[]PankajKumar []HeikeVogel []JuliaBruckert []LisaJanina Muth []Gholam
AliHoshyaripour

]Institute of Meteorology and Climate Research, Karlsruhe Institute of
Technology, Karlsruhe, Germany

Pankaj Kumar<EMAIL_ADDRESS>
# MieAI: A neural network for calculating optical properties of internally
mixed aerosol in atmospheric models

###### Abstract

Aerosols influence weather and climate by interacting with radiation through
absorption and scattering. These effects heavily rely on the optical
properties of aerosols, which are mainly governed by attributes such as
morphology, size distribution, and chemical composition. These attributes
undergo continuous changes due to chemical reactions and aerosol micro-
physics, resulting in significant spatio-temporal variations. Most atmospheric
models struggle to incorporate this variability because they use pre-
calculated tables to handle aerosol optics. This offline approach often leads
to substantial errors in estimating the radiative impacts of aerosols
alongwith posing significant computational burdens. To address this challenge,
we introduce a novel and computationally efficient machine learning approach
called MieAI. It allows for online calculation of the optical properties of
internally mixed aerosols with a log-normal size distribution. Importantly,
MieAI fully incorporates the variability in aerosol chemistry and
microphysics. Our evaluation of MieAI against traditional Mie calculations,
using number concentrations from ICOsahedral Nonhydrostatic model with Aerosol
and Reactive Trace gases (ICON-ART) simulations, demonstrates that MieAI
exhibits excellent predictive accuracy for aerosol optical properties. MieAI
achieves this with errors well within 10%, and it operates more than 1000
times faster than the benchmark approach of Mie calculations. Due to its
generalized nature, the MieAI approach can be implemented in any chemistry
transport model which represents aerosol size distribution in the form of log-
normally distributed internally mixed modes. This advancement has the
potential to replace frequently employed lookup tables and play a substantial
role in the ongoing attempts to reduce uncertainties in estimating aerosol
radiative forcing.

##  1 Introduction

Aerosol particles have a significant impact on Earth’s radiation balance due
to their interactions with solar radiation and clouds. Particles’ ability to
scatter and absorb radiation, known as the aerosol direct effect, is
influenced by their mixing state – how different aerosol types are distributed
within the population (Riemer et al., 2019; Jacobson, 2001). This mixing state
can range from external mixing (single species) to internal mixing (mixture of
species). Newly emitted aerosols usually have external mixing, while aging
processes lead to internal mixing. Aerosol particles consist of diverse
organic and inorganic components, showing significant variability in
composition and abundance across time and space. Previous studies emphasize
the importance of mixing state in understanding aerosols’ optical properties
(AOPs) (Yao et al., 2022; Koike et al., 2014; Wang et al., 2021; Riemer et
al., 2019). For example, studies demonstrate a greater positive forcing for
internally mixed black carbon aerosols under the assumption of core-shell
mixing in contrast to homogeneous volume-mixing and external mixing scenarios
(Jacobson, 2001; Yu et al., 2006; Bond et al., 2006b; Zhang et al., 2008;
Moffet and Prather, 2009; Lack and Cappa, 2010).

Accurately modeling aerosol populations and predicting their impact on air
quality, weather, and climate has long been a major challenge. Despite a good
understanding of the underlying physics, resolving many small-scale processes,
especially within atmospheric models, remains difficult. Precise
quantification of AOPs, including mass extinction coefficient ($k_{e}$),
single-scattering albedo ($\omega$), and asymmetry factor ($g$) are crucial
for improving the forecasting capabilities of the atmospheric composition
models (ACMs).

Accurate representation of AOPs of internally mixed particles remains a
significant challenge in ACMs (Riemer et al., 2019; Brown et al., 2021).
Currently, many ACMs use large database of pre-calculated AOPs, in the form of
look-up tables (Dubovik et al., 2006; Ghan and Zaveri, 2007; Meng et al.,
2010; Gasteiger and Wiegner, 2018; Stromatas et al., 2012; Gasch et al., 2017;
Ghosh et al., 2021; Geiss et al., 2023; Wang et al., 2022). These AOPs are
often archived using Mie calculations for a discrete set of chemical and
micro-physical attributes (such as particle size and refractive index) (Brown
et al., 2021; Tuccella et al., 2020). But aerosol size and composition have
large spatial and temporal variability in model simulations. An interpolation
is inevitable whenever AOPs in the model are queried for a set of input
parameters different from the archived values. The interpolation may lead to
non-trivial errors due to non-linearity. Such errors can be reduced by adding
more parameters to the interpolation. For example, Gasch et al. (2017)
implemented polynomial fits to account for the variability of median diameter
of log-normal modes during atmospheric transport (due to faster sedimentation
of large particles). The database, however, grows larger as the number of
parameters increases, making the offline AOPs less convenient for use with
diverse applications (Yu et al., 2022; Geiss et al., 2023).

Due to the significant computational burden, online calculation of AOPs using
Mie code is only feasible for specific applications and impractical for real-
time use in ACMs (Wang et al., 2022). To tackle this issue, several attempts
have been made for the online calculation of AOPs that often parameterize the
Mie calculations for variable aerosol size and composition (Ghan et al., 2001;
Ghan and Zaveri, 2007; Fast et al., 2006; Klingmüller et al., 2014; Curci et
al., 2015). Yet, these methods are also subjected to large uncertainties and
errors stemming from underlying assumptions and interpolations. This
highlights the immediate demand for accurate and computationally efficient
tools for online calculaton of AOPs consistent with the aerosol chemical and
microphysical characteristics in ACMs (Stier et al., 2007; Geiss et al.,
2023).

In recent years, the application of Machine Learning (ML) and, more
specifically, Deep Learning (DL), has garnered significant prominence within
the domain of weather and climate research. This prominence is reflected in
its diverse applications, spanning across various aspects including weather
prediction (Bi et al., 2023; Zhang et al., 2023), the refinement of numerical
model outputs through post-processing (Sayeed et al., 2022), and even the
substitution of pivotal model physics (Mishra and Molinaro, 2021) and
parameterizations (Yuval et al., 2021; Rasp et al., 2018). The methodologies
employed encompass a variety of techniques, spanning from emulation (Sharma et
al., 2023) to the resolution of partial differential equations (PDEs) via
widely adopted ML algorithms (Huang and Seinfeld, 2022; Goswami et al., 2023).
ML has undergone significant advancements in recent years, particularly after
2010, as a result of the development of effective techniques for training a
neural network (NN) of considerable size. NNs excel in learning knowledge
representation in very high-dimensional spaces; in forms of connecting weights
in between neurons of the networks. The organisation of the networks or the
network architecture is thus a mapping of the knowledge space of various
domains. As demonstrated in recent studies, it is feasible to predict the
optical properties of aerosol particles by means of a NN, rather than solving
Maxwell’s equation as in Mie calculations (Lamb and Gentine, 2023; Geiss et
al., 2023; Yu et al., 2022).

Therefore, the objective of this study is to develop a NN based emulator to
replace the current aerosol optics parameterization for internally mixed
aerosols used in ACMs such as ICON-ART. We present a multi-layer fully
connected feed-forward NN to derive optical properties for spherical particles
covering a large size range accurately and efficiently; thereby meeting the
emergent requirements in both remote sensing and atmospheric modeling of
aerosol particles. This study builds upon prior endeavors that employed ML
techniques for emulating aerosol optics and radiative transfer modeling
(Belochitski and Krasnopolsky, 2021; Mishra and Molinaro, 2021; Ukkonen, 2022;
Pal et al., 2019; Lamb and Gentine, 2023; Geiss et al., 2023; Yu et al.,
2022). The overarching objective here is to devise an approach capable of
robust generalization.

##  2 Materials and Methods

###  2.1 Mie Calculation of Aerosol Optical Properties

Optical properties are a function of the particle size and the wavelength-
dependent refractive indices (RIs) of the constituents of the aerosol
particles (Gordon et al., 2017). Both relative RI of the particle with respect
to surrounding medium and particle shape should be accounted for in radiation
interaction studies (Lamb and Gentine, 2023). If the particle shape is
spherical, Mie theory can be used to calculate the optical properties. The Mie
theory uses Maxwell’s equations to solve a 3-D electromagnetic wave equation
whose solution can be written as an infinite series of products of orthogonal
functions (Bohren and Huffman, 2008). As per Mie theory, the extinction
($Q_{ext}$) / scattering ($Q_{sca}$) efficiencies and g of a spherical
particle can be written as:

$Q_{ext}=\frac{2}{x^{2}}\sum_{n=1}^{\infty}(2n+1)\mathbb{R}(a_{n}+b_{n})$ (1)

$Q_{sca}=\frac{2}{x^{2}}\sum_{n=1}^{\infty}(2n+1)(\lvert
a\rvert_{n}^{2}+\lvert b\rvert_{n}^{2})$ (2)

$g=\frac{4}{Q_{scat}x^{2}}\Bigg{[}\sum_{n=1}^{\infty}\frac{n(n+2)}{n+1}\mathbb{R}(a_{n}a_{n+1}^{*}+b_{n}b_{n+1}^{*})\\\
+\sum_{n=1}^{\infty}\frac{2n+1}{n(n+1)}\mathbb{R}(a_{n}b_{n}^{*})\Bigg{]}$ (3)

Here, $a_{n}$ and $b_{n}$ are the Mie scattering coefficients and $x$ is the
size parameter which, in turn, is given by:

$x=\frac{\pi d_{p}}{\lambda}$

where $d_{p}$ is the particle diameter.

An approximate solution for $Q_{ext}$, $Q_{sca}$ and $g$ can be obtained by
truncating the infinite series as explained by Bohren and Huffman (2008). Mie
codes calculate the Mie scattering coefficients ($a_{n}$ and $b_{n}$), which
are solely dependent on particle diameter ($d_{p}$), incident wavelength
($\lambda$) and RI ($B_{\lambda}$), followed by determination of the number of
terms required before truncation and calculation of the series. Mass
extinction coefficient ($k_{e}$) is obtained from $Q_{ext}$ (Muser, 2022):

$k_{e}(l,\lambda,B_{\lambda})=\frac{\int_{0}^{\infty}\frac{\pi}{4}d_{p}^{2}Q_{ext}(d_{p},\lambda,B_{\lambda})\psi_{0,l}(d_{p})dd_{p}}{\int_{0}^{\infty}\rho_{p}[\frac{\pi}{6}d_{p}^{3}]\psi_{0,l}(d_{p})dd_{p}}$
(4)

where $\psi_{0,l}$ and $\psi_{3,l}$ are the parameters of the log-normal
distribution for aerosol mode $l$.

Figure 1: Coated internally mixed aerosol particle. It is assumed to be
composed of a core that is insoluble and a shell that is soluble. The core
consists of black carbon, volcanic ash, sea salt and dust whereas the shell
consists of organic, inorganic matter (such as ammonia (NH4), nitrate (NO3),
chlorine (Cl), sulphate (SO4) and sodium (Na) and water (H2O)). Here, $D_{c}$
represensts the diameter of the core and $D_{t}$ is the total diameter of
coated, mixed aerosol particle that consists of both core and shell.
Refractive indices (RI) for all chemical species constituting the mixed
aerosol particle except dust are obtained from Gordon et al. (2022) whereas
those for dust are obtained from Di Biagio et al. (2019).

To calculate the optical properties of the internally mixed aerosol particle
using Mie calculations, some assumption are required. The Mie theory assumes
that the particles have spherical shapes. In reality, the majority of aerosol
particles are non-spherical. However, the liquid coating can lead to spherical
particle surfaces, which justifies the assumption of the particle sphericity
in the mixed mode. In this study, aerosol particles were assumed to be
spherical in a core-shell configuration, with solid phase as the core and
liquid species as the shell. Both core and shell are considered as ternary
systems of different chemical species. For example, core is the ternary system
consisting of dust, sea salt and soot whereas the shell is constituted by
water, inorganic and organic species as shown in Fig. 1. This assumption does
not imply the one-to-one existence of such mixtures in nature. Rather, it
covers a wide range of the possible RIs for core and shell accruing in the
atmosphere which is shown in Fig. 8. We employ the PyMieScatt Python library
for computing the optical characteristics of a coated sphere using Mie theory
(Sumlin et al., 2018). This library is built on Mie codes originally written
by Mätzler (2002) and Bond et al. (2006a), rooted in the concepts presented by
Bohren and Huffman (2008).

Figure 2: MieAI Architecture. MieAI is a NN based model with multiple hidden
layers. The first and last layers represent input and output of MieAI
respectively. Here, size parameter (x), wavelength ($\lambda$), coating
fraction (f), real and imaginary parts of refractive indices for both core
($RI^{c}_{re}$ and $RI^{c}_{im}$) and shell ($RI^{s}_{re}$ and $RI^{s}_{im}$)
constitute the input of MieAI whereas the extinction (Ext), scattering (Sca)
Efficiency and asymmetry parameter (Asy) are the output.

###  2.2 Emulation of the Mie Calculation: MieAI

In this study, we propose a multi-layer fully connected NN popularly known as
multi-layer perceptron (MLP) to emulate the calculation of AOPs using Mie
calculation i.e. MieAI. As a universal function approximator, the feed-forward
NN is ideally suited for modeling nonlinear processes. The schematic diagram
of the MLP is shown in Fig. 2. Specifically, it is used here to establish the
relationships between the micro-physical parameters of aerosol particles and
corresponding single-scattering properties (Chen et al., 2022; Wang et al.,
2022; Yu et al., 2022; Geiss et al., 2023). Its feature is the interconnection
of neurons with all nodes in the front and rear hidden layers. The output
$O^{(l)}_{i}$ of the $i$-th node in the fully connected layer $l$ can be
calculated from the output of the previous layer $l-1$ with a non-linear
activation function ($\phi$).

$O^{(l)}_{i}=\phi\Bigg{(}\sum^{N^{(l-1)}}_{j=1}w^{(l)}_{i,j}O_{j}^{(l-1)}+b^{(l)}_{i}\Bigg{)}$
(5)

Here, $w^{(l)}_{i,j}$ represents the weight of the $j$-th neuron in the layer
$l-1$ to the $i$-th neuron in the layer $l$ and $b^{(l)}_{i}$ represents the
bias term of the $i$-th neuron in the layer $l$. $N^{(l-1)}$ is the number of
neurons in layer $l-1$.

For estimating AOPs using MieAI, 7 aerosol micro-physical parameters are
regarded as input features (X = [ $x_{1}$, $x_{2}$, …, $x_{7}$ ]) and 3
single-scattering properties (Y = [ $Q_{ext}$, $Q_{sca}$, g ]) are output
targets as shown in Fig. 2. Here the input features are the size parameter
($x$), wavelength ($\lambda$), coating fraction ($f$) for coated, internally
mixed aerosol and RIs for both core ($RI^{c}$) and shell ($RI^{s}$). Using a
dataset comprising known input and output matrices, denoted as X and Y
respectively, the model undergoes training to optimize its parameters –
weights ($w$) and biases ($b$). This optimization is achieved via back-
propagation, which minimizes the cost function $C_{y}$:

$C_{y}=\sum_{i=1}^{N}(y_{true}-y_{pred})^{2}$ (6)

This function quantifies the error between the predicted values ($y_{pred}$)
generated through forward propagation in the NN and the actual values
($y_{true}$). The cost function $C_{y}$ is differentiable with respect to the
model parameters ($w$ and $b$), enabling the application of various gradient
descent techniques for efficient optimization.

###  2.3 Training Data and its preprocessing

To facilitate MieAI training, a total of 30 distinct combinations of core and
shell chemical compositions are considered, as outlined comprehensively in
Table 4. The computation of optical characteristics relies on wavelength-
dependent RI. As emphasized by Rieger et al. (2017), distinct peaks in the
real component of the RI manifest as prominent maxima in $Q_{ext}$.
Simultaneously, the $\omega$ and, consequently, the absorption efficiency
($Q_{abs}$) is governed by the imaginary component of the RI. In Fig. 8(a), we
present the real and imaginary components of RI for the chemical species
composing both the core and shell of aerosol particles (Rieger et al., 2017;
Muser et al., 2020; Hoshyaripour et al., 2019). Fig. 8(b) illustrates the
variations in the real and imaginary components of the RI for internally mixed
and coated particles as a function of changes in the chemical composition of
the core and shell across various wavelengths of solar radiation. The real
part of the RI exhibits a range from 1.1 to 2.75 for the core and 1.2 to 2 for
the shell, contingent upon the specific chemical compositions of the core and
shell. Meanwhile, the imaginary component varies from values as low as
$10^{-8}$ to 0.5 for the core and from $10^{-9}$ to 1 for the shell. It’s
noteworthy that the core is characterized as a volume-averaged ternary system
involving mineral dust, sea salt, and soot, while the shell is likewise
modeled as a ternary system, featuring water, inorganic, and organic
constituents.

Figure 3: Variation of target AOPs before and after quantile mapping. Quantile
mapping transforms input features to a Gaussian distribution with mean 0 and
standard deviation 1.

The training, test and validation datasets for MieAI are generated by randomly
selecting 600,000 samples (about 2 %) from more than 45 million possible
combinations of input features arising from varying wavelength (0.2 to 100
$\mu$m), shell thickness (from 0 to 40% of total diameter with a step wise of
0.1%), core diameter (from 10 nm to 20 $\mu$m) and RI by considering 30
different combinations for core and shell as discussed before. Randomly
selected samples were divided into training (70%), validation (15%) and test
datasets (15%) while optimising the NN architecture and parameters.

Both input and target datasets have a large variability; hence it is important
to normalize them before feeding to NN for training in order to improve the
model learning ability. Hence, input and target data to NN model is
transformed using Min-Max normalization before being fed to NN model.
Afterwards, the output from the NN model is denormalized to its original
optical properties space. We first normalized the training dataset and used
the same normalization scale to transform validation and test datasets to
avoid data leakage during model training.

Due to the non-normal distribution of the target AOPs in training dataset, we
perform a quantile distribution mapping over the raw target AOPs to a normal
distribution. Quantile mapping transforms all input features to the same
target distribution (Gaussian distribution in this case) based on the formula
$G^{-1}(F(X))$ where $F$ is the cumulative distribution function (CDF) of the
input feature and $G^{-1}$ is the quantile function of the target distribution
G (Jakob Themeßl et al., 2011). Quantile mapping smooths out uneven
distribution and is influenced less by outliers unlike scaling methods like
min-max transformation. Quantile mapping has been used extensively in
meteorology for bias correction (Hertig et al., 2019) and statistical
downscaling (Abatzoglou and Brown, 2012). We use the python library scikit-
learn for performing quantile mapping in this study. As shown in Fig. 3, the
raw training dataset for g is bi-modal with one peak over 0 and another over
1. While non-linear algorithms like MieAI may not have a Gaussian distribution
assumption, however they perform better if variables have a Gaussian
distribution. Thus, mapping to the normal distribution improves the
generalization of the trained network. During inference, the predicted AOPs
are transformed back to the original distribution using inverse quantile
transform with the same parameters used during the training.

###  2.4 Optimisation and assessment of MieAI

In addition to the model parameters optimized by the NN training procedure,
there are hyper-parameters that define the model architecture and control the
learning process, such as the number of hidden layers, the number of neurons
in each layer, the activation function, batch size and the learning rate of
the optimizer. The mean squared error (eq. 6) is employed as the loss function
for the optimizer to minimize. We apply a non-linear activation function to
all of the layers except the output where we apply linear activation to
restrict the NN output between 0 and 1 (Cachay et al., 2021). After the
training, the weight matrices in the NN are saved and used afterwards for
evaluation using ICON-ART simulations.

To assess the performance of the network, we used the coefficient of
determination ($R^{2}$) and Mean Absolute Percentage Error (MAPE) as metrics
to evaluate the fitness of the predictions with the true values. $R^{2}$ is
defined as:

$R^{2}=1-\frac{\Sigma^{M}_{i=1}(y_{i}-f_{i})^{2}}{\Sigma^{M}_{i=1}(y_{i}-\bar{y})^{2}}$
(7)

Here, $f_{i}$ is the value predicted by MieAI and $y_{i}$ is the true value.
$\bar{y}$ is the average of all true values. The closer $R^{2}$ is to 1, the
higher the performance of MieAI. The MAPE metrics is defined as:

$MAPE=\frac{100}{N}\sum^{N}_{i=1}\bigg{|}\frac{Y_{mie}-Y_{MieAI}}{Y_{mie}}\bigg{|}$
(8)

Here, $Y_{MieAI}$ is AOP prediction from MieAI, $Y_{mie}$ is the AOP estimated
using Mie theory and N is the number of times AOPs are predicted using MieAI.

Table 1: Hyper-parameter tuning of MieAI model. We use the keras-tuner hyper-parameter optimization library to tune the hyper-parameters of MieAI. | Tried value ranges | Best value
---|---|---
Number of Hidden Layers | 1–5 | 4
Neurons per Hidden layer | 8, 16, 32,64 | 64
Minibatch Size | 128, 256, 512, 1024 | 128
Activation Function | relu, gelu, sigmoid, tanh | gelu
Optimizer | adam, sgd, rmsprop | adam
Initial Learning rate |  $10^{-1}$, $10^{-2}$, $10^{-3}$, $10^{-4}$ | $10^{-2}$
|  |

To avoid over-fitting and other training related issues, we chose our NN
hyper-parameters using keras-tuner hyper-parameter optimization library and
apply early stopping with patience parameter set as 50 (Shawki et al., 2021).
The hyper-parameters of the model have been meticulously optimized through the
application of Bayesian optimization. The entire hyper-parameter tuning
procedure is executed in a two-stage approach, wherein each stage serves to
fine-tune distinct aspects of the model. In the first stage, we focus on
optimizing critical architectural components, including the number of hidden
layers, the neuron count in each hidden layer, activation functions, and the
choice of optimizer. Subsequently, the second stage hones in on further
enhancements by fine-tuning the learning rate and the batch size of the
training data, for the NN selected in the first stage. During hyper-parameter
optimization, we trained various NN architectures for 200 epochs. The
corresponding MSE values for these diverse NN architectures are presented in
Table 5 (first stage) and Table 6 (second stage). The resultant optimal values
for all hyper-parameters are shown in Table 1.

As depicted in Table 5, the MieAI model with Adam optimizer having 5 hidden
layers with 64 neurons in each layers and GELU activation function performed
the best with the lowest MSE. With the aim to select the most accurate NN with
smallest possible number of trainable parameters, we performed the second
stage of tuning wherein we varied the number of hidden layers and the number
of neurons in each layer along with the bath size and learning rate of Adam
optimiser selected after the first stage tuning. As shown in Table 6, the
MieAI model with 4 hidden layers outperformed the 5 layer NN as selected in
first stage when batch size and learning rate were also optimised. We apply
early stopping with patience set as 50 and reduce the learning rate of the
optimizer by one-fifth if the validation loss plateaus during both hyper-
parameter tuning and training of the network. Therefore, the best NN after
hyper-parameter optimisation is a MLP with 4 hidden layers each having 64
neurons trained using Adam optimizer with learning rate of 0.01 and training
batch size of 128.

###  2.5 ICON-ART model system

In addition to evaluating the trained MieAI using test datasets, we conducted
three reference ICON-ART simulations for real-world events to validate the
MieAI prediction of AOPs against Mie calculations. The ICON modelling
framework excels in solving the nonhydrostatic and compressible Navier-Stokes
equations on an icosahedral-triangular grid (Zängl et al., 2015). This model
exhibits versatility in predicting various processes across scales, from
global to local, as highlighted by Giorgetta et al. (2018) and Heinze et al.
(2017). Complementing the ICON model, the ART module forms an integral part
responsible for simulating trace gases and aerosols in both the troposphere
and stratosphere. This module encompasses processes spanning emission,
transport, physicochemical transformation, removal of gases and aerosols as
well as their interactions with clouds and radiation (Rieger et al., 2015;
Gasch et al., 2017; Schröter et al., 2018). Deutscher Wetterdienst (DWD) uses
ICON and ICON-ART for operational weather and mineral dust forecasting and
pollen, respectively.

ICON-ART uses the European Centre for Medium-Range Weather Forecasts (ECMWF)
radiation scheme ecRad (Hogan and Bozzo, 2018) as the standard radiation
scheme for numerical weather prediction (Rieger et al., 2019; Seifert et al.,
2023). To calculate the local radiative transfer parameters, ecRad needs the
$k_{el,j}$, $\omega_{l,j}$ and $g_{l,j}$ for every mode $l$ and every waveband
$j$ for 30 wavelength bands between 0.2 and 100 µm. These are often obtained
using Mie calculations. Together with the local aerosol mass mixing ratios
($\psi_{3,l}$) from ART and air density ($\rho_{a}$), they allow for
calculation of the volume specific extinction coefficient (Muser, 2022):

$\beta_{ext,l,j}=k_{el,j}\cdot\rho_{a}\cdot\psi_{3,l}\cdot 10^{-6}$ (9)

$\omega$ gives the scattering coefficient:

$\beta_{scat,l,j}=\omega_{l,j}\cdot\beta_{ext,l,j}$ (10)

These volume specific properties are then converted to values per model layer
by multiplying with the respective layer height ($\Delta z$), followed by
summation across all model layers to calculate total aerosol optical depth
(AOD) for the ART aerosol within a specific waveband. These computed values
then serve as input parameters for the radiation scheme (Gasch et al., 2017).
This approach ensures full coupling and feedback between aerosol processes,
radiation, and the atmospheric state (Hoshyaripour et al., 2019; Shao et al.,
2011).

The present study focuses on the interaction of internally mixed aerosols with
radiation, which is comprehensively addressed through the use of the AEROsol
DYNamic module (AERODYN) in ICON-ART. This module enables examination of
aerosol dynamics processes, including nucleation, condensation and coagulation
that generate internally mixed aerosols. AERODYN comprises flexible number of
log-normal modes (up to 10) that accounts for Aitken, accumulation, and coarse
particles in soluble, insoluble, and mixed states, alongside a giant insoluble
mode (Muser et al., 2020). The term "mixed state" here pertains to an aerosol
that is composed of an insoluble core and a soluble shell, and the latter
constitutes no less than 5% of the overall mass of the aerosol. The prognostic
equations for number density and mass concentration are solved for each
species and each mode while maintaining constant standard deviations. There
exist two distinct circumstances that result in the alteration of particle
modes. The first circumstance is when the mass threshold of soluble coating on
insoluble particles surpasses 5%, leading to a transition from insoluble to
mixed mode. The second circumstance is when the diameter threshold of the
soluble and mixed mode is exceeded, resulting in a shift to a larger mode.
Alterations in the particle modes can modify the optical properties of
particles with consequential impacts on both the atmospheric state and
radiation (Muser et al., 2020; Bruckert et al., 2023).

In ICON-ART, each aerosol component was assigned a RI, and the RI values were
obtained from Di Biagio et al. (2019) for dust and Gordon et al. (2022) for
other species. The volume-average mixing rule is used to compute the complex
RI of both core and shell, which then serves as input for the core–shell
calculation. To facilitate a comparison between Mie calculations and MieAI
predictions, we initially derived bulk AOPs for each aerosol mode by
aggregating optical properties across individual aerosol population bins. To
achieve this, we initially mapped each aerosol mode, based on its median
diameter, to 15 log-normal bins, as illustrated in Fig. 9. Both Mie
calculations and MieAI emulation were then applied to these bins, and the
results were subsequently integrated to obtain bulk optical properties for
each mode. For our validation, we employed RI values at a wavelength
($\lambda$) of 550 nm.

###  2.6 Case Studies

In order to validate accuracy and computational efficiency of the MieAI, we
apply both MieAI and Mie code to the outputs of three different case studies
with different aerosol species. In the following, we briefly explain the
experiments. Table 2 summarizes the relevant aerosol characteristics in each
experiment. It is noteworthy that the MieAI was exclusively trained on a
dataset featuring shell thicknesses up to 40%, while the comparisons in all
three cases encompass shell thicknesses beyond 40%, reaching up to 50%. This
extension aims to demonstrate the generalization capability of MieAI.
Additionally, it is imperative to recognize that the stability of the Mie code
output diminishes as the coating exceeds 50%. Furthermore, we hypothesize that
particles undergo a transition into an optically soluble mode beyond a coating
threshold of 0.5. Importantly, our focus is not to validate the model
simulations in these events. Rather, we aim at evaluating the MieAI
performance with real model data.

Table 2: Summary of the mixed mode properties in case studies. Case | Bin diameter range (nm) | Core components | Shell components
---|---|---|---
2021 La Soufrière eruption | 10 – 1200 | Volcanic ash | $H_{2}O/SO_{4}^{2-}$
2019–20 Australian Wildfire | 3 – 1000 | Soot | $H_{2}O/SO_{4}^{2-}/NO_{3}^{-}$
Summer 2019 dust event | 100 – 5000 | Dust, sea salt and soot | $H_{2}O/SO_{4}^{2-}/NO_{3}^{-}/NH_{4}^{+}$
|  |  |

####  2.6.1 2021 volcanic eruption of La Soufrière

The first numerical experiment (case volcano) is a simulation of the last La
Soufrière eruption in April 2021 and was performed by Bruckert et al. (2023).
Located on the St Vincent island in the Caribbean, the La Soufrière volcanic
eruption occurred during 09–21 April 2021 and emitted volcanic material such
as ash and SO2 in 49 eruption phases. The simulation covered the initial four
days of the eruption, encompassing 43 of the 49 eruption phases starting from
09 April at 12 UTC. The simulation had a grid spacing of 13 km with 2 nested
grids around the volcano with 6.6 and 3.3 km grid spacing, respectively. The
model employed 90 vertical levels to resolve the atmosphere up to 75 km. The
experiment accounts for aging of volcanic ash (ash coated by sulfate-water
mixture) due to aerosol dynamics. More details on this experiment is provided
by Bruckert et al. (2023).

####  2.6.2 2019–20 Australian wildfire

The second case (case wildfire) investigated the catastrophic 2019–20
Australian wildfires in Queensland, which severely affected over 7.5 million
hectares and caused a decline in air quality. A 1-day simulation was performed
on November 23rd in an area on the eastern coast of Australia (150°E-160°E,
23°S-33°S). The model featured a grid spacing of 6.6 km, extending vertically
to 20 km with 125 levels in a limited area setting. The emission fluxes are
taken from the Global Fire Assimilation System (GFAS). 25% of the particle
mass is emitted in the Aitken mode and 75% in the accumulation mode. The
emission height is parameterized with the plume rise model according to
Freitas et al. (2005, 2007, 2010); Walter et al. (2016).

####  2.6.3 Summer 2019 dust event over central Europe

The third case study (case dust) centered around a dust event over central
Europe during 22–27 June 2019, involving global-scale simulations with a 40 km
grid. The simulation considers comprehensive aerosol emissions (including sea
salt, dust, and soot) and their dynamic processes (such as nucleation,
condensation and coagulation), with simplifications made in gas-phase
chemistry for operational forecasting. Similar to the wildfire case study,
chemical species were reinitialized daily using CAM-Chem data.

##  3 Results and discussion

###  3.1 MieAI training and testing

((a)) MieAI predictions against true AOPs estimated using Mie calculations for
the test dataset.

((b)) Distribution of NN errors for the test dataset. Here, error reported is
the percentage error of MieAI with respect to true AOPs estimated using Mie
calculations.

Figure 4: MieAI training and evaluation.

In this study, we use a MLP with multiple hidden layers (named MieAI) to
emulate the calculation of AOPs using Mie theory. MieAI takes particle mixing
state and morphology (which include size parameter, shell thickness and RI of
core and shell) as inputs and outputs three AOPs such as $Q_{ext}$, $Q_{sca}$
and g. $\omega$ is calculated from $Q_{ext}$ and $Q_{sca}$ using eq. 10.

MieAI model, selected after hyper-parameter tuning, is trained for 5000 epochs
until the loss function is minimized, resulting in an optimized network. The
NN was trained on 500,000 training data samples whereas verification was done
on 100,000 test samples randomly chosen from 600,000 Mie samples. Fig. 10
provides a comprehensive visualization of the dynamic evolution of two crucial
loss metrics, the MSE and the MAPE, throughout the training process of our NN
model. These loss metrics are pivotal for assessing the model’s performance,
particularly in its capacity to accurately approximate AOPs. The observed
trends in this figure offer profound insights into the model’s convergence and
its effectiveness in learning from the training data. Notably, the continuous
and monotonic decrease in validation losses, both in terms of MSE and MAPE,
serves as a strong indicator of the model’s robust fitting to the data. This
persistent reduction in validation losses underscores the model’s consistent
improvement in its ability to predict AOPs accurately as the training
progresses. Such a trend is highly promising, as it demonstrates the model’s
capacity to continually refine its representations and effectively grasp the
intricate relationships inherent in the AOP data.

It’s important to highlight that our model’s training incorporates an early
stopping mechanism, with a patience parameter set at 50. This strategy ensures
that the model training halts at the 2548th epoch, optimizing the MieAI model
with a validation MSE of 0.01187. This early stopping mechanism is a prudent
approach to prevent overfitting and ensure that the model generalizes well to
unseen data.

To verify the optimized network, we evaluated its performance by comparing its
AOP predictions against the true AOP values estimated using Mie calculations
and the $R^{2}$ values for different AOPs modelled in this study are shown in
Fig. 4(a). As depicted in the figure, our trained NN model demonstrates a
commendable ability to model AOPs effectively, as evidenced by high $R^{2}$
values of 0.994, 0.994, and 0.997 for $Q_{ext}$, $Q_{sca}$, and g,
respectively. These results underscore the robust learning capability of the
selected NN model, affirming its aptitude for capturing the intricate
relationships within the data.

However, it is worth noting that while MieAI excels in predicting these three
AOPs overall, there are specific regions where it encounters challenges. In
particular, these challenges become apparent in the case of $Q_{ext}$ and
$Q_{sca}$, especially when these values fall below 2 and 1, respectively. In
these regions, MieAI appears to struggle, leading to more substantial
discrepancies between its predictions and the actual values.

To gain deeper insights into these discrepancies, we examine the distribution
of relative errors in MieAI predictions, as illustrated in Fig. 4(b). This
analysis reveals that MieAI tends to underestimate $Q_{ext}$ and $Q_{sca}$
slightly when compared to those calculated using the Mie theory. However,
notably, there is no such bias observed in the case of g. These findings
provide valuable insights into the performance characteristics of MieAI and
highlight specific areas where further model refinement may be warranted.

###  3.2 MieAI validation using ICON-ART simulations

Figure 5: Comparison of AOPs predicted by MieAI against those estimated using
Mie theory for coarse mode internally mixed aerosol particles at an altitude
of 15 km above sea level for the La Soufrière volcanic eruption (denoted by
the plus symbol) event simulated using ICON-ART. Here, left column (a, d, g,
j) shows the AOPs estimated using Mie theory, middle column (b, e, h, k) shows
the same predicted from MieAI and right column (c, f, i, l) shows the relative
error of MieAI AOPs prediction against Mie calculations. Panel (m) shows the
geographical distribution of the aerosol median diameter simulated using ICON-
ART whereas Panel (n) shows the geographical variation of shell thickness, as
a fraction of the total particle diameter (in percentage), of the coated
aerosol.

Next, we compare the AOP predictions of MieAI against the same estimated using
Mie theory for the outputs of ICON-ART simulations and are shown in Figs. 5, 6
and 7 for different real events examined in this study. For this purpose, the
number concentrations for the constituent species of mixed modes were taken
from ICON-ART output. We first map the ICON-ART number concentrations to RIs
for core and shell as shown in Fig. 8(b) and ICON-ART modes to bins assuming
log-normal distributions as illustrated in Fig. 1. MieAI predictions for the
bins are integrated back to modes and then compared with the Mie calculations.

Fig. 5 shows the spatial distribution of AOPs for simulated internally mixed
volcanic aerosols in accumulation mode, obtained from both Mie and MieAI (See
Fig. 11 for the comparison in coarse mode). The illustration focuses on the
derived AOPs after the La Soufrière volcanic eruption in April 2021,
specifically showcasing the comparison at an altitude of 15 km above sea level
27 hours after the start of the simulation. The median diameters, shown in
Fig. 5(m), exhibit a range spanning from 100 nm to 1200 nm. Concurrently, the
shell (coating) thickness, depicted in Fig. 5(n), varies from 10 to 80% of the
total diameter. It is notable that a majority of particles possess median
diameters exceeding 500 nm and exhibit thick coatings (more than 50% coating
fraction). In this case, volcanic ash constitute the core whereas water and
inorganic species (sulphate, nitrate and ammonium) are the constituents of the
coating/shell. A discernible trend emerges in the figure, where $Q_{sca}$
(Fig. 5(d)) and consequently $Q_{ext}$ (Fig. 5(b)) appear to align with the
distribution of both median diameter and coating fraction. Higher values of
$Q_{ext}$ and $Q_{sca}$ are observed in regions characterized by lower median
diameters and coating fractions. Conversely, both $\omega$ (negative
correlation; Fig. 5(g)) and g (positive correlation; Fig. 5(j)) show a more
pronounced correlation with changes in median diameter, with a lesser
influence from the coating fraction. Impressively, MieAI, shown in Fig. 5(b,
e, h, k), effectively captures these dependencies, showcasing an impressive
agreement between its predictions and Mie theory estimates. The comparison
between MieAI predictions and Mie theory estimates reveals a very good
agreement, with relative errors (depicted in Fig. 5(c, f, i)) generally
staying within 10% for all AOPs, except for g (Fig. 5(l)), where the relative
error reaches up to 12%. This suggests that the NN model effectively captures
the intricate relationships between particle morphology, mixing state, and
optical properties. Interestingly, it’s worth noting that network errors
exhibit a degree of dependency on the coating fraction for all AOPs, except g.
In the case of g, network errors closely track the distribution of median
diameter, with higher relative errors occurring in regions where the median
diameter is smaller.

Figure 6: Same as Fig. 5 but for coarse mode internally mixed aerosols at an
altitude of 850 m above sea level in ICON-ART simulation of Australian biomass
burning event. Figure 7: Same as Fig. 5 but for coarse mode internally mixed
aerosols at an altitude of 5 km above sea level in ICON-ART simulation of a
dust event over central Europe.

Fig. 6 shows a comparison of the bulk AOPs estimated from MieAI and Mie for
case wildfire. This case study centers on an Australian wildfire event from
2019, specifically examining the comparison at an average altitude of 850 m
above sea level after 23 hours of simulation (23rd of November 2019, 23:00
UTC). The selected altitude corresponds to the mass weighted height of the
plume, further the plume at that level is wide spread with a high
concentrations compared to other model levels. The time step selected is
towards the end of the one day simulation, enabling transport and aging of the
aerosol. In this case, soot constitutes the core whereas water, organic and
inorganic species (sulphate, nitrate and ammonium) are the constituents of the
coating/shell. Here, the median diameter (Fig. 6(m)) for the internally mixed
aerosol in coarse mode ranges from 50nm to 1000nm whereas the shell (coating)
thickness (Fig. 6(n)) varies from 35 to 50$\%$ of the total diameter (See Fig.
12 for the comparison in accumulation mode). It’s worth noting that this
simulation predominantly features aerosol particles with total diameters
exceeding 900 nm. Similar to Case volcano, intriguing patterns emerge wherein
all four optical properties exhibit alignment with the distribution of median
diameters. In particular, changes in $Q_{ext}$ (Fig. 6(a)), $Q_{sca}$ (Fig.
6(d)), and $\omega$ (Fig. 6(g)) showcase a negative correlation with the
variations in median diameters, while g (Fig. 6(j)) demonstrates a positive
correlation with the same. Intriguingly, none of the optical properties appear
to exhibit sensitivity to variations in the shell thickness. Remarkably, the
comparison between MieAI, shown in Fig. 6(b, e, h, k), and Mie calculations,
shown in Fig. 6(a, d, g, j), underscores an excellent agreement, reaffirming
the robustness of the employed NN model in effectively emulating Mie theory
for internally mixed aerosols. The relative errors for all optical properties,
shown in Fig. 6(c, f, i, l), in this case remain within the 10%. Notably, in
contrast to the La Soufrière case study, network errors in this instance
appear to be particularly responsive to changes in median diameters rather
than variations in the coating fraction.

Finally, Fig. 7 shows a comparison of the MieAI predictions using the model
trained with quantile transformation and Mie calculation for coarse mode
internally mixed particles using ICON-ART simulation for case dust. The
investigation focuses on a dust event occurring over central Europe, wherein
the simulation encompasses a comprehensive range of aerosol species emissions,
including sea salt, dust, and soot. The figure exclusively showcases these
comparisons at an altitude of 5 km above sea level. In terms of particle
characteristics, the median diameter (Fig. 7(m)) for mixed-phase aerosols
within the coarse mode exhibits a range spanning from 200 nm to 2300 nm.
Notably, the majority of these particles possess a median diameter of less
than 500 nm. Concurrently, the shell (coating) thickness varies from 0 up to
50% of the total diameter as shown in Fig. 7(n). However, it is important to
note that a substantial proportion of the particles feature a shell thickness
of less than 10%. As anticipated, the optical properties (Fig. 7(a, d, g, j))
display sensitivity to changes in median diameter, mirroring the patterns
observed in previous cases. While akin to the previous case, the influence of
shell thickness remains relatively limited. As expected, MieAI (Fig. 7(b, e,
h, k)) excels in capturing the variations in AOPs, with relative errors (Fig.
7(c, f, i)) staying below 10% for $Q_{ext}$ and $Q_{sca}$ as well as $\omega$.
The prediction accuracy for g (Fig. 7(l)) is also reasonably strong, with
errors generally remaining within 15%. Importantly, it is noteworthy that the
magnitude of errors in g is sensitive to the coating fraction, a
characteristic distinguishing it from the other three optical properties. For
a complementary comparison in the accumulation mode, please refer to Fig. 13.

Table 3: Timing results (in seconds) of MieAI and Mie calculations for different real cases investigated in this study. Here, MieAI results are shown for the prediction batch size of 8192. Case | Number of grid cells | Mie | MieAI | Computational Gain
---|---|---|---|---
2021 La Soufrière Volcanic eruption | 73,500 | 423.3126 s | 0.2136 s | 1981.80x
2019–20 Australian Wildfire | 74,865 | 398.1323 s | 0.2108 s | 1888.67x
Summer 2019 dust event over central Europe | 28,800 | 132.9224 s | 0.1971 s | 0674.39x
|  |  |  |

The comparisons between AOPs estimated using Mie theory and the predictions
made by MieAI, employing a model trained without quantile transformation, are
presented in Fig. 14. As clearly evident from the figure, the MieAI model,
when trained without quantile transformation, exhibits notable shortcomings in
capturing the variations in AOPs, with the exception of $Q_{ext}$. This
discrepancy becomes particularly conspicuous despite the model’s impressive
performance on the test dataset, where correlation coefficients (R) exceeded
0.98 for all AOPs examined, including $Q_{ext}$, $\omega$, and g, as
demonstrated in Fig. 15.

The divergence between the model’s performance on the test dataset and its
application to real-world data underscores a critical limitation in its
ability to generalize beyond the training context. Implications of this
observation are far-reaching and offer valuable insights into the complexities
of emulating intricate physical mechanisms using NNs, particularly when not
validated against real-world scenarios. Consequently, it underscores the
critical importance of comprehensive preprocessing of datasets before their
integration into ML models, serving as a precautionary measure against
potential pitfalls in model generalization.

In summation, this comprehensive analysis underscores the robustness of the
MieAI model (with quantile transformation) in reproducing the optical
properties of internally mixed aerosols. Note that the MieAI was trained using
a dataset which had shell thickness up to 40% only whereas the comparisons in
all three cases include shell thickness beyond 40% (up to 50%). Thus, the
comparisons clearly demonstrate the extrapolating capability of MieAI. The
fact that the model successfully extrapolates its predictions beyond the
training data’s confines is a testament to its inherent capacity to generalize
and capture the underlying physical principles governing the interactions
between aerosol particles and light. This characteristic is particularly
valuable in real-world scenarios where aerosol properties can exhibit a wide
range of variability, often extending beyond the confines of training data.
MieAI’s capacity to accurately predict optical properties for aerosols with
shell thicknesses up to 50% highlights its versatility and reliability as a
Mie emulator.

###  3.3 Computational efficiency of MieAI

In addition to its high fidelity in modeling AOPs, MieAI offers a remarkable
advantage in computational efficiency, showcasing significant computational
enhancements in comparison to traditional Mie calculations employed for the
same purpose. As indicated in Table 3, MieAI demonstrates a computational
speedup exceeding 500 times that of Mie calculations across all scenarios
investigated in this study.

The extent of performance gain is particularly noteworthy; for instance,
during the 2019 dust event over central Europe with 28,800 ICON grid cells,
MieAI exhibited a speedup of approximately 500 times. As the number of grids
increases, this gain becomes more pronounced, with speedups surpassing three
orders of magnitude when compared to Mie calculations. This phenomenon is
exemplified in the ICON-ART simulations for events such as the La Soufrière
volcanic eruption with 73,500 ICON grid cells, where MieAI achieved a speedup
of around 1900 times, and the Australian wildfire event with 74,865 ICON grid
cells, boasting a remarkable speedup of around 1800 times.

Furthermore, the computational cost associated with MieAI training is
exceedingly minimal, taking approximately 3 hours and 20 minutes. This stands
in stark contrast to the runtime requirements of ICON-ART simulations.
Notably, MieAI training utilized a single computing node from a high-
performance computing (HPC) cluster equipped with multiple nodes, each housing
36 Intel Xeon CPUs. It is pertinent to mention that both MieAI predictions and
Mie calculations were executed utilizing a single CPU core.

##  4 Conclusions

This study endeavors to introduce an innovative and computationally efficient
framework, aptly named MieAI, specifically designed for calculating the bulk
optical properties of internally mixed and coated aerosols characterized by a
log-normal size distribution. Our approach leverages a straightforward multi-
layer perceptron, a type of artificial neural network, to unravel the
intricate relationship between AOPs and their physico-chemical
characteristics, such as particle size distribution, mixing state, and
chemical composition. Central to MieAI is the representation of both core and
shell as ternary systems, subsequently linked to RIs via a volume mixing
approach.

In order to validate the efficacy of our approach, we subjected it to rigorous
evaluation against the gold standard method of Mie calculations – a technique
renowned for its precision albeit its notably sluggish computational speed.
Our comparative evaluation unveiled that the NN-based MieAI approach not only
attains remarkable accuracy -— with errors confined within 10% -— but also
exhibits an excellent computational efficiency, boasting a speed improvement
of three orders of magnitude.

Furthermore, our study underscores the paramount significance of meticulous
pre-processing in enhancing the accuracy and generalizability of NN-based
methodologies. We emphasize the necessity for rigorous evaluations of novel
ML-based approaches prior to their widespread deployment in scientific
applications.

With its generic design, the approach presented herein holds versatile
applicability, seamlessly integrating into CTMs that adopt either bin or modal
frameworks for representing aerosols and their optical properties. Moreover,
the same framework can be extended to accommodate externally mixed aerosols
and aerosol models featuring non-spherical shapes.

The substantial precision achieved through our developed approach bears the
potential to significantly contribute to the ongoing efforts aimed at
mitigating uncertainties in aerosol forcing estimations. By bridging the gap
between precision and computational efficiency, MieAI emerges as a valuable
asset in the realm of physics-based weather and climate models, especially
CTMs; poised to contribute substantially to advancing our understanding of
aerosol-climate interactions and fostering more robust climate models.

The ICON-ART code is license protected and can be accessed upon request. The
training data and output from ICON-ART simulations generated in this study
will be uploaded on Radar4KIT after completion of the review processes. MieAI
model and python codes used for performing analyses can be accessed here:
https://github.com/pankajkarman/MieAI.

GAH conceived the idea. PK and GAH designed the research. PK wrote the NN
code, performed the data analyses. GAH, JB, LMH, HK, and AR developed the
ICON-ART code and carried out simulations. PK and GAH prepared the paper with
significant contributions and comments on the original draft from all authors.
GAH supervised the research at KIT.

The authors declare no conflict of interest.

###### Acknowledgements.

This work contributes to and is partly funded by the project PermaStrom (grant
no. 03EI4010A) within the seventh Energieforschungsprogramm of the German
Federal Ministry of Economic Affairs and Climate Action (Bundesministerium für
Wirtschaft und Klimaschutz, BMWK). This work used resources of the Deutsches
Klimarechenzentrum (DKRZ) granted by its Scientific Steering Committee (WLA)
under project ID 1070. Part of this research has been funded by the Deutsche
Forschungsgemeinschaft (DFG) as part of the Research Unit VolImpact (FOR2820,
DFG Grant 398006378). Open Access funding enabled and organized by Projekt
DEAL.

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Table 4: Various combinations (in percentage) of chemical species used to constitute the ternary systems for both core and shell to generate the dataset used for training MieAI. | Constituent1 | Constituent2 | Constituent3
---|---|---|---
1 | 100 | 0 | 0
2 | 90 | 10 | 0
3 | 90 | 0 | 10
4 | 90 | 5 | 5
5 | 80 | 20 | 0
6 | 80 | 0 | 20
7 | 80 | 10 | 10
8 | 70 | 30 | 0
9 | 70 | 0 | 30
10 | 70 | 15 | 15
11 | 60 | 40 | 0
12 | 60 | 0 | 40
13 | 60 | 20 | 20
14 | 50 | 50 | 0
15 | 50 | 0 | 50
16 | 50 | 25 | 25
17 | 40 | 60 | 0
18 | 40 | 0 | 60
19 | 40 | 30 | 30
20 | 30 | 70 | 0
21 | 30 | 0 | 70
22 | 30 | 35 | 35
23 | 20 | 80 | 0
24 | 20 | 0 | 80
25 | 20 | 40 | 40
26 | 10 | 90 | 0
27 | 10 | 0 | 90
28 | 10 | 45 | 45
29 | 0 | 0 | 100
30 | 0 | 100 | 0
|  |  |
Table 5: Stage 1 of hyper-parameter tuning. Here, top 5 combinations of optimizer, number of neurons in a hidden layer, activation functions and number of hidden layers in the order of decreasing MSE values are shown. | optimizer | units | activation | number of layers | mse
---|---|---|---|---|---
1 | adam | 64 | gelu | 5 | 0.03209
2 | adam | 32 | tanh | 3 | 0.03307
3 | adam | 64 | tanh | 3 | 0.03372
4 | adam | 32 | gelu | 4 | 0.03562
5 | adam | 64 | gelu | 4 | 0.03632
|  |  |  |  |
Table 6: Stage 2 of hyper-parameter tuning. Here, top 5 combinations of number of neurons in a hidden layer, learning rate, number of hidden layers and Minibatch size in the order of decreasing MSE values are shown. | units | learning rate | number of layers | batch size | mse
---|---|---|---|---|---
1 | 64 | 0.010 | 4 | 128 | 0.014102
2 | 64 | 0.010 | 5 | 512 | 0.017238
3 | 64 | 0.010 | 5 | 256 | 0.017359
4 | 32 | 0.010 | 4 | 128 | 0.017747
5 | 64 | 0.001 | 5 | 128 | 0.017767
|  |  |  |  |

((a)) Refractive indices of the constituents of coated, internally mixed
aerosol particles i.e. Mineral dust, sea salt, soot, water, inorganic and
organic species.

((b)) Real (upper row) and imaginary (lower row) part of refractive indices
for 30 different chemical compositions of core and shell. Here, the left
column shows the RI for core and right column shows RI for shell.

Figure 8: Refractive indices of the internally mixed aerosol particle. Mineral
dust, sea salt and soot constitute the core whereas water, Organic and
Inorganic species constitute the shell of the particle. Figure 9: Mapping of
aerosol mode to bins. Here, 15 log-normal bins are used to represent each
aerosol mode. Figure 10: Variation of NN loss with epochs. Here, the green
colored curve shows mean squared error (MSE) and mean absolute percentage
error (MAPE) loss for the training samples whereas the orange colored curves
represent the same for testing samples. Intended to train for 5000 epochs, the
training of MieAI finishes at 2548 epoch due to earling stopping (patience
parameter set to 50). Figure 11: Same as Fig. 5 but for accumulation mode
internally mixed aerosols in ICON-ART simulation of the La Soufrière volcanic
eruption event. Here, the plus symbol shows the location of volcanic eruption
site. Figure 12: Same as Fig. 5 but for accumulation mode internally mixed
aerosols in ICON-ART simulation of Australian Biomass Burning event. Figure
13: Same as Fig. 5 but for accumulation mode internally mixed aerosols in
ICON-ART simulation of a dust event over central Europe. Figure 14: Same as
Fig. 5 but for ICON-ART simulation of Australian Biomass Burning event. Here,
quantile mapping was not used for pre-processing the training data fed to
MieAI.

((a)) x >= 0.5

((b)) x < 0.5

Figure 15: MieAI predictions with model trained without quantile mapping
against true AOPs estimated using Mie calculations for the test dataset. Here,
we trained 2 separate NN models: one for size parameter upto 0.5 (Rayleigh
scattering regime) and another for size parameter more than 0.5 (Mie
Scattering regime).