Tractable Generative Modelling of Cosmological Numerical Simulations

Amit Parag

1 a

and Vaishak Belle

1,2 b

The University of Edinburgh, U.K.

Alan Turing Institute, U.K.

Keywords:

Cosmological Simulations, Generative Models, Sum-Product Networks.

Abstract:

Cosmological simulations aim to understand the matter distribution in the universe by employing either semi-

analytic methods or hydrodynamical models of matter distribution. These simulations describe the evolution

of baryonic structures within dark matter potential wells, where dark matter is modeled as a self-gravitating,

collisionless system. Despite advances in reducing computational costs, these simulations still require millions

of CPU hours to achieve stable solutions. This raises the question: can generative models predict galaxy

properties from a partial history of their dynamical evolution? Tractable probabilistic models, such as sum-

product networks, enable efﬁcient computation of conditional probabilities, allowing conditional marginals

to be computed in time linear to the model size. In this work, we investigate the application of sum-product

networks to compactly represent and learn distributions for predictions in concordance cosmology. Using

the Eagle suite of cosmological hydrodynamical simulations, we demonstrate that these graphical models can

effectively reproduce mock galaxy catalogs, capturing the relationship between baryonic and dark matter with

promising accuracy.

1 INTRODUCTION

Probabilistic models, like Bayesian and Markov net-

works, are key in statistical machine learning for ex-

pressing dependencies compactly. However, infer-

ence in such models is intractable, making learn-

ing difﬁcult (Koller et al., 2009). Tractable learn-

ing, which enables efﬁcient probabilistic querying,

addresses this issue, with early work focusing on low

tree-width models (Bach and Jordan, 2002) and later

efforts utilizing local structures (Chavira and Dar-

wiche, 2008), leading to arithmetic circuits (ACs) that

support exact inference in polynomial time.

Sum-product networks (SPNs) (Poon and Domin-

gos, 2011) are notable instances of ACs with a recur-

sive structure, where an SPN is a weighted sum of

products of SPNs, and its leaves represent tractable

distributions (e.g., univariate Bernoulli). SPNs are of-

ten seen as tractable deep architectures due to their

hierarchical nature. While deep learning models face

challenges in structure learning (Bengio et al., 2009),

SPNs inherently provide a reliable structure learning

framework. Although SPNs can be manually speci-

ﬁed, weight learning and adherence to conditions like

https://orcid.org/0009-0001-6597-3976

https://orcid.org/0000-0001-5573-8465

completeness and decomposability make automated

structure learning preferable.

Since their introduction, various structure learning

methods for SPNs and related models have emerged,

such as LearnSPN (Gens and Domingos, 2013), OSL

(Hsu et al., 2017), and ID-SPN (Liang et al., 2017).

Extensions like RAT-SPNs (Peharz et al., 2020), deep

tractable models (Vergari et al., 2021), and hybrid

SPN frameworks (Rahman and Gogate, 2014) further

demonstrate their adaptability. Although related mod-

els may differ in properties and features (Liang et al.,

2017), SPNs remain attractive for generative model-

ing due to their simplicity and scalability.

In this work, we study how SPNs can be used to

model a novel and challenging problem related to the

evolution of galaxies, particularly in the context of

concordance cosmology with hydrodynamical simu-

lations. Since actual experiments in cosmology are

not feasible, simulations are crucial for testing cosmo-

logical theories, properties, and parameters. Numeri-

cal simulations are indispensable due to the inability

to derive analytic solutions for gravitationally inter-

acting particles, which form the basis of all cosmo-

logical simulations. Dark matter, modeled as particles

interacting solely through gravity, plays a pivotal role

in these simulations. When combined with baryonic

Parag, A. and Belle, V.

Tractable Generative Modelling of Cosmological Numerical Simulations.

DOI: 10.5220/0013229100003890

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 3, pages 901-908

ISBN: 978-989-758-737-5; ISSN: 2184-433X

901

physics, numerical simulations validate cosmological

models and provide a general picture of structure for-

mation in the universe.

Simulations attempt to describe the cosmological

structure by modeling galaxies inside dark matter ha-

los, which involves making unveriﬁable assumptions

(Somerville and Dav

e, 2015) and incurs high com-

putational costs, as shown by the millions of CPU

hours required by simulations like Eagle and Illus-

tris (Schaye et al., 2014; Llinares, 2017). Several

algorithms aim to reduce computation times while

improving accuracy (Llinares, 2017; Gheller et al.,

2015). Machine learning has been used in cosmol-

ogy for particle tracing and classiﬁcation (Guest et al.,

2018), but its potential for enhancing cosmological

research is still underexplored. This work takes initial

steps toward using recent advances in tractable proba-

bilistic models for generative modeling in cosmology.

The focus of the paper is on applying these models

rather than extending existing algorithms. As demon-

strated, this involves understanding concordance cos-

mology and overcoming challenges in feature selec-

tion. Ultimately, we hope our work inspires further

interdisciplinary research, enabling machine learning

to tackle signiﬁcant cosmological problems.

2 BACKGROUND

In this section, we proceed by ﬁrst by giving a fairly

informal picture of the cosmological model, before

turning to the equations driving the computational

task. We then discuss works where machine learning

has been used to model structure formation.

2.1 Concordance Cosmology

The concordance cosmological model, the ΛCDM

model, is based on the Copernican principles of

isotropy and homogeneity (Ryden, 2016). This model

assumes that observers in any location in the universe

cannot be the central observers, implying that the uni-

verse is isotropic and homogeneous on the largest

scales. These properties, however, only hold on cos-

mological scales where the differences between bary-

onic features are smoothed over.

Under these assumptions, the current cosmolog-

ical model describes a universe with rotational and

translational symmetry, dominated by dark matter.

Dark energy comprises the majority of the universe’s

energy, while dark matter plays a secondary role.

Baryonic matter, forming structures like galaxies, is

a small fraction of the universe. The ΛCDM model

(Bertschinger, 1994) describes the curvature of space-

time using the Robertson-Walker metric, and the evo-

lution of the universe follows the Friedmann equa-

tions (M

ortsell, 2016).

The concordance cosmology posits that the large-

scale structure of baryonic matter today originated

from seeds in dark matter halos. Galaxies form within

these halos, with their morphology largely determined

by the properties of the surrounding halo, merger his-

tory, and feedback effects. The presence of a dark

matter halo is essential for galaxy formation, with

massive subhalos at the center of halos containing

the central galaxies. Modeling the universe involves

painting a temporal picture of the cosmic web.

There are broadly two ﬂavors of simulations that

model matter distribution in the universe (Dolag et al.,

2008): semi-analytic simulations and hydrodynami-

cal simulations. The predictive power of both of these

approaches is in agreement with actual observations.

We refer the reader to (Benson et al., 2001) for a

comparison between semi-analytic methods and hy-

drodynamical modelling. In particular, physical pro-

cesses critical to galaxy formation and evolution such

as core collapse supernovae, accretion shocks, stel-

lar winds, involve multiple sets of partial differen-

tial equations (Somerville and Dav

e, 2015) such that

modeling structure formation through either approach

becomes extremely difﬁcult. The already intractable

complexity of this problem is further compounded by

the addition of approximations of physical phenom-

ena which cannot be derived ab initio.

2.2 Learning Structure Formation

Using machine learning algorithms to model structure

formation has inevitably resulted in varying degrees

of efﬁcacy. Algorithms like k-nearest neighbors and

support vector machines used in (Xu et al., 2013) have

conclusively shown that machine learning galaxy-

halo relation is not unsuccessful. The work was fur-

ther extended in (Kamdar et al., 2016), (Agarwal

et al., 2018), and (Cavuoti et al., 2018) by including

other discriminative or ensemble algorithms like de-

cision trees and/or random forests. Recent advances

in deep learning have shown signiﬁcant potential in

this domain. For instance, (Villaescusa-Navarro et al.,

2021) introduced the CAMELS project, leveraging

convolutional neural networks (CNNs) and varia-

tional autoencoders (VAEs) to predict galaxy proper-

ties. Similarly, (Lucie-Smith et al., 2023) employed

graph neural networks (GNNs) to model complex

galaxy-halo interactions. (Ho et al., 2022) explored

the application of foundation models in astrophysics,

focusing on tasks like large-scale structure model-

ing. Additionally, (Heitmann et al., 2021) emphasized

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

902

the use of emulators for precision cosmology, and

(Kobayashi et al., 2022) introduced machine learn-

ing frameworks for predicting baryonic effects on the

matter power spectrum. Works like (Davies et al.,

2021) further demonstrate how simulations combined

with machine learning can enhance our understand-

ing of galaxy morphologies, while (Tamosiunas et al.,

2023) applied semi-supervised learning to improve

galaxy classiﬁcation in limited data scenarios.

However, focusing on the algorithmic aspects of

the task is equally important since the choice of algo-

rithms usually involves some trade-offs between scal-

ability and accuracy, while certain algorithms like de-

cision trees are prone to overﬁtting. To our knowl-

edge, tractable graphical models have never been ap-

plied to this problem. Our contribution in this paper is

to apply a deep architecture with probabilistic seman-

tics, sum product networks (SPNs) (Poon and Domin-

gos, 2011), to estimate a generative model for the data

such that a mock catalog of galaxies can be built. The

added advantage of using SPNs is that they guaran-

tee that inference will always be in time linear in the

model size.

3 METHOD

Making machines learn to recognize halos and their

corresponding baryonic content broadly involves two

steps.

The ﬁrst step is ﬁnding features which are good

representatives of a halo-galaxy system and indicate a

strong correlation between the potential well of host

dark matter halo and the galaxy inside it. This is usu-

ally followed up by providing a merger history of the

galaxy-halo system to the machine. The choice of the

depth of history to be provided is generally the pre-

rogative of the machine learning practitioner. How-

ever, this choice comes with a few caveats. Since

galaxy clusters generally formed in a very early uni-

verse, their merger histories usually cover billions of

years and involve thousands of progenitors. Provid-

ing a description of all the progenitors of any galaxy

is simply an impractical task. A good way to approach

this choice is by constraining the number of progen-

itors of a galaxy (subhalo) and providing their corre-

sponding properties only for a subset of the cosmic

time. This is done keeping in mind that even though a

subhalo may have thousands of progenitors and con-

tinuously morphs through multiple collisions and ac-

cretions, only a few of its progenitors play an over-

whelming role in its overall shape and so only these

few progenitors are sufﬁcient to indicate the overall

lineage of the subhalo. A partial merger history is

choosing how far to travel along the main branch of

a galaxy. As shown in Figure 1, the morphology

of a galaxy at some redshift, is the result of evolu-

tion along many branches, but its protogalaxies along

the main branch can adequately trace the history of a

galaxy.

An alternate approach is to provide the algorithm

with only a few random snapshots of the universe cor-

responding to different look-back times. In this ap-

proach, merger history need not be provided. The al-

gorithm learns the underlying generative model which

can be subsequently used to infer the morphology of

galaxies at different redshifts. The motivation behind

this is the drastic reduction in the dimensionality of

the dataset. This then reduces the computation time.

Table 1: Features of dark and baryonic matter.

Dark Matter Features

Feature Description

Halo Group

Mass

Aggregate Group Mass of all subhalos

within a larger halo

Mass Critical

200, M

200

Deﬁnes the mass of a halo

Radius Critical

200, R

200

The Radius that bounds Mass Critical

200

Number of

Subhalos

Representative of the number of smaller

subhalos that make up a larger halo

Baryonic Matter Features

Feature Description

Black Hole

Mass

The mass of the central black hole in a

halo

Stellar Mass Representative of the stellar content of

a galaxy

Velocity Dis-

persion

Provides a measure of velocity of a

galaxy

Maximum of

Circular Veloc-

ity, Vmax

Maxima of the circular velocity curve

of a galaxy

In this paper, we ﬁnd the set of progenitors of a

galaxy along the main branch between redshift 0 and

0.5 sufﬁcient for our purposes. We provide progenitor

history in our ﬁrst approach. In our second approach

to model the relation between dark and baryonic mat-

ter, we do not provide progenitor history at all. The

dataset construction, as well as a reporting of the re-

sults in discussed in a subsequent section.

The added advantage of using a graphical model is

the greater interpretability. SPNs augment this by al-

lowing probabilistic semantics even when there are no

conditional dependencies present (Butz et al., 2017),

while guaranteeing inference in time linear in tree

width of the network. We generate the dataset using

the results of the Eagle suite of smoothed particle hy-

drodynamical simulations (Schaye et al., 2014).

In the interest of space, we do not go into the de-

tails of SPNs, and refer interested readers to (Poon

Tractable Generative Modelling of Cosmological Numerical Simulations

903

and Domingos, 2011). These data structures allow

the modes and marginals of a probability distribution

to be computed efﬁciently. (See (Kisa et al., 2014)

for other data structures with such properties.) More-

over, the size, shape and the weights of the network

can also be learned from the data, either discrimi-

natively or generatively (Gens and Domingos, 2012;

Gens and Domingos, 2013). In this work, we learn

generatively with the leaf nodes of our SPN explicitly

encoded to contain univariate Bernoulli distributions.

This is however not a strict requirement: as shown

in works such as (Molina et al., 2017; Molina et al.,

2018; Rashwan et al., 2016; Hsu et al., 2017; Bueff

et al., 2018), SPNs can also be learned online with

Gaussian and other distributions, which might be use-

ful for future work on modelling physical phenomena

via generative models.

4 EMPIRICAL EVALUATIONS

Simulation Overview. The suite of Eagle simula-

tions (Schaye et al., 2014) uses a modiﬁed version

of Gadget3 hydrodynamical code, last described in

(Springel, 2005), to evolve resolution elements in

boxes of size 12, 25, 50 and 100 comoving mega

parsecs (cMpc) on a side. The cosmology employed

in the simulations is consistent with the results of

(Planck et al., 2014), where Ω

= 0.693, Ω

= 0.307,

Ω

= 0.04825, σ

= 0.8288, n

= 0.9611, h = 0.677,

where, Ω

,Ω

,σ

, n

, h stand for the contribu-

tions to matter/energy content of the universe from

cosmological constant, matter, baryons respectively,

h is the dimensionless Hubble parameter, n

is the

spectral index of the primordial power spectrum while

is the rms amplitude of the linear mass ﬂuctua-

tions. High resolution simulations correspond to sim-

ulations with an initial baryonic particle mass of m

2.26 x 10

⊙

while intermediate resolution simula-

tions have a higher initial baryonic particle mass, m

= 1.81 ∗ 10

⊙

, where M

⊙

is 1 solar mass.

The key run of the simulations, which we use in

this paper, the Fiducial Ref-L0100N1504 simulation

is an intermediate resolution simulation with periodic

box with a volume of (100cMpc)

, initially containing

1504

gas particles, with an initial mass of 1.81 ∗ 10

⊙

and the same amount of dark matter particles with

9.70 ∗ 10

⊙

Substructures, including galaxies, in Eagle simu-

lations were identiﬁed using the SUBFIND algorithm

(developed in (Springel et al., 2001)). First, halos

were detected with the Friends-of-Friends (FOF) al-

gorithm (More et al., 2011) on dark matter particles,

with a linking length of 0.2 times the mean interpar-

ticle separation. Gas and star particles were assigned

to the same halo as their nearest dark matter particles.

Next, SUBFIND identiﬁed substructure candidates by

locating overdense regions within halos, deﬁned by

saddle points in the density distribution. Finally, grav-

itationally unbound particles were removed, and the

remaining substructures were classiﬁed as galaxies.

The simulations themselves have a ﬁnite resolu-

tion and are generally not reliable on lower mass

range of satellite galaxies and dwarf halos; the

physics on lower scales is more inﬂuenced by feed-

back effects and stellar winds which are poorly un-

derstood and have no analytic solutions. In general,

many galaxy properties are unreliable below a stellar

mass of 10

⊙

. Thus, we only select central galaxies

with halo mass above 10

⊙

. For a comprehensive

discussion on the parameters of the simulation, we re-

fer readers to (Schaye et al., 2014).

4.1 Feature Engineering

Modeling galaxy formation involves complex physi-

cal processes like supernovae, accretion shocks, and

stellar winds, which require approximations and par-

tial differential equations. Probabilistic machine

learning can help model the interactions between

baryonic and dark matter, capturing their joint dynam-

ical evolution. However, feature selection remains

challenging due to the high-dimensional nature of

cosmological datasets and the inﬂuence of non-linear

processes.

Feature selection in cosmology relies heavily on

domain knowledge, particularly because most of the

universe’s energy and matter is dark. A simpliﬁed

galaxy model includes four components: dark matter

halo, stellar halo, central black hole, and stellar bulge.

The virial radius, which represents the galaxy’s size,

is often used heuristically but may not always be ac-

curate, especially for galaxies undergoing tidal stress

or collisions.

A generative model of dark and baryonic mat-

ter is crucial for understanding their distribution and

the mapping between them. For instance, predicting

baryonic content based on a halo’s merger history is

challenging, particularly considering the peak of star

formation at redshift 1–2.

The baryonic features we model as random vari-

ables are:

• Black hole mass: Modeled in simulations with

feedback from active galactic nuclei, where black

holes grow through mergers and accretion.

• Stellar mass: Determined within a 30 kpc aper-

ture, aligning with observed data from the Galaxy

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904

And Mass Assembly (Baldry et al., 2012) and

SDSS (Li and White, 2010).

• Velocity dispersion: Calculated as

/3M,

with E

being kinetic energy and M the stellar

mass within the aperture.

• Maximum circular velocity: Derived using

(r) =

M(<r)

, where M(< r) is the enclosed

mass at radius r.

Dark matter features include:

• Halo mass (M

200

) and radius (R

200

), deﬁned at the

virial radius.

• Halo group mass, referring to the total mass of

dark matter subhalos within a group, as identiﬁed

by the SUBFIND and FOF algorithms.

4.2 Dataset Construction

Since our method involves two different approaches,

we construct four datasets by querying both the ﬁdu-

cial and dark matter-only models in the database for

the properties of sub-halos (galaxies) with their cor-

responding dark matter halos and halos only.

The ﬁrst approach, where we provide a merger

history, corresponds to Dataset 1 and Dataset 3. With

Dataset 1, we provide SPNs with a selection of prop-

erties of the central galaxy at zero redshift in each halo

along with a description of their corresponding cen-

tral subhalo merger history from redshift 0 to redshift

0.50. This is equivalent to providing the halo history

for approximately the last 5 billion years. The merger

tree was traversed only along the main branch, see

Figure 1, of every galaxy.

The galactic properties we model are the mass of

its central black hole, stellar mass, velocity dispersion

of the stars and the maximum of the circular velocity

rotation curve of the galaxy. Dataset 3 was generated

in a similar way through the Dark Matter-Only snap-

shots in the Eagle simulations. In Dataset 3, we only

use halo properties and halo merger histories, from

redshift 0 to redshift 0.50, as inputs and query for

properties redshift 0. The common factors in Dataset

1 and Dataset 3 are the halo properties and merger

histories.

In the second approach, applied to Datasets 2 and

4 where halo history was not provided, SPN generates

models of matter distribution from snapshots. These

datasets, created from the ﬁducial and dark matter-

only runs, focus on galaxy-halo systems and halos

between redshifts 3.5 and 1.7. This redshift range

was chosen to model the universe more accurately, as

star formation peaked during this period. The genera-

tive model trained on this data was tested by querying

Figure 1: Merger history of a galaxy with stellar mass,

star

> 10

⊙

. Figure from the Eagle Database (McAlpine

et al., 2016). A galaxy’s present state results from mergers

over billions of years. Redshift 0 represents the present,

while redshift 10 corresponds to a 12 Gigayear lookback

time. The galaxy’s merger history follows a main pro-

genitor branch, shown as a thick black line in the ﬁgure.

The Descendant ID represents a galaxy at a speciﬁc time,

while the TopLeafID indicates the ﬁrst progenitor along the

main branch. All other branches are indicated with a thin

line. The merger history is traced along the main progenitor

branch.

properties at redshift 0 to evaluate how well SPN cap-

tures the matter distribution. The use of dark matter-

only simulations helps assess how well SPNs approx-

imate N-body calculations, given the lack of an ana-

lytic solution and the reliance on energy and momen-

tum conservation for convergence.

5 ANALYSIS

In this section, we present and discuss the results

obtained when applying the algorithm to the Eagle

data. Using dark matter internal halo properties as

inputs, we predict the following baryonic features:

black hole mass, stellar mass, velocity dispersion, and

max

. These attributes result from billions of years

of evolution through dissipative, nonlinear baryonic

processes. While large-scale structure formation fol-

lows the ΛCDM model, smaller scales are vastly more

complex.

Once the SPN captures the joint distribution over

all the variables at its root node, it can be queried for

conditional and marginal likelihoods of any random

variable, such as stellar mass or central black hole

mass. The SPN’s root node can also generate syn-

thetic datasets following the learned joint distribution.

Out trained SPN has 143 edges with 144 nodes in 20

layers. The network has 19 sum nodes, 40 product

nodes and 85 leaf nodes each modeling a univariate

Bernoulli distribution. The SPN took 149 seconds to

be learned.

Tables 2 and 3 show minimal difference in errors

Tractable Generative Modelling of Cosmological Numerical Simulations

905

Table 2: Dataset 1: The structure of SPN for this dataset was learned in 847.6 seconds. Progenitor history was provided.

Feature MSE R

Accuracy Score PearsonR

Central Black Hole

Mass

0.041714 0.464182 0.958286 0.743506

Stellar Mass 0.019964 0.732150 0.980036 0.870518

Velocity Dispersion 0.118812 0.464540 0.881188 0.727086

max

0.065533 0.680239 0.934467 0.837789

Table 3: Dataset 2: The structure of SPN for this dataset was learned in 144.7 seconds. Only random snapshots were provided.

Feature MSE R

Accuracy Score PearsonR

Central Black Hole

Mass

0.039717 0.469593 0.960283 0.735701

Stellar Mass 0.019542 0.727792 0.980458 0.867607

Velocity Dispersion 0.107178 0.512861 0.892822 0.751796

max

0.055159 0.728921 0.944841 0.863211

between the two approaches. However, the computa-

tion time to learn the joint distribution is much shorter

when only snapshots are provided. Merger histories

do not signiﬁcantly enhance model richness. Tables

4 and 5 present similar results for dark matter proper-

ties.

The results show that SPNs can recreate mock cat-

alogs with properties similar to those from hydrody-

namic codes. Baryonic properties, which are mass-

dependent, are predicted accurately, with the central

black hole and stellar mass linearly related to halo

mass M

200

, and velocity dispersion and V

max

governed

by mass and radius. The predicted and true distri-

butions for stellar mass and central black hole mass

match closely.

A key observation is that progenitor history does

not improve prediction accuracy, even with increased

computation time. For example, the mean squared

errors for stellar mass are nearly identical with and

without progenitor history, but computation time in-

creases with progenitor history.

This trend also holds for dark matter-only runs,

where errors in subhalo number and halo group mass

are similar with or without progenitor history, though

training with progenitor history takes longer.

Overall, the results are surprising given the com-

plexity of numerical simulations. While our model

cannot replace numerical simulations, it provides a

useful tool for exploring the galaxy-halo connection

and the impact of different simulation physics, as seen

in semi-analytic modeling.

6 DISCUSSION AND

CONCLUSIONS

We conducted an empirical study to explore the rela-

tionship between dark matter halos and their enclosed

galaxies using a Sum-Product Network (SPN), a prob-

abilistic graphical model, in the context of a large

cosmological hydrodynamic simulation. Our study

demonstrates that SPNs offer signiﬁcant computa-

tional savings, making predictions in minutes com-

pared to the millions of CPU hours required by hy-

drodynamical simulations. The model accurately pre-

dicts baryonic properties like stellar mass and central

black hole mass, with strong R² and Pearson correla-

tion metrics. Additionally, SPNs generate synthetic

datasets, enabling further exploration of galaxy-halo

relationships. Comparing different approaches, we

found that using random snapshots instead of progen-

itor histories does not greatly affect accuracy. SPNs’

hierarchical structure and probabilistic nature provide

enhanced interpretability over other machine learning

models.

However, the model is insensitive to progenitor

histories, questioning its ability to capture complex

baryonic feedback. Its phenomenological nature lim-

its its ability to simulate processes like AGN feed-

back or star formation. The need for extensive domain

knowledge and the ﬁnite resolution of the Eagle sim-

ulations also limit its generalizability. Future work

could combine SPNs with physics-based constraints,

test on other simulations or observational data, and

develop methods to better model temporal dependen-

cies. Despite these challenges, SPNs show promise

for tractable generative modeling in cosmology, of-

fering efﬁciency and accuracy while complementing

traditional simulations.

The aim of this work was to assess how dark mat-

ter properties can inform the evolutionary properties

of galaxies, rather than replicating a numerically iden-

tical population. The results suggest SPNs can ef-

fectively mimic galaxy evolution in a hydrodynamic

context, with runtimes in the order of minutes versus

the millions of hours required by simulations. This

highlights the potential of probabilistic models in an-

alyzing complex physical phenomena and their role

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

906

Table 4: Dataset 3: The structure of SPN for this dataset was learned in 1890.15 seconds. Dark Matter Only run with halo

history.

Feature MSE R

Accuracy Score PearsonR

Number of Subhalos 0.053304 0.442150 0.946696 0.701084

Halo group Mass 0.014672 0.799276 0.985328 0.905383

200

0.005449 0.929433 0.994551 0.965560

200

0.012702 0.938336 0.987298 0.969134

Table 5: Dataset 4: The structure of SPN for this dataset was learned in 149.5 seconds. Dark Matter Only run with just

snapshots.

Feature MSE R

Accuracy Score PearsonR

Number of Subhalos 0.051744 0.464274 0.948256 0.714493

Halo group Mass 0.015195 0.793547 0.974805 0.914319

200

0.004964 0.933783 0.994036 0.963175

200

0.010756 0.947684 0.989244 0.973857

in testing machine learning models. Future work will

explore advanced algorithms to further integrate ma-

chine learning in cosmology.

ACKNOWLEDGEMENTS

This work is partly supported by the EPSRC grant To-

wards Explainable and Robust Statistical AI: A Sym-

bolic Approach

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