\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\newlabel{FirstPage}{{}{1}{}{Doc-Start}{}}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}\protected@file@percent }
\newlabel{sec:intro}{{1}{1}{Introduction}{section.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Methods}{2}{section.2}\protected@file@percent }
\newlabel{sec:methods}{{2}{2}{Methods}{section.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Latent Space Data Preparation}{2}{subsection.2.1}\protected@file@percent }
\citation{ferreras2006principalcomponentanalysistool,ugwoke2023principalcomponentanalysisground,nandi2025impactcosmicwebgalaxy}
\citation{ferreras2006principalcomponentanalysistool,mcgurk2010principalcomponentanalysissdss}
\citation{ferreras2006principalcomponentanalysistool}
\citation{ferreras2006principalcomponentanalysistool,mcgurk2010principalcomponentanalysissdss}
\citation{paraficz2016pcalensfinderapplicationcfhtls,hatipo\unhbox \voidb@x \bgroup \let \unhbox \voidb@x \setbox \@tempboxa \hbox {g\global \mathchardef \accent@spacefactor \spacefactor }\let \begingroup \let \typeout \protect \begingroup \def \MessageBreak {
(Font)              }\let \protect \immediate\write \m@ne {LaTeX Font Info:     on input line 54.}\endgroup \endgroup \relax \let \ignorespaces \relax \accent 21 g\egroup \spacefactor \accent@spacefactor lu2022utilitypcadatatransformation,cretignier2022stellaractivitycorrectionusing,sharma2024inferencecosmologicalmodelsprincipal}
subsubsection}{\numberline {2.2.1}Principal Component Analysis (PCA)}{3}{subsubsection.2.2.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.2.2}Subspace Similarity Measures}{3}{subsubsection.2.2.2}\protected@file@percent }
\citation{shukla2021geometricpriorsscientificgenerative,rissaki2024reconstructingmidinfraredspectragalaxies}
\citation{ferreras2006principalcomponentanalysistool,nandi2025impactcosmicwebgalaxy}
\citation{damiano2019principalcomponentanalysisbasedmethod}
\citation{nandi2025impactcosmicwebgalaxy}
\citation{ferreras2006principalcomponentanalysistool,damiano2019principalcomponentanalysisbasedmethod,nandi2025impactcosmicwebgalaxy}
\citation{ferreras2006principalcomponentanalysistool,nandi2025impactcosmicwebgalaxy}
\citation{nandi2025impactcosmicwebgalaxy}
\citation{ferreras2006principalcomponentanalysistool,nandi2025impactcosmicwebgalaxy}
\citation{nandi2025impactcosmicwebgalaxy}
\citation{nandi2025impactcosmicwebgalaxy}
\citation{diclemente2025explainableautoencoderneutronstar,diclemente2025explainableautoencoderneutronstar}
\citation{cuomo2022scientificmachinelearningphysicsinformed}
\HyPL@Entry{3<</P(\376\377\0004)>>}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Analysis Workflow}{4}{subsection.2.3}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.1}Initial Exploratory Data Analysis}{4}{subsubsection.2.3.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.2}Characterization of Individual Manifolds}{4}{subsubsection.2.3.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.3}Comparative Analysis Across Initial Conditions}{4}{subsubsection.2.3.3}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.4}Relation to Global Latent Space Structure}{4}{subsubsection.2.3.4}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.5}Synthesis}{4}{subsubsection.2.3.5}\protected@file@percent }
\HyPL@Entry{4<</P(\376\377\0005)>>}
\@writefile{toc}{\contentsline {section}{\numberline {3}Results}{5}{section.3}\protected@file@percent }
\newlabel{sec:results}{{3}{5}{Results}{section.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Global structure of the latent space}{5}{subsection.3.1}\protected@file@percent }
\newlabel{fig:global_pca_scree}{{1}{5}{Global structure of the latent space}{figure.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Intrinsic dimensionality of per initial condition manifolds}{5}{subsection.3.2}\protected@file@percent }
\HyPL@Entry{5<</P(\376\377\0006)>>}
\newlabel{fig:per_ic_scree}{{2}{6}{Intrinsic dimensionality of per initial condition manifolds}{figure.2}{}}
\newlabel{fig:intrinsic_dim}{{3}{6}{Intrinsic dimensionality of per initial condition manifolds}{figure.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Geometric arrangement of manifold centroids}{6}{subsection.3.3}\protected@file@percent }
\newlabel{fig:centroid_pca_scree}{{4}{6}{Geometric arrangement of manifold centroids}{figure.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4}Comparison of manifold orientations}{6}{subsection.3.4}\protected@file@percent }
\HyPL@Entry{6<</P(\376\377\0007)>>}
\newlabel{fig:centroid_pca_2d}{{5}{7}{Geometric arrangement of manifold centroids}{figure.5}{}}
\newlabel{fig:centroid_pca_3d}{{6}{7}{Geometric arrangement of manifold centroids}{figure.6}{}}
\newlabel{fig:subspace_similarity}{{7}{7}{Comparison of manifold orientations}{figure.7}{}}
\HyPL@Entry{7<</P(\376\377\0008)>>}
\newlabel{fig:dot_product_heatmaps}{{8}{8}{Comparison of manifold orientations}{figure.8}{}}
\newlabel{fig:orientation_pca}{{9}{8}{Comparison of manifold orientations}{figure.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.5}Relationship between per initial condition structures and global structure}{8}{subsection.3.5}\protected@file@percent }
\newlabel{fig:variance_in_global_subspace}{{10}{8}{Relationship between per initial condition structures and global structure}{figure.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.6}Synthesis and interpretation}{8}{subsection.3.6}\protected@file@percent }
\HyPL@Entry{8<</P(\376\377\0009)>>}
\newlabel{fig:projected_centroids_2d}{{11}{9}{Relationship between per initial condition structures and global structure}{figure.11}{}}
\newlabel{fig:projected_centroids_3d_global}{{12}{9}{Relationship between per initial condition structures and global structure}{figure.12}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.7}Limitations and future directions}{9}{subsection.3.7}\protected@file@percent }
\bibdata{bibliography}
\HyPL@Entry{9<</P(\376\377\0001\0000)>>}
\@writefile{toc}{\contentsline {section}{\numberline {4}Conclusions}{10}{section.4}\protected@file@percent }
\newlabel{sec:conclusions}{{4}{10}{Conclusions}{section.4}{}}
\bibcite{diclemente2025explainableautoencoderneutronstar}{{1}{2025}{{Clemente {et~al.}}}{{Clemente, Scialpi, \& Bejger}}}
\bibcite{cretignier2022stellaractivitycorrectionusing}{{2}{2022}{{Cretignier {et~al.}}}{{Cretignier, Dumusque, \& Pepe}}}
\bibcite{cuomo2022scientificmachinelearningphysicsinformed}{{3}{2022}{{Cuomo {et~al.}}}{{Cuomo, di~Cola, Giampaolo, Rozza, Raissi, \& Piccialli}}}
\bibcite{damiano2019principalcomponentanalysisbasedmethod}{{4}{2019}{{Damiano {et~al.}}}{{Damiano, Micela, \& Tinetti}}}
\bibcite{ferreras2006principalcomponentanalysistool}{{5}{2006}{{Ferreras {et~al.}}}{{Ferreras, Rogers, Lahav, \& .}}}
\bibcite{hatipoğlu2022utilitypcadatatransformation}{{6}{2022}{{Hatipoğlu}}{{}}}
\bibcite{hatipoğlu2023pcabaseddatareductionsignal}{{7}{2023}{{Hatipoğlu}}{{}}}
\bibcite{mcgurk2010principalcomponentanalysissdss}{{8}{2010}{{McGurk {et~al.}}}{{McGurk, Kimball, \& Ivezic}}}
\bibcite{nandi2025impactcosmicwebgalaxy}{{9}{2025}{{Nandi \& Pandey}}{{}}}
\bibcite{paraficz2016pcalensfinderapplicationcfhtls}{{10}{2016}{{Paraficz {et~al.}}}{{Paraficz, Courbin, Tramacere, Joseph, Metcalf, Kneib, Dubath, Droz, Filleul, Ringeisen, \& Schäfer}}}
\bibcite{rissaki2024reconstructingmidinfraredspectragalaxies}{{11}{2024}{{Rissaki {et~al.}}}{{Rissaki, Pavlou, Fotakis, Lesta, \& Efstathiou}}}
\bibcite{sharma2024inferencecosmologicalmodelsprincipal}{{12}{2024}{{Sharma \& Jassal}}{{}}}
\bibcite{shukla2021geometricpriorsscientificgenerative}{{13}{2021}{{Shukla {et~al.}}}{{Shukla, Anirudh, Kur, Thiagarajan, Bremer, Spears, Ma, \& Turaga}}}
\bibcite{ugwoke2023principalcomponentanalysisground}{{14}{2023}{{Ugwoke {et~al.}}}{{Ugwoke, Ubachukwu, Urama, Okike, Alhassan, \& Chukwude}}}
\bibcite{çakır2024megsmorphologicalevaluationgalactic}{{15}{2024}{{Çakır \& Buck}}{{}}}
\HyPL@Entry{10<</P(\376\377\0001\0001)>>}
\bibstyle{aasjournal}
\newlabel{LastPage}{{}{11}{}{}{}}
\gdef \@abspage@last{11}
