODUCTION

In this paper, we consider the optimal control of complex

and high DoF (Degree of Freedom) material micro-structure

dynamics, by doing which we aim at developing a robust

tool for exploring the fundamental limits of materials micro-

structure control during processing. The micro-structure

dynamics process is governed by the so-called Phase Field

(PF) model [1], which is a powerful tool for modeling

micro-structure evolution of materials. It represents the

spatial distribution of physical quantities of interest by an

order parameter ﬁeld governed by one or a set of partial

differential equations (PDEs). The Phase Field method

can naturally handle material systems with complicated

interface, thanks to which it has been successfully applied to

the simulation of a wide range of materials micro-structure

evolution processes, such as e.g., alloy solidiﬁcation [2][3],

alloy phase transformation [4][5], grain growth[6][7], and

failure mechanism of materials [8][9] etc.

The phase ﬁeld model is complex, nonlinear, and high

DOF, and does not admit analytical solutions. Thus, data

based approaches are natural to consider for the control of

these systems. In recent work [10], we proposed a novel

decoupled data based control (D2C) algorithm for learning

to control an unknown nonlinear dynamical system. Our

approach introduced a rigorous decoupling of the open-loop

(planning) problem from the closed-loop (feedback control)

problem. This decoupling allows us to come up with a highly

efﬁcient approach to solve the problem in a completely data

based fashion. Our approach proceeds in two steps: (i) ﬁrst,

we optimize the nominal open-loop trajectory of the system

using a ‘blackbox simulation’ model, (ii) then we identify

the linear system governing perturbations from the nominal

trajectory using random input-output perturbation data,

and design an LQR controller for this linearized system.

We have shown that the performance of D2C algorithm

is approximately optimal, in the sense that the decoupled

design is near optimal to second order in a suitably deﬁned

noise parameter [10]. In this work, we apply the D2C

approach to the optimal control of the evolution of material

micro-structures governed by the phase ﬁeld model. For

comparison, we consider RL techniques [11]. The past

several years have seen signiﬁcant progress in deep neural

networks based reinforcement learning approaches for

controlling unknown dynamical systems, with applications

in many areas like playing games [12], locomotion [13]

and robotic hand manipulation [14]. A number of new

algorithms that show promising performance are proposed

[15][16][17] and various improvements and innovations have

been continuously developed. However, despite excellent

performance on a number of tasks, RL is highly data

intensive. The training time for such algorithms is typically

very large, and high variance and reproducibility issues mar

the performance [18]. Thus, materials microstructure control

is intractable for current RL techniques. Recent work in

RL has also focused on PDE control using techniques

like the Deep Deterministic Policy Gradient(DDPG) with

modiﬁcations such as action descriptors to limit the large

action spaces in inﬁnite dimensional systems [19], as such

algorithms tend to degrade in performance with an increase

in dimensionality of the action-space. However, such an

approach, in general, is not feasible for the Microstructure

problem.

The control design of systems governed by PDEs are

known to be computationally intractable [20]. Owing to

the inﬁnite dimensionality of the system, typically model

reduction techniques are used to reduce the size of the

state space. The most widely used method is the Proper

Orthogonal Decomposition (POD) method that ﬁnds a

reduced basis for the approximation of the nonlinear PDE

using a simulation of the same [21]. However, in general,

the basis eigenfunctions can be quite different around the

optimal trajectory when compared to an initial trajectory,

and an approximation using the latter’s reduced basis can

lead to unacceptable performance [22],[23]. In our approach,

the nominal optimization problem is directly solved which

does not need any reduced order modeling. Furthermore,

the feedback design is accomplished by identifying a linear

time-varying (LTV) model which automatically produces

a reduced order model (ROM) of the system based purely

on the input output data while facilitating the use of

linear control design techniques such as LQG/ILQR on the

problem as opposed to having to solve a nonlinear control

problem in the methods stated previously. Furthermore, the

entire control design for the problem is done purely in a

data based fashion since the technique only queries a black

box simulation model of the process.

arXiv:2210.06734v1 [eess.SY] 13 Oct 2022

(a) Initial State (b) Goal state

Fig. 1: Material Micro-structure Control: the goal is to take

the micro-structure from the initial to the goal state (top row).

The micro-structure evolution is governed by a nonlinear PDE

which makes its control computationally intractable. A comparison

between a state of the art data based RL technique is shown versus

the proposed D2C technique (bottom row) for the reliable and

tractable solution of this control problem.

The contributions of the paper are as follows: we show

how to model the dynamics of a multi-phase micro-

structure and unveil its underlying structure. We present the

application of the D2C approach, and a state of the art RL

technique (DDPG) to the control of such microstructure

dynamics for the ﬁrst time in the literature, to the best

of our knowledge. Our results show that the local D2C

approach outperforms the global RL approach such as

DDPG when operating on higher dimensional phase ﬁeld

PDEs. We also show that the D2C approach is robust to

noise in the practical regimes, and that global optimality can

be recovered in higher noise levels through an open-loop

replanning approach. Furthermore, we can exploit the

peculiar physical properties of material micro-structure

dynamics to optimize the control actuation architecture

that leads to highly computationally efﬁcient control that

does not sacriﬁce much performance. We envisage that

this work is a ﬁrst step towards the construction of a

systematic feedback control synthesis approach for the

control of material micro-structures, potentially scalable to

real applications.

The rest of the paper is organized as follows: In Section II,

the micro-structure dynamics are expanded upon, describing

the different PDEs tested. We propose the decoupled data

based control (D2C) approach in Section III. In Section

IV, the proposed approach is illustrated through custom-

deﬁned test problems of varying dimensions, along with

testing robustness to noise, followed by improvements to the

algorithms which exploit the physics of the system.

II. MICROSTRUCTURE DYNAMICS: CLASSES OF PDES

IMPLEMENTED

This section provides a brief overview of the non-linear

dynamics model of a general multi-phase micro-structure.

We assume an inﬁnitely large, 2-dimensional structure satis-

fying periodical boundary conditions.

A. The material system

The evolution of material micro-structures can be repre-

sented by two types of partial differential equations, i.e.,

the Allen-Cahn equation [24] representing the evolution of

a non-conserved quantity, and the Cahn-Hilliard equation

[25] representing the evolution of a conserved quantity. The

Allen-Cahn equation has a general form of

∂φ

∂t =−M(∂F

∂φ −γ∇2φ)(1)

while the Cahn-Hilliard equation has the form

∂φ

∂t =∇ · M∇(∂F

∂φ −γ∇2φ)(2)

where φ=φ(x, t)is called the ‘order parameter’, which

is a spatially varying quantity. In Controls parlance, φis

the state of the system, and is inﬁnite dimensional, i.e., a

spatio-temporally varying function. It reﬂects the component

proportion of each phase of material system. For a two-phase

system studied in this work, φ= -1 represents one pure phase

and φ= 1 represent the other, while φ∈(-1, 1) stands for a

combination state of both phases on the boundary interface

between two pure phases; Mis a parameter related to the

mobility of material, which is assumed constant in this study;

Fis an energy function with a non-linear dependence on φ;

γis a gradient coefﬁcient controlling the diffusion level or

thickness of the boundary interface.

In essence, the Phase Field Method is one of gradient

ﬂow methods, which means the evolution process follows

the path of steepest descent in an energy landscape starting

from an initial state until arriving at a a local minimum.

So, the behavior of micro-structures in phase ﬁeld modeling

highly depends on the selection of energy density function

F. For instance, the double-well potential function owing to

two minima at φ=±1 and separated by a maximum at φ=

0 (as plotted in Fig. 2) will cause the ﬁeld φto separate into

regions where φ≈ ± 1, divided by the boundary interface,

while the single-well potential function owing to a single

minimum at φ= 0 (see Fig. 2) predicts a gradual smoothing

of any initial non-uniform φﬁeld, yielding a uniform ﬁeld

with φ= 0 in the long-time limit.

Accordingly, the evolution of material micro-structure can

be governed by selecting proper form of energy density func-

tion F. Controlling the evolution process of micro-structures

represented by Allen-Cahn equation and Cahn-Hilliard dif-

fusion equation are completely distinct. The former one is

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OptimalControlofMaterialMicro-StructuresAayushmanSharma,ZiruiMao,HaiyingYang,SumanChakravorty,MichaelJDemkowicz,DileepKalathilAbstractInthispaper,weconsidertheoptimalcontrolofmaterialmicro-structures.Suchmaterialmicro-structuresaremodeledbytheso-calledphaseeldmodel.Westudytheunderlyingphysicalstru...

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Optimal Control of Material Micro-Structures Aayushman Sharma Zirui Mao Haiying Yang Suman Chakravorty Michael J Demkowicz Dileep Kalathil Abstract In this paper we consider the optimal control

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