# Machine learning based surrogate models for microchannel heat sink
optimization

Ante Sikirica<EMAIL_ADDRESS>Luka Grbčić<EMAIL_ADDRESS>Lado
Kranjčević<EMAIL_ADDRESS>
###### Abstract

Microchannel heat sinks are an efficient cooling method for semiconductor
packages. However, to properly cool increasingly complex and thermally dense
circuits, microchannel designs should be improved and expanded on. In this
paper, microchannel designs with secondary channels and with ribs are
investigated using computational fluid dynamics and are coupled with a multi-
objective optimization algorithm to determine and propose optimal solutions
based on observed thermal resistance and pumping power. A workflow that
combines Latin hypercube sampling, machine learning-based surrogate modeling
and multi-objective optimization is proposed. Random forests, gradient
boosting algorithms and neural networks were considered during the search for
the best surrogate. We demonstrated that tuned neural networks can make
accurate predictions and be used to create an acceptable surrogate model.
Optimized solutions show a negligible difference in overall performance when
compared to the conventional optimization approach. Additionally, solutions
are calculated in one-fifth of the original time. Generated designs attain
temperatures that are lower by more than 10% under the same pressure limits as
a convectional microchannel design. When limited by temperature, pressure
drops are reduced by more than 25%. Finally, the influence of each design
variable on the thermal resistance and pumping power was investigated by
employing the SHapley Additive exPlanations technique. Overall, we have
demonstrated that the proposed framework has merit and can be used as a viable
methodology in microchannel heat sink design optimization.

###### keywords:

Microchannel heat sink , Machine learning , Multi-objective optimization ,
Computational fluid dynamics

††journal: Journal

[inst1]organization=Center for Advanced Computing and Modelling, University of
Rijeka, addressline=Radmile Matejčić 2, city=Rijeka, postcode=51000 Rijeka,
country=Croatia

[inst2]organization=Faculty of Engineering, University of Rijeka,
addressline=Vukovarska 58, city=Rijeka, postcode=51000 Rijeka, country=Croatia

##  1 Introduction

Microfabrication technology improvements over the last several decades have
enabled the production of dense semiconductor packages. However, the resulting
increase in thermal density poses a problem as it leads to higher device
temperatures, which can cause failure. As technology advances, heat
dissipation will become a major performance-limiting factor, thus the
development of adequate cooling is of paramount importance.

Tuckerman and Pease [1] introduced microchannel heat sinks (MCHS) as a method
for ensuring effective cooling of integrated circuits. These early MCHS
designs featured rectangular channels embedded in a thermally conductive
substrate. MCHS with rectangular, trapezoidal, and triangular cross-sections
were studied by Gunnasegaran et al. [2] and Wang et al. [3]. Rectangular
cross-sections were deemed optimal, particularly when the aspect ratio of the
cross-section is $\approx$ 10\. Moreover, it was demonstrated that by reducing
the hydraulic diameter, their effectiveness can be increased even further.
Recently, Vasilev et al. [4] investigated rectangular MCHS designs. The
authors have concluded that by modifying the spatial orientation of channels
while maintaining the same pumping power, the thermal resistance can be
lowered. Xia et al. [5] investigated various avenues for improving the
conventional rectangular MCHS. Different cross-sections, header shapes, and
inflow/outflow locations were evaluated. Due to flow disturbance, channels
with reentrant cavities performed better in terms of thermal performance.
Later, Xia et al. [6] numerically and experimentally evaluated designs with
corrugated channels.

Li et al. [7] investigated MCHS with rectangular ribs and triangular cavities.
Lower observed temperatures were attributed to flow disturbance and the
interruption and redevelopment of the hydrodynamic and thermal boundary
layers. A paper by Ghani et al. [8] explored the properties of MCHS with
sinusoidal cavities and ribs. Bayrak et al. [9] presented a comparative
numerical analysis that included conventional as well as designs with cavities
and ribs. Bidirectional ribs were evaluated in a paper by Wang et al. [10].
The combination of vertical and spanwise ribs provided better thermal
performance, albeit with a notable pressure drop. The use of slanted passages
to induce secondary flow was investigated by Kuppusamy et al. [11], Ghani et
al. [12] and Memon et al. [13]. Introduced topological changes produced flow
disturbance. All authors reported a reduction in thermal resistance and
pressure drop. A study by Japar et al. [14] combined cavities, ribs and
secondary channels. The proposed composite design, according to the research,
has excellent thermal performance.

Recent design improvements include the use of wavy microchannels [15, 16],
double-layered [17, 18, 19] and porous MCHS [20, 21]. A combined double-
layered semi-porous MCHS design was investigated using a multi-objective (MO)
optimization approach in a paper by [22] which additionally included an
overview of previous design improvement studies. A comparative overview of
proposed MCHS designs has been conducted by Lu and Vafai [23] and Sidik et al.
[24]. Ramesh et al. [25] conducted an overview of experimental and numerical
studies in the field. Recently, a review by Gao et al. [26] summarized all
advancements and methodological approaches used for microchannel modeling.

Applied geometrical and topological modifications to MCHS are typically
arbitrarily chosen to provide an improvement. Consequently, optimization is
required to provide balanced designs for thermal performance and pressure
drop. The simplest MCHS optimization workflows combine analytical expressions
or surrogate models with single objective iterative processes or optimization
algorithms [27, 28]. A contemporary alternative is topology optimization [29,
30, 31, 32].

Currently, the dominant workflow employed to optimize MCHS designs
incorporates response surface method (RSM) and Non-dominated Sorting Genetic
Algorithm (NSGA-II). Based on an initial set of numerical (or analytical)
results, a relationship between variables and objectives is established and
used in a multi-objective analysis. Papers by Normah et al. [33], Xia et al.
[34], Shi et al. [35] and Yang and Cao [36] employed analogous methodology
with pumping power and thermal resistance designated as optimization
objectives. Alperen and Sertac [37] and Kose et al. [38] considered the use of
pumping power and Nusselt number. In a comparative study, Hadad et al. [39]
noted that the Jaya algorithm outperforms Faced Centered Central Composite
Design and NSGA-II-based approaches. Kwanda [40] recommended using the
Improved Augmented $\epsilon$-Constraint Method as it was able to outperform
The Strength Pareto Evolutionary Algorithm 2 and NSGA-II. A recent study by
Polat and Cadirci [41] suggested the use of artificial neural networks (ANN)
coupled with NSGA-II algorithm to generate the Pareto front of an MCHS with
diamond-shaped pin fins.

Balachandar et al. [42] combined ANN and genetic algorithm to determine the
optimal fin geometry. The network was trained using CFD results. A paper by
Yang et al. [43] considered a conditional generative adversarial network
pix2pix trained on a CFD-generated dataset to propose optimized geometries of
micro-pin fin heat sinks. Both studies employed small datasets and lack
extensive assessment of developed ANN surrogates. Studies by Mohammadpour et
al. [44, 45] combined CFD and machine learning algorithms to generate optimal
designs. In [44], support vector regression and PSO were combined and led to a
design that reduces temperature by 4 K. Subsequently, the k-nearest neighbors
algorithm was used to improve performance in MCHS with double synthetic jets.
Improvements of up to 15 K were reported. Multi-objective workflow combining
CFD, ANN and NSGA-II algorithm was considered by Sui et al. [46] to optimize
the absorber geometries in membrane-based microchannels. Optimized geometry
reduced pressure by more than six times. Recently, Bayer et al. [47] used
artificial neural networks trained on CFD data to optimize wavy, double-
layered MCHS with porous ribs. Authors reported an improvement of up to 1.2 K.
It is important to note, however, that the majority of papers dealing with ML
and MCHS lack the assessment depth needed to properly validate a model. This
includes dataset partitioning, analysis of the influence of dataset size, and
often detailed model tuning that addresses overfitting. Although there has
been progress in this aspect, a concise set of guidelines should be defined
and adhered to, to ensure the validity of proposed ML models.

As the complexity of electronic chips grows, ensuring adequate heat
dissipation will be an increasingly challenging task. Thus, the purpose of
this work is to present and propose a methodology for optimizing MCHS designs
that is based on high-performance computing and machine learning (ML). The
proposed approach should allow for the rapid generation of optimized MCHS
designs for a given purpose. The use of machine learning instead of
traditional RSM enables a computationally less expensive procedure that, if
conducted per accepted guidelines, should ensure overall validity with
increased accuracy. The inclusion of learning curves should ensure validity
for any considered regression models. Furthermore, ML allows seamless analysis
of feature importance thus defining key variables that affect the performance.
Initially, a workflow is defined that combines three-dimensional computational
fluid dynamics (CFD) simulations with the NSGA-II optimization algorithm.
Latin hypercube sampling (LHS) is subsequently used to define a set of design
points for which numerical simulations are conducted. Obtained data are
thereafter used to train and fine-tune a machine learning model. Common
machine learning algorithms such as Random Forests, Gradient Boosting
algorithms and ANNs are evaluated. Based on observations, recommendations on
appropriate ML choices are given. Finally, the trained machine learning model
is coupled with the NSGA-II algorithm, and the resulting Pareto front is
compared to the front obtained using the traditional methodology. Microchannel
heat sinks with secondary channels and ribs are considered in this paper.
Relevant design variables in the optimization process are determined using the
novel SHapley Additive exPlanations (SHAP) [48] technique. Thus obtained
insights are noteworthy even if the ML-based approach is not employed and
should hence facilitate and direct design optimization studies to consider and
optimize the most relevant design parameters. Optimal designs based on thermal
resistance and pumping power are reported. The presented methodology should be
considered as a guideline for future ML implementations when optimizing MCHS.

##  2 Problem definition and methodology

###  2.1 MCHS models

Two distinct MCHS models were considered in this paper. They are based on a
conventional microchannel heat sink defined by Xia et al. [5]. The adoption of
a design was done to ensure comparability. The adopted design was modified to
include slanted passages and ribs. Figure 1 shows the geometric configurations
of the proposed microchannel variants. The model with secondary channels will
henceforth be referred to as SC-MC, while the model with ribs will be referred
to as R-MC.

(a)

(b)

Figure 1: Considered MCHS designs; (a) MCHS with secondary channels (SC-MC),
(b) MCHS with ribs(R-MC).

The overall dimensions of the heat sinks are 10 mm $\times$ 10 mm $\times$ 0.9
mm. The coolant, water, enters the MCHS at the inlet, located centrally in the
header region as shown in Figure 2(c). The location of the outlet is
symmetrical to the inlet. The height of the channels is 0.3 mm. Depending on
the channel width, the hydraulic diameter of the main channels is 0.15 mm
$\leq d_{h}\leq$ 0.3 mm. Parameters depicted in Figure 1 correspond to
optimization variables and will hence be discussed later. All other relevant
geometric characteristics are given in Table 1.

Table 1: Geometric characteristics of the MCHS. | $l$ | $w$ | $h$ | $l_{x}$ | $l_{y}$ | $l_{z}$ | $l_{1}$ | $l_{2}$ | $l_{i}$ | $h_{i}$ | $d_{i}$
---|---|---|---|---|---|---|---|---|---|---|---
Value (mm) | 10 | 10 | 0.9 | 6 | 6 | 0.3 | 2 | 3 | 1 | 0.3 | 1

Sectional views of the three-dimensional models of the MCHS are given in
Figure 2. These figures include all noted geometric characteristics and are
representative of the computational domain.

(a)

(b)

(c)

Figure 2: Sectional views at the $xy$ plane where $z=0$; (a) SC-MC, (b) R-MC.
$xz$ cross section at $y=0$ (c).

###  2.2 Numerical modelling and assumptions

The primary assumption in this study is that the flow inside the MCHS is
laminar and incompressible. A single Newtonian fluid was studied. Radiation
heat transfer, gravity and other body forces were neglected. Viscous
dissipation has been taken into account. The problem is three-dimensional and
all the results were observed at a steady state. The heat sink is made of
silicon. The working fluid is water. Both materials were assumed to have
constant thermophysical properties i.e. values at 298 K for density, specific
heat capacity, thermal conductivity and dynamic viscosity were used. This was
an intentional simplification. Material properties are given in Table 2.

Table 2: Thermophysical properties of materials in this study. |  $\rho$ (kg m-3) |  $c_{p}$ (J kg-1 K-1) |  $k$ (W m-1 K-1) |  $\mu$ (kg m-1 s-1)
---|---|---|---|---
Solid | 2329 | 712 | 148 | -
Fluid | 996.6 | 4178 | 0.609 |  9.027 $\cdot 10^{-4}$

The adopted inflow flow rate ranges from 100 ml$\cdot$min-1 to 200
ml$\cdot$min-1. For the purposes of the optimization procedure, however,
$Q=150$ ml$\cdot$min-1 has been chosen as a reference and if not stated
otherwise, is the flow rate in the reported results. The bottom of the
computational domain (MCHS) has been subjected to a uniform heat flux
$q_{w}=2\cdot 10^{6}$ W m-2. All remaining heat sink walls were considered
adiabatic.

Based on noted assumptions, governing equations, namely the continuity,
momentum and energy equations can be defined:

$\displaystyle\nabla\cdot\bm{u}=0$ (1)
$\displaystyle\rho_{f}(\bm{u}\cdot\nabla)\bm{u}=-\nabla
p+\mu_{f}\nabla^{2}\bm{u}$ (2) $\displaystyle\rho_{f}c_{p,f}(\bm{u}\cdot\nabla
T_{f})=k_{f}\nabla^{2}T_{f}$ (3) $\displaystyle k_{s}\nabla^{2}T_{s}=0$ (4)

where $\bm{u}$ is velocity, $p$ pressure, $T$ temperature, $k_{s}$ thermal
conductivity of the solid, while $\rho_{f}$, $\mu_{f}$, $c_{p,f}$ and $k_{f}$
are density, specific heat capacity, thermal conductivity and dynamic
viscosity of the fluid, respectively.

CFD simulations of conjugated heat transfer have been conducted using Ansys
Fluent 20 [49]. A coupled algorithm was used to solve the momentum and
continuity equations. Second order accurate scheme was employed for the
viscous and second-order upwind scheme for the remaining terms. The maximum
number of iterations was 5000. Convergence was assumed if the residuals for
the energy fall below $10^{-9}$ and below $10^{-4}$ for the remaining
variables. Additionally, an external convergence-control mechanism was used.
The mechanism ensured that the standard derivation of the last 25% of
iterations, for converged cases, was below 3%. A constant volume flow rate,
$Q$, and water temperature $T_{f}=298$ K were set at the inlet:

$\displaystyle u_{in}=\frac{Q}{A_{in}},~{}T_{f,in}=298K.$ (5)

At the outlet, a constant gauge pressure $p=0$ Pa was used:

$\displaystyle p_{out}=p_{atm}.$ (6)

The bottom was assigned a constant heat flux $q_{w}$:

$\displaystyle k_{s}\frac{\partial T_{s}}{\partial z}=-q_{w}.$ (7)

The remaining external walls were considered thermally insulated i.e. $q=0$ W
m-2. The interface between the solid and the fluid was treated as a thermally
coupled wall and a no-slip condition for the fluid was specified:

$\displaystyle-k_{s}\frac{\partial T_{s}}{\partial n}=-k_{f}\frac{\partial
T_{f}}{\partial n}$ (8) $\displaystyle u_{f,w}=0.$ (9)

###  2.3 Automated mesh generation

An automated mesh generation procedure was employed in this paper. Since the
accuracy of numerical results varies greatly depending on the numerical grid,
finalized grids were the result of several iterative processes. Candidate
geometry was created using the FreeCAD [50] Python API for each MCHS case
considered in multi-objective and machine learning workflows. Initial meshing
was done using TetGEN’s [51] Delaunay tetrahedral generator. This deliberate
step was implemented to ensure adequate feature capture. An external control
algorithm ensured that each grid contained fewer than $10^{7}$ cells and that
the mesh generation procedure was completed in less than 300 s. Automated
coarsening and re-meshing functions were used for complex geometries that did
not satisfy noted requirements. The finalized polyhedral mesh was created
using the Ansys Fluent mesher. The default cell size was set to
$l_{g,c}=r_{f}\cdot 5\cdot 10^{-5}$ m, where $r_{f}$ is a mesh scaling factor.
Three layers of prismatic cells with an arbitrary height $l_{g,w}=5\cdot
10^{-6}$ m were created at the fluid-solid interface to ensure a better mesh
transition. Examples of numerical grids for SC-MC and R-MC are shown in Figure
3.

(a)

(b)

Figure 3: Automated grid generation using TetGen and Fluent; (a) mesh for an
SC-MC case, (b) mesh for an R-MC case.

###  2.4 Multi-objective optimization

Thermal resistance and pumping power, two conflicting objective functions,
were used to evaluate the performance of the microchannels studied in this
paper. Thermal resistance, $R_{t}$, defines the MCHS’s heat transfer
capabilities and can be expressed as:

$\displaystyle R_{t}=\frac{T_{max}-T_{f,in}}{q_{w}A_{sub}}$ (10)

where $T_{max}$ is the measured maximal temperature and $A_{sub}$ surface area
of the substrate exposed to the heat flux $q_{w}$. Pumping power, $P_{p}$,
denotes the force required to push the fluid through the microchannels:

$\displaystyle P_{p}=Q\cdot\Delta p$ (11)

where $\Delta p$ is the pressure drop across the microchannel heat sink. It is
evident that, by increasing the flow to dissipate the heat quickly, an
increase in pumping power is inevitable. Complex and elaborate channel designs
that facilitate heat transfer, on the other hand, produce a greater pressure
drop, which has the same effect. The goal of the considered multi-objective
optimization procedure is to minimize the objective functions, $R_{t}$ and
$P_{p}$, at a constant flow rate. In general, a constrained multi-objective
optimization problem can be defined as:

$\displaystyle\underset{\bm{x}}{\text{minimize}}$ $\displaystyle
F(\bm{x})=\bigl{(}R_{t}(\bm{x}),P_{p}(\bm{x})\bigl{)}^{T}$ (12) subject to
$\displaystyle d\leq d_{m}$

where $\bm{x}=(x_{1},...,x_{n})^{T}$ is an $n$-dimensional vector of design
variables in decision space $\mathbb{R}^{n}$ while $F(\bm{x})$ is an
$m$-dimensional objective vector such that $F(\bm{x})\in\mathbb{R}^{m}$.
Conducted multi-objective optimizations were constrained. For both designs,
the constraint $d\leq d_{m}$ was set, where $d$ and $d_{m}$ are geometric
characteristics shown in Figure 1. All solutions that satisfy set constraint
are considered feasible. If they are non-dominated, they are considered Pareto
optimal solutions. Solutions presented in this paper are chosen Pareto optimal
solutions that satisfy one of these conditions:

  * 1.

$R_{t}$ and $P_{p}$ must be improved; relative improvement for $R_{t}$ is
larger (variant A),

  * 2.

$R_{t}$ and $P_{p}$ must be improved; relative improvement for $P_{p}$ is
larger (variant B),

  * 3.

lowest possible $R_{t}$ where $P_{p}$ is, at worst, equal to the original
design (variant C).

For the SC-MC case, seven design variables have been considered in the
optimization procedure. These variables are: the number of microchannels $n$,
the number of secondary channels per fin $m$, microchannel width $d$,
secondary channel width $d_{m}$, secondary channel angle $\alpha$, and two
rotational parameters, $r_{s}$ and $r_{f}$. Noted variables are shown in
Figure 1(a) in Section 2. Unlike other variables, $r_{s}$ and $r_{f}$ are
binary. $r_{s}$ controls the orientation of neighboring secondary channels.
Secondary channels can thus be parallel or converging, resulting in fin
segments that are parallelogram-like or trapezoid-like. The variable $r_{f}$
provides the option to flip the entire secondary channel segment so that two
neighboring sequences are symmetrical about the bounded microchannel’s
centerline. All design variables and bounds are included in Table 3.

Design variables for the R-MC can be seen in Figure 1(b) in Section 2. Six
variables were examined: the number of microchannels $n$, the number of ribs
$m$, microchannel width $d$, base width of the rib $d_{m}$, top width of the
rib $d_{w}$ and rib height $d_{h}$. Ribs are generated alternately on the
opposite sides of the microchannel. An overview of all design variables with
included upper and lower limits is given in Table 3.

Table 3: Design variables and respective bounds. | SC-MC | R-MC
---|---|---
$\bm{x}$ | $n$ | $m$ |  $d$ (mm) |  $d_{m}$ (mm) |  $\alpha$ (∘) | $r_{s}$ | $r_{f}$ | $n$ | $m$ |  $d$ (mm) |  $d_{m}$ (mm) |  $d_{w}$ (mm) |  $d_{h}$ (mm)
LB | 5 | 0 | 0.1 | 0.1 | 10 | 0 | 0 | 5 | 0 | 0.1 | 0.1 | 0.02 | 0.02
UB | 30 | 30 | 0.3 | 0.5 | 90 | 1 | 1 | 30 | 30 | 0.3 | 0.5 | 0.3 | 0.5

The upper bounds of noted design variables were arbitrarily chosen; the number
of microchannels (and accordingly ribs and secondary channels) was to be less
than in the original design. Other variables had sensibly large yet feasible
values for upper bounds. Lower bounds were either set low to ensure constant
hydraulic diameter (widths) or were set low enough so that they should never
affect the optimization process.

In all considered cases, the geometric parameters defined in Table 1 have been
kept constant. This includes MCHS thickness, $h$, as well as microchannel
height, $l_{z}$. This was done deliberately to reduce complexity and
computational effort as well as ensure comparability with Xia et al. [5].
Implemented design choices should not affect the overall workflow/methodology.

Python was used to carry out the optimization procedure. Implementation of the
NSGA-II algorithm from the jMetalPy v1.5.5 [52] framework was employed to
determine the Pareto optimal solutions i.e. Pareto front. The population size
was set to 100. The optimization procedure ran for 200 generations. Simulated
binary crossover probability, $p_{c}$, and polynomial mutation probability,
$p_{m}$, were set to $0.9$ and $0.15$, respectively. All other parameters were
kept at their default values. Optimization workflow is depicted in Figure 4.
For each optimization instance, a candidate i.e. vector of design variables
$\bm{x}$ defined in Table 3, is passed to the geometry generation routine. To
improve the performance, parallel geometry generation and meshing instances
for the fluid and solid parts were defined for each candidate. The resulting
numerical grids were subsequently integrated into a unified model which was
evaluated in an HPC environment according to the numerical methodology defined
in Section 2.2.

Figure 4: Multi-objective optimization workflow based on CFD and NSGA-II
algorithm. The geometry generation and meshing routines are given a candidate
comprised of design variables as an input.

###  2.5 Machine learning surrogates

Utilized machine learning algorithms were trained on data obtained through
numerical simulations. Latin hypercube sampling technique was employed to
generate near-random samples from the decision space $\mathbb{R}^{n}$.
Sampling using LHS ensures that the respective ranges for each variable in an
$n$-dimensional problem are split into $m$ intervals of equal probability.
Each interval is then sampled and the result is combined using random grouping
with other components to form an $n$-dimensional sample (i.e. candidate). In
total, 1500 candidates per design were generated. Candidates were generated
randomly from a decision space bounded by the upper and lower bounds defined
in Table 3. This means that each candidate adheres to the same limits set for
the optimization process. Thus defined candidates were evaluated using CFD.
The outliers were filtered using the interquartile range filter. Filtered
datasets were subsequently used to train multi-output regression models.

Four machine learning algorithms were evaluated in this study: random forest
(RF) implementation in scikit-learn v1.1.1 library [53], gradient boosting
algorithms CatBoost (CB) v1.0.6 [54] and LightGBM (LGBM) v3.3.2 [55], and
TensorFlow v2.8.1 based ANNs [56]. Ensemble algorithms (RF, CB, LGBM) were
chosen as they allow seamless and quick development of respectable ML models.
Noted algorithms rely on a similar concept; baseline models (i.e. decision
trees) are trained and subsequently, a collection of models is combined to
provide a prediction. The fundamental difference between these algorithms is
in the manner they are created and how they tackle the problem of overfitting;
RF utilizes fully grown decision trees (which can overfit) and reduces
variance, whereas boosting algorithms use weak learners (thus minimizing
overfitting), are built additively and aim to reduce the bias. ANNs represent
a traditional approach to machine learning whereby the predictive accuracy and
overfitting are dependent on the network design and its parameters. All models
were trained on an 80/20 split dataset. Input features were independently
standardized by subtracting the mean and scaling to unit variance using the
StandardScaler function. EarlyStopping was employed to prevent overfitting.
For RF, CB and LGBM, rough estimates of the hyperparameters were obtained
using a randomized search that ran for 20 iterations. scikit-optimize’s [57]
Bayesian search was thereafter used to fine-tune the hyperparameters. The
search ran 500 iterations. Robustness during tuning was assessed using a
5-fold cross-validation on the training dataset.

ANN models were tuned using the fireworks algorithm (FWA) in the indago v0.2.4
module [58]. Use of metaheuristic algorithms such as particle swarm
optimization (PSO) for hyperparameter tuning is becoming more prominent [59,
60, 61]. Since FWA shows better overall performance than PSO [62], it was
chosen for this task. Five separate optimization runs per ANN were carried
out, each lasting 200 iterations. Optimization variables were the number of
hidden layers, number of neurons, batch size and learning rate. Additionally,
different activation functions and initializers were considered in the
optimization procedure. Variables are briefly summarized in Table 4. A
stochastic gradient-descent-based optimizer, Adam, was used to minimize the
loss.

Table 4: Optimization variables for the FWA-based ANN tuning. No. hidden layers | $L=\\{x\in\mathbb{N}~{}|~{}1\leq x\leq 4\\}$
---|---
No. neurons per layer | $2^{N},~{}N=\\{x\in\mathbb{N}~{}|~{}0\leq x\leq 8\\}$
Batch size | $2^{B},~{}B=\\{x\in\mathbb{N}~{}|~{}5\leq x\leq 9\\}$
Learning rate | $\\{0.0005,0.00075,0.001,0.0025,0.005\\}$
Activation functions | ELU, ReLU, LeakyReLU
Initializers | GlorotNormal(Uniform), HeNormal(Uniform), TruncatedNormal

The robustness of fully tuned models was investigated by employing a 10-fold
cross-validation. The prediction accuracy was assessed using several metrics,
namely, mean absolute percentage error (MAPE), root mean squared error (RMSE)
and the coefficient of determination (R2). MAPE is a common statistical method
for determining prediction accuracy. It is calculated according to:

$\displaystyle\text{MAPE}=\frac{1}{n}\sum\limits_{i=1}^{n}\left|\frac{y_{i}-\hat{y}_{i}}{y_{i}}\right|\cdot
100$ (13)

where $n$ is the sample size, $\hat{y}_{i}$ the predicted value and ${y_{i}}$
observed value. Similarly, RMSE is often used to measure the predictive error
of regression models:

$\displaystyle\text{RMSE}=\sqrt{\sum\limits_{i=1}^{n}\frac{\bigl{(}y_{i}-\hat{y}_{i}\bigl{)}^{2}}{n}}.$
(14)

R2 indicates the percentage of variance in the dependent variable that can be
explained by the model and is calculated as:

$\displaystyle\text{R}^{2}=1-\frac{\sum\limits_{i=1}^{n}\bigl{(}y_{i}-\hat{y}_{i}\bigl{)}^{2}}{\sum\limits_{i=1}^{n}\bigl{(}y_{i}-\bar{y}_{i}\bigl{)}^{2}}$
(15)

where $\bar{y}_{i}$ is the mean of all values.

##  3 Results and discussion

###  3.1 Grid convergence and validation

Three grids were created for the grid convergence study: coarse, medium and
fine, with $1.8\cdot 10^{5}$, $3.9\cdot 10^{5}$ and $7.9\cdot 10^{5}$ cells,
respectively. Cell sizing was selected according to $l_{g,c}=r_{f}\cdot 5\cdot
10^{-5}$ m. The scaling factor, $r_{f}$, was equal to 1 for fine, 1.4 for
medium and 1.96 for the coarse grid. Assessment is conducted for a
conventional MCHS design i.e. design without secondary channels and ribs. Grid
convergence index (GCI) [63] was calculated for two variables: velocity $u$
measured at the centerline of the MCHS and temperature $T_{max}$ at the MCHS.
Results are presented in Table 5. As is evident, solutions for both variables
and noted grids are in the asymptotic range of convergence. Consequently, the
coarse grid was employed for future calculations.

Table 5: Results of the grid convergence study. | $p_{GCI}$ |  $\varepsilon_{c,m}$ (%) |  $\varepsilon_{m,f}$ (%) |  $GCI_{c,m}$ (%) |  $GCI_{m,f}$ (%) | $\frac{GCI_{m,f}}{r^{p}\cdot GCI_{c,m}}$
---|---|---|---|---|---|---
$u$ | 2.209 | 0.986 | 0.471 | 1.117 | 0.534 | 0.995
$T_{max}$ | 1.548 | 0.050 | 0.030 | 0.091 | 0.054 | 1.000

To ensure the validity of the numerical procedure, the results were compared
to those of a conventional MCHS design presented in a study by Xia et al. [5].
Despite the introduced simplification (fluid properties are not temperature
dependent), the overall agreement is acceptable. The relative error between
average and maximum temperatures for all flow rates is below 2%. The results
for pressure show a rather large discrepancy of 26% at the flow rate range’s
limits. However, as demonstrated in a later experimental study by Xia et al.
[6], experimental results produce pressure drops larger than what is typically
reported in numerical studies. Furthermore, since all subsequent simulations
were performed at $Q=150$ ml$\cdot$min-1, where the temperature disparity is
below 0.5% and the pressure disparity is less than 10%, and since the pressure
trendlines match well, it is reasonable to assume that these comparative
results are adequate for a MO workflow. The observed difference can be
partially attributed to the numerical grids and setup as well as to the
simplification of the physics of the problem. Comparison between CFD results
for different grids and results from the literature are given in Figure 5.

(a)

(b)

Figure 5: Results obtained for different grids compared to the results by Xia
et al. [5]; (a) $T_{max}$, (b) $\Delta P$. The considered model is a
conventional rectangular MCHS with the dimensions 10 mm $\times$ 10 mm
$\times$ 0.9 mm. MCHS design has 30 microchannels with a hydraulic diameter of
0.15 mm. Remaining parameters are given in Table 1. Additional information can
be found in the original paper [5].

###  3.2 CFD-based multi-objective optimization

Multi-objective optimizations for each design were executed 10 times. In
addition to the previously described partitioning of the workflow, concurrent
candidate evaluation was employed to accelerate the optimization process; 25
candidates were evaluated at any time. On average, the complete workflow with
20000 case evaluations took 55 h on an Intel Haswell-based cluster. Obtained
Pareto fronts for the SC-MC and R-MC cases are given in Figure 6. As can be
seen, as the normalized thermal resistance increases, the pumping power
decreases; at the upper limit, an increase in resistance of roughly 50% is
accompanied by a 50% reduction in pumping power. Additionally, at equal or
lower pumping power ($P_{p}/P_{p,ref}\leq 1$) compared to the baseline design,
improvements in thermal resistance are feasible ($R_{t}/R_{t,ref}\leq 1$). As
the thermal resistance decreases, a sharp increase in pumping power occurs;
this is particularly pronounced for the R-MC where pumping power can increase
by more than 4 times. This behavior, however, is expected since complex
channel designs which are beneficial for dissipation also induce larger
pressure drops.

(a)

(b)

Figure 6: Pareto fronts obtained through the CFD-NSGA-II coupling; (a) front
for the SC-MC design, (b) front for the R-MC design.

Optimal designs based on criteria presented in Section 2.4 for SC-MC and R-MC
are given in Table 6. Pareto solution A considers thermal resistance and
pumping power improvements, with a focus on thermal resistance. SC-MC shows
improvements of up to 4.5% and 15.6% for $T_{max}$ and $\Delta p$ at the
design point $Q=150$ ml$\cdot$min-1. The lowest reduction in temperature of
3.1% is noted for $Q=100$ ml$\cdot$min-1, while for pressure, at $Q=200$
ml$\cdot$min-1, observed reduction is roughly 11.7%. The R-MC model at the
design point shows temperature and pressure drop improvements of 6% and 17.9%.
Across the entire flow rate range, R-MC achieves better thermal and pressure
results; temperatures and pressure are comparatively reduced by approximately
20% and 12%, respectively.

Reduction in pumping power is emphasized in Pareto B variants. Obtained
results for the SC-MC show a negligible improvement of 0.5% in temperature at
the design point. At the same time, the pressure is reduced by more than 25%.
Improvements in $T_{max}$ are marginal and below 1% at all flow rates.
Pressure drop is reduced by 21.4% at worst and 31.5% at best for $Q=200$
ml$\cdot$min-1 and $Q=100$ ml$\cdot$min-1, respectively. For the R-MC,
temperature improvement at the design point is 2.6%, with a 23% reduction in
pressure drop. As the flow rate increases, the reduction in $T_{max}$ improves
from 0.6% to 3.7%. Pressure drops are lower for all flow rates when compared
to SC-MC. At $Q=100$ ml$\cdot$min-1 pressure is reduced by 28.6%; this value
drops and reaches 19.1% for $Q=200$ ml$\cdot$min-1.

Pareto optimal solutions denoted as C variants provide the greatest reductions
in thermal resistance while maintaining comparable pumping power to the
baseline design. At the design point, $T_{max}$ is reduced by 9.2% for SC-MC
and by 8.1% for R-MC. Similarly, $\Delta p$ is reduced by 2.5% and 7.1% for
SC-MC and R-MC, respectively. Pressure drop is well below the baseline value
for all cases (up to 10% when $Q=100$ ml$\cdot$min-1). As the flow rate
approaches $Q=200$ ml$\cdot$min-1, the pressure drop is around 1% and 4.2% for
SC-MC and R-MC.

Table 6: Model parameters for chosen optimal designs. | SC-MC | R-MC
---|---|---
$\bm{x}$ | $n$ | $m$ |  $d$ (mm) |  $d_{m}$ (mm) |  $\alpha$ (∘) | $r_{s}$ | $r_{f}$ | $n$ | $m$ |  $d$ (mm) |  $d_{m}$ (mm) |  $d_{w}$ (mm) |  $d_{h}$ (mm)
A | 24 | 4 | 0.14 | 0.33 | 18 | 1 | 0 | 22 | 20 | 0.17 | 0.17 | 0.07 | 0.05
B | 21 | 4 | 0.18 | 0.24 | 23 | 1 | 0 | 22 | 11 | 0.17 | 0.18 | 0.06 | 0.07
C | 26 | 6 | 0.13 | 0.13 | 39 | 1 | 0 | 26 | 15 | 0.13 | 0.13 | 0.03 | 0.13

A includes three-dimensional models of all optimal solutions presented in
Table 6. Additionally, velocity, pressure and temperature fields at the
midplane ($z=0$) for SC-MC and R-MC designs that achieve the largest
temperature reduction (C variant) are given in Figure 7. Pressure and velocity
fields are mostly symmetrical about the $yz$ plane due to the positioning of
the inlet and outlet. The bulk of the flow takes place near the centerline and
is accompanied by a substantial pressure drop. The secondary channels of the
SC-MC only account for a small portion of the flow, but, like ribs, they
introduce enough disturbance and increase the overall surface area to
facilitate heat dissipation.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: Results for the SC-MC and R-MC C Pareto optimal variants at the
midplane ($z=0$); (a) SC-MC velocity field, (b) SC-MC pressure field, (c)
temperature distribution for the SC-MC, (d) R-MC velocity field, (e) R-MC
pressure field, (f) temperature distribution for the R-MC.

###  3.3 Multi-objective optimization using ML surrogate

Input datasets for surrogate modeling were made up of numerical results for
distinct candidates defined through Latin hypercube sampling. Candidates that
violated the MO constraints noted in Section 2.4 were excluded from datasets
as they could hinder the training procedure. Additionally, since these were
simple and known (user-defined) relations that would be used in the subsequent
MO procedure, they did not need to be considered in the ML model. Following
this, interquartile filter and feature scaling were employed. The tuning
procedure for all considered ML models has been described in Section 2.5.
Metrics for ML models trained using the SC-MC LHS data are given in Table 7.

Table 7: Metrics and standard deviations for different ML models trained on the SC-MC dataset. Noted values represent the worst results from both output features ($R_{t}$ and $P_{p}$). Values in bold are the most accurate results for a given metric. | RF | CB | LGBM | ANN
---|---|---|---|---
MAPE |  7.762 $\pm$ 0.679 |  4.568 $\pm$ 0.330 |  6.850 $\pm$ 0.649 | 1.409 $\pm$ 0.123
RMSE |  0.023 $\pm$ 0.004 |  0.016 $\pm$ 0.003 |  0.020 $\pm$ 0.003 | 0.007 $\pm$ 0.001
R2 |  0.911 $\pm$ 0.006 |  0.956 $\pm$ 0.004 |  0.933 $\pm$ 0.005 | 0.983 $\pm$ 0.001

Based on the presented data, it is evident that the tuned ANN model achieves
the overall best results. In terms of MAPE, the error is below 1.5%. Given
that the values of the output features are larger than 0.5, an RMSE of 0.007
is respectable. A high R2 score indicates that the trained ANN model
sufficiently explains the dependent variables. Low standard deviations are
indicative of the robustness of the model. Metrics for models trained on the
R-MC LHS dataset are given in Table 8.

Table 8: Metrics and standard deviations for different ML models trained on the R-MC dataset. Noted values represent the worst results from both output features ($R_{t}$ and $P_{p}$). Values in bold are the most accurate results for a given metric. | RF | CB | LGBM | ANN
---|---|---|---|---
MAPE |  8.000 $\pm$ 0.633 |  5.503 $\pm$ 0.466 |  7.217 $\pm$ 0.644 | 1.925 $\pm$ 0.248
RMSE |  0.031 $\pm$ 0.005 |  0.024 $\pm$ 0.006 |  0.027 $\pm$ 0.005 | 0.012 $\pm$ 0.006
R2 |  0.939 $\pm$ 0.015 |  0.964 $\pm$ 0.010 |  0.952 $\pm$ 0.012 | 0.980 $\pm$ 0.004

Metrics for the R-MC ML models are comparatively slightly worse than for SC-MC
models. Still, ANN achieves the best results with a high R2 score. The
presented data is indicative of the suitability of considered ML models as
design surrogates. ANN models were hence used in the multi-objective
optimization procedure. Relevant hyperparameters for tuned ANNs are given in
Table 9. Graphical depictions of the neural network designs are included in A.

Table 9: Tuned parameters for ANN models. All other parameters are kept at default values. | SC-MC | R-MC
---|---|---
No. hidden layers | 4 | 3
No. neurons per layer | 256,128,8,4 | 256,128,64
Batch size | 64 | 64
Learning rate | 0.001 | 0.00075
Activation functions | ELU | ReLU
Initializers | TruncatedNormal | TruncatedNormal

Learning curves for both ANN models have been calculated and included in A
(Figure 7) to further validate the models and determine the influence of the
dataset size on the model accuracy. It is evident that for $N>1000$, obtained
R2 score for 5-fold validation is larger than 95%. This implies that the
sample size of 1500 is sufficient for the next stage of the workflow. It is
important to note that for $N<500$, accuracy for both models drops
significantly hence it is sensible to conclude that smaller datasets would be
inadequate to properly model the MCHS.

A simplified overview of the proposed machine-learning-based MCHS optimization
workflow is given in Figure 8. Numerical results for LHS-defined candidates
are filtered, scaled, and used to train an ANN model. The model is fine-tuned
using the FWA algorithm and subsequently integrated into an NSGA-II-based
optimization procedure. Pareto optimal solutions of the multi-objective
optimization are considered when proposing the best designs, per methodology
defined in Section 2.4.

Figure 8: Machine learning-based workflow.

The Pareto fronts obtained using the aforementioned workflow are shown in
Figure 9 and exhibit similar trends to those obtained using the conventional
optimization approach. The asymptotic behavior of normalized thermal
resistance is in accordance with the previous results. Fronts, however, fail
to capture the magnitude of the increase in pumping power following the
decrease in thermal resistance. As the LHS sampling provides a reduced dataset
which may lack information on correlation under these circumstances, the
observed behavior is reasonable. Still, this disparity can be partially
attributed to the stochastic nature of the MO algorithm.

(a)

(b)

Figure 9: Pareto fronts obtained using the ML-based workflow; (a) front for
the SC-MC design, (b) front for the R-MC design.

Proposed designs based on the ANN-NSGA-II approach, that meet the criteria
outlined in Section 2.4, are given in Table 10. Respective three-dimensional
models are included in the A. Optimal designs for both microchannel models
show similar performance improvements. At the design point, variants A achieve
a $T_{max}$ reduction of 4.6% for SC-MC and 4.7% for R-MC. The pressure drop
is reduced by 18.6% and 18%, respectively. The Pareto B variants provide a
greater reduction in pumping power at the expense of thermal resistance.
Consequently, the pressure drop is reduced by 23.2% and 24%, while the maximum
temperature is lowered by 1.8% and 1.2% for SC-MC and R-MC designs.
Temperature reduction for C variants is around 9.3% and 8.7% for considered
MCHS models, with pressure drop reduced by approximately 4.7%.

Table 10: Model parameters for chosen optimal designs generated through the ANN-NSGA-II coupling. | SC-MC | R-MC
---|---|---
$\bm{x}$ | $n$ | $m$ |  $d$ (mm) |  $d_{m}$ (mm) |  $\alpha$ (∘) | $r_{s}$ | $r_{f}$ | $n$ | $m$ |  $d$ (mm) |  $d_{m}$ (mm) |  $d_{w}$ (mm) |  $d_{h}$ (mm)
A | 22 | 8 | 0.17 | 0.17 | 31 | 1 | 0 | 22 | 16 | 0.17 | 0.17 | 0.04 | 0.12
B | 22 | 6 | 0.17 | 0.27 | 20 | 1 | 0 | 21 | 13 | 0.18 | 0.18 | 0.04 | 0.14
C | 26 | 6 | 0.13 | 0.13 | 37 | 1 | 0 | 26 | 14 | 0.13 | 0.13 | 0.03 | 0.09

###  3.4 Feature importance

SHAP is a novel technique that expands on cooperative game theory’s Shapely
values in order to interpret ML models and determine which input features have
the most influence on the output features [48]. The fundamental idea behind
the technique is to distribute the credit for an output feature among the
inputs. Based on this and Shapely values properties, it is possible to
calculate the relevance of each input feature.

Since LHS was used to generate the data for training, the inherent
applicability is not immediately apparent. However, as it was demonstrated
that proposed ANNs accurately model the relationship between the inputs and
outputs, SHAP methodology can be employed to determine features i.e. design
variables with the most influence on thermal resistance and pumping power.
These variables are thus the most important when optimizing MCHS design.
Employed KernelExplainer method uses a weighted linear regression to ascertain
the feature importance [48]. Figure 10 shows SHAP values for each of the model
outputs (for both SC-MC and R-MC). Feature importance is ranked in a
descending manner. Graphs can be interpreted as follows: as the feature value
for a given feature (variable, $y$-axis) increases (red) or decreases (blue),
it can have a positive or a negative influence (SHAP value, $x$-axis) on the
output. Microchannel width $d$ and the number of secondary channels/number of
ribs $m$ have the biggest influence on thermal resistance for SC-MC and R-MC.
Pumping power is predominantly determined by microchannel width $d$ and the
number of microchannels $n$ i.e. with their increase $\Delta p$ is reduced. On
the other hand, an increase in $m$ leads to an increase in pumping power,
which is expected. These conclusions are valid for the conventional
optimization approach too.

(a)

(b)

(c)

(d)

Figure 10: SHAP feature impact on ML model outputs; (a) feature impact on
thermal resistance for the SC-MC, (b) feature impact on pumping power for the
SC-MC, (c) feature impact on thermal resistance for the R-MC, (d) feature
impact on pumping power for the R-MC.

The limitations, however, of the SHAP methodology stem from the machine
learning approach itself, in which assessed surrogates rely on data that may
be limited or not representative, making it impossible to fully understand or
interpret the physics of the problem. Recently, however, there have been
strides to incorporate well-known physics into machine learning models [64] as
well as to use machine learning to discover the underlying physics [65].

###  3.5 Comparative analysis

Pareto optimal solutions obtained at the design point and presented in
previous chapters will thus be assessed for a range of flow rates and compared
to respective counterparts, i.e. best solutions obtained using the CFD-NSGA-II
approach will be compared to solutions obtained using the proposed ML-based
workflow. Presented graphs include CFD-verified curves for generated
solutions.

Proposed A variants for the SC-MC design show negligible variance with respect
to the temperature. At best, results obtained using the ML approach are 0.3%
lower. Pressure drop is slightly larger for the CFD-NSGA-II solution, up to
4.9% at $Q=100$ ml$\cdot$min-1. Noted results are shown in Figure 11. When
compared to the initial design, the CFD-NSGA-II variant shows improvements of,
on average, 1.6∘C and 7.3 kPa, for $T_{max}$ and $\Delta p$.

(a)

(b)

Figure 11: Comparison between proposed SC-MC A variants for different flow
rates; (a) temperature, (b) pressure drop.

ANN-NSGA-II solutions are more convincing when comparing B variants for the
SC-MC model. Compared to the initial design, $T_{max}$ and $\Delta p$ are
lowered by 0.7∘C and 11 kPa, respectively. This is to be expected, given that
the emphasis in B variants is on pressure drop reduction. Temperatures between
variants differ by up to 1.4% and for $Q=100$ ml$\cdot$min-1, the temperature
limit of the initial design is respected. In terms of pressure, the CFD-NSGA-
II solution shows up to 3.1% lower drops. Comparison is shown in Figure 12.

(a)

(b)

Figure 12: Comparison between proposed SC-MC B variants for different flow
rates; (a) temperature, (b) pressure drop.

Both MO optimization approaches produce similar results when determining the
lowest attainable temperatures for the SC-MC model. The temperature difference
is insignificant. However, CFD-NSGA-II solutions achieve this temperature
equivalence due to an increase in pressure drop of up to 2.6%. Still, the
pressure is approximately 0.9 kPa lower than that of the initial design, while
the temperatures are improved, on average, by 4.4∘C. These observations can be
seen in Figure 13.

(a)

(b)

Figure 13: Comparison between proposed SC-MC C variants for different flow
rates; (a) temperature, (b) pressure drop.

The temperature difference between presented A variants of the R-MC model is
less than 1.5%. The CFD-NSGA-II solution achieves lower temperatures but with
a 1% increase in pressure drop. A slight improvement in thermal performance is
thus obtained at the expense of an increase in pressure drop. When compared to
the initial design, temperatures and pressure are decreased by, on average,
2.7∘C and 8.3 kPa. These results are given in Figure 14.

(a)

(b)

Figure 14: Comparison between proposed R-MC A variants for different flow
rates; (a) temperature, (b) pressure drop.

The deviations for the B solutions are equally small. Temperature disparity is
below 1.5%. The difference in pressure drop is slightly increased, now up to
1.6%, but the overall conclusion remains the same; the CFD-NSGA-II solution
provides lower temperatures at the expense of increased pressure. $T_{max}$
and $\Delta p$ for the ANN-NSGA-II solution are on average 0.4∘C and 11.4 kPa
lower than the initial design. Figure 15 depicts this observation.

(a)

(b)

Figure 15: Comparison between proposed R-MC B variants for different flow
rates; (a) temperature, (b) pressure drop.

The results shown in Figure 16 indicate that the proposed ANN-NSGA-II variant
C solution provides the largest temperature decrease. The difference between
considered variants is less than 0.7%, however. The pressure drop is larger
for the noted solution, yet it is still well below the initial design
guideline. This means that if the pumping power is limited i.e.
$P_{p}/P_{p,ref}\approx 1$, the ML-based approach is more flexible. When
compared to the initial design, temperature and pressure drop are on average
improved by 4.2∘C and 2 kPa, respectively.

(a)

(b)

Figure 16: Comparison between proposed R-MC C variants for different flow
rates; (a) temperature, (b) pressure drop.

Overall, the presented optimal solutions, while different, provide comparable
thermal performance. The difference is, at worst, 1.5%, which is negligible.
Observed pressure differences are slightly larger, up to 4.9%, but still
acceptable. Based on this, it is reasonable to conclude that both approaches
have merit and are applicable. Differences in obtained and proposed optimal
solutions can be attributed to stochasticity, ML model inadequacies (small
dataset) and general CFD-induced inaccuracy (numerical errors).

When compared to the original, conventional MCHS design, the proposed
solutions are clear improvements. Depending on the interpretation of the
results on the Pareto front, different balanced optimums can be found;
however, if we focus on the temperature and limit the pressure, it has been
shown that the $T_{max}$ can be reduced by more than 4∘C.

It should be noted that proposed solutions stem from a dataset and
optimization procedure conducted for a flow rate $Q=150$ ml$\cdot$min-1. Since
MCHS are sensitive to flow rate, the behavior of the trendlines and overall
results might have been different if the optimization process had been
conducted at a different flow rate.

##  4 Conclusion

In this paper, multi-objective optimization techniques used to improve the
performance of microchannel designs were investigated. The study focused on
two different and widely used MCHS design concepts, namely microchannel heat
sinks with secondary channels and ribs. A numerical procedure was implemented
to evaluate MCHS performance. Validated numerical models were coupled with an
NSGA-II-based multi-objective optimization workflow. In such a defined
workflow, conflicting goal functions, thermal resistance and pumping power,
were used to determine a series of optimal design solutions.

A computationally efficient and effective methodology was hence proposed to
generate optimal designs. It uses Latin Hypercube Sampling and CFD to generate
a dataset of unique design solutions. Thus obtained data are used to train ANN
models which can subsequently be used for a variety of purposes, but primarily
in conjunction with an optimization method to provide optimal designs. The use
of ML-based surrogates has several advantages over traditional approaches. One
important aspect is that the validity of the obtained surrogate model can be
easily determined by analyzing the learning curve. Additionally, ML-based
surrogates can be seamlessly assessed to determine key aspects/parameters that
govern the physics of the problem, even if they do not fully explain the
physics of the problem. Key findings are as follows:

  * 1.

Ensemble learning and gradient boosting methods provide respectable results
without tuning. However, tuned ANNs were shown to outperform tuned
alternatives. The proposed ML-based methodology can generate acceptable
solutions while requiring only about 20% of the computational time. Solutions
obtained using CFD-based and ML-based methodologies have different topological
designs but similar thermal performance.

  * 2.

Optimal ML-generated solutions provide lower maximum temperatures (by more
than 4∘C) under the same pressure drop limit. A significant reduction in
pressure drop (more than 25%) can be achieved without an increase in
temperature. The SC-MC MCHS design outperforms the R-MC design. The
differences are, however, below 0.5∘C.

  * 3.

Conducted SHAP analysis indicated that the channel width $d$ and the number of
secondary channels/ribs $m$ are the main variables governing the thermal
resistance. Pumping power is mainly driven by the width $d$ and the number of
microchannels $n$. The relevance of every design variable in the optimization
process is presented.

As demonstrated, the proposed methodology provides acceptable solutions and
allows for quick turnaround times in the design process. It combines
previously established principles and methods with novel machine learning
concepts, which should improve the overall understanding of the design
variable influence and thus facilitate the optimization and design process, in
addition to improving performance. The applicability of various ML algorithms
is also evaluated, emphasizing the importance of proper assessment in the
context of a given problem. Applicability should be further investigated by
assessing different designs, combining varying models into a unified model,
and investigating the influence of the dataset size on predictive accuracy.

## Appendix A

(a)

(b)

(c)

Figure 1: Cross-sectional view of the optimal SC-MC designs obtained using
coupled CFD-NSGA-II approach; (a) variant A, (b) variant B, (c) variant C.
Relevant parameters for each design are included in Table 6. Performance
specifics of each design are given in Section 3.2. Position on the Pareto
front for each variant is given in Figure 6(a).

(a)

(b)

(c)

Figure 2: Cross-sectional view of the optimal R-MC designs obtained using
coupled CFD-NSGA-II approach; (a) variant A, (b) variant B, (c) variant C.
Relevant parameters for each design are included in Table 6. Performance
specifics of each design are given in Section 3.2. Position on the Pareto
front for each variant is given in Figure 6(b).

(a)

(b)

(c)

Figure 3: Cross-sectional view of the optimal SC-MC designs obtained using
ANN-NSGA-II coupling; (a) variant A, (b) variant B, (c) variant C. Relevant
parameters for each design are included in Table 10. Performance specifics of
each design are given in Section 3.3. Position on the Pareto front for each
variant is given in Figure 9(a).

(a)

(b)

(c)

Figure 4: Cross-sectional view of the optimal R-MC designs obtained using ANN-
NSGA-II coupling; (a) variant A, (b) variant B, (c) variant C. Relevant
parameters for each design are included in Table 10. Performance specifics of
each design are given in Section 3.3. Position on the Pareto front for each
variant is given in Figure 9(b). Figure 5: Topology of the tuned ANN employed
in the SC-MC optimization workflow. An input layer, four hidden layers, and an
output layer comprise the network. The number of neurons per layer is shown.
In total, 7 features are considered. The model outputs thermal resistance
$R_{t}$ and pumping power $P_{p}$. Figure 6: Topology of the tuned ANN
employed in the R-MC optimization workflow. An input layer, three hidden
layers, and an output layer comprise the network. The number of neurons per
layer is shown. In total, 6 features are considered. The model outputs thermal
resistance $R_{t}$ and pumping power $P_{p}$.

(a)

(b)

Figure 7: Learning curves obtained using the tuned ANN models; (a) for the SC-
MC design, (b) for the R-MC design. Learning curves depict the change in
$R_{2}$ score as the size of the dataset used to train the model increases. It
is evident that for $N>1000$, both considered models are sufficiently
accurate.

(a)

(b)

Figure 8: Comparison of obtained Pareto fronts; (a) fronts for the SC-MC
design, (b) fronts for the R-MC design. Each figure includes fronts obtained
using CFD-based and ANN-based approaches as well as optimal variants for each
case. The initial design is depicted with a black dot.

## Appendix B

Supplementary data on Pareto fronts discussed in this paper can be found
online at https://gitlab.com/sikirica_a/mchs_data.

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