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<!DOCTYPE HTML>
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<h1><a href="index.html"> The <strong>EXP</strong> Collaboration </a></h1>
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<li><a href="#footer" class="icon solid fa-info-circle">About</a></li>
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<article class="thumb">
<a href="images/fulls/plain.png" class="image"><img src="images/thumbs/1.png" alt="" /></a>
<h2> What is a basis function expansion? </h2>
<p> In the Milky Way, the density, potential, velocity, energy, angular momentum and chemical abundance
patterns fluctuate across the disk. More generally, fields in galaxies – and other physical systems –
often vary in complex manners that cannot be captured with simple, analytic formulae. In simulations,
these fluctuations are described in simulations by particles representing random draws from the full
field. In the Milky Way, the fluctuations are captured in the spatial density of stars. In
observations of other galaxies, the fluctuations are reflected in the properties of light captured in
a pixel. A full description often requires millions or even hundreds of millions of numbers. </p>
<p> A well-designed Basis Function Expansion can typically capture the salient features of a complex
field in 1000’s of numbers, providing a succinct description. It does so by representing the full
field in a series of basis functions that span function space. The information in the field is
captured in the coefficients of each basis function in the expansion. For large enough numbers of
basis functions, any field can be represented. However, BFE’s become truly powerful if they can be
tailored to the application, so the information can be captured in as few terms as possible. </p>
<p> The best-known examples of a BFE are Fourier series of sines and cosines, which are very good at
capturing variations in time or space that repeat with a small set of regular periods. In a galactic
dynamics context, the optimal basis will resemble the galaxy, with additional functions to capture
variance or deviations from the simplest axisymmetric model. In N-body simulations of galaxies, BFEs
have been used to derive potential fields at each timestep from particle data at computational effort
proportional to the number of particles – drastically less computationally intense than other
techniques to determine potentials
<a href="https://ui.adsabs.harvard.edu/abs/1992ApJ...386..375H/abstract">Hernquist (1992)</a>. </p>
<p> More generally: in theoretical analyses, BFE have been partnered with mathematical tools of
perturbation theory and linear algebra to solve equations, to describe interactions and identify
physical mechanisms such as in the interaction of the Milky Way and Large Magellanic Cloud
(<a href="https://ui.adsabs.harvard.edu/abs/2006ApJ...641L..33W/abstract">Weinberg & Blitz 2006</a>).
Additionally, BFE have been used in observations to compress vast data sets and allow interpretation,
such as in the power spectrum of fluctuations in the temperature of the Cosmic Microwave Background
(<a href=”https://ui.adsabs.harvard.edu/abs/2007ApJS..170..377S/abstract”>Spergel et al. 2007</a>). </p>
</article>
<article class="thumb">
<a href="images/fulls/plain.png" class="image"><img src="images/thumbs/2.png" alt="" /></a>
<h2> <b> Exp </b> = Adaptive BFEs: precision and concision in the language</h2>
<p> <b> Exp </b> provides numerical tools that derive efficient representations of BFEs from linear combinations
of an initial set of functions based on the character of the data, providing a concise description that
minimizes the degrees of freedom while efficiently capturing the properties of the fields. At the same
time, the description is more precise in its representation of the fields. These distillations provide
opportunities to store and reuse key dynamical content in easy-to-reconstruct field form. Applications
include resampling phase space at higher resolution than the original simulation, replaying the
time-evolving fields to study their influence on ensembles of orbits that may represent stellar streams,
star clusters, dwarf galaxies, dark matter substructure, just to name a few. </p>
</article>
<article class="thumb">
<a href="images/fulls/plain.png" class="image"><img src="images/thumbs/3.png" alt="" /></a>
<h2> <b> Exp </b> = BFE+mSSA: finding the story being told by BFE’s </h2>
<p> <b> EXP </b> also provides tools that correlate the morphology and time dependence of dominant features contributing
to the evolution of a field from multiple sets of expansion coefficients. By adding correlations in the time
domain to the correlations represented by the BFEs, the dynamical content of temporal variation becomes manifest.
This automatic spatio-temporal discovery is a form of unsupervised learning and has already led to
discovery of new, previously unknown, dynamics in our simulations. <b> EXP </b> implements multivariate Singular
Spectrum Analysis (SSA) – an unsupervised machine learning algorithm – tailored to basis-function expansions.
SSA decomposes the BFE variation in time into interpretable components and provides for spectral estimation
without specific assumptions about the time dependence of the system. We also provide some preliminary
support for dynamical mode decomposition (DMD) and other Koopman-related techniques. </p>
</article>
<article class="thumb">
<a href="images/fulls/plain.png" class="image"><img src="images/thumbs/4.png" alt="" /></a>
<h2> <b> EXP </b>: applications to cosmology</h2>
<p> <b> EXP </b> can be used to analyze structure in cosmological simulations. Members of the <b> EXP </b> Collaboration are applying
these tools to snapshots from simulations of galaxy formation to: </p>
<ul>
<li> Compare and contrast the signatures of filamentary accretion from halo deformation in the FIRE simulation suite
(<a “https://ui.adsabs.harvard.edu/abs/2025ApJ...988..190A/abstract” Arora et al, 2025 </a>); </li>
<li> Describe the deformation of dark matter halos as they respond to infalling satellites in the MWest simulation suite
(Darragh-Ford et al 2025, in prep)</li>
<li> Characterise the effect of deforming dark matter halos on the structural properties of disks in the Auriga simulation
suite (Lavin et al 2025, in prep); </li>
<li> Investigate the interplay between dynamical structure formation, dark matter physics, and feedback mechanisms in the
<a “https://dreams-project.readthedocs.io/en/latest/index.html” DREAMS<a/> suite of cosmological simulations
(Filion et al 2025, in prep) </li>
</ul>
</article>
<article class="thumb">
<a href="images/thumbs/center-2.png" class="image"><img src="images/thumbs/5.png" alt="" /></a>
<h2> <b> Exp </b>: the code and the collaboration </h2>
<p><b> Exp </b> is designed to connect: (1) theoretical descriptions of dynamics, (2) N-body simulations, and (3) data-efficient descriptions of
their natural consequences. <b> Exp </b> provides recent developments from applied mathematics and numerical computation to represent time
series of BFEs that describe the temporal variation of any field in space. In the context of galactic dynamics, these fields may be density,
potential, force, velocity fields or any intrinsic field produced by simulations such as chemistry data. By combining the coefficient
information through time using spectral analysis, we hope to discover the dynamics of galaxy evolution directly from simulations and by
predictive comparison with observed data. </p>
<p> The <b> Exp Collaboration</b> is exploring the applications of these tools to galactic dynamics, from the Milky Way disk to cosmological
simulations of galaxy formation. Our team combines expertise in (i) analytic models, (ii) numerical simulations and (iii) data analysis.
We aim to build a language that unites all three. We use <b> Basis Function Expansions (BFE) </b> to compactly summarize spatial or
velocity features in simulations and <b> multivariate Singular Spectrum Analysis (mSSA)</b> to discover the non-linear dynamics of their
interaction. This allows us to detect deep dynamical relationships in our simulations. The approach promises multiple connections: from
observations to simulations to theoretical descriptions; between galactic components; and across phase-space dimensions. </p>
</article>
<article class="thumb">
<a href="images/fulls/plain.png" class="image"><img src="images/thumbs/5.png" alt="" /></a>
<h2> <b> EXP</b>: applications to dynamical systems </h2>
<p> Density/potential pairs of BFEs that are solutions to Poisson’s equation provide a natural
language for dynamics. They are used both in simulations and analytic calculations, and
hence can be used to link the two. </p>
<p> In addition to its analysis framework, <b> EXP </b> includes an N-body gravitational code. The resulting
simulations output both particle-based snapshot data and BFE information, including the basis and
time-evolving coefficients. The theory underpinning <b> EXP </b> simulations and the implementation are
discussed in more detail in the
<a href="https://exp-docs.readthedocs.io/en/latest/topics/multistep.html"readthedocs></a>, as well
as these papers (<a href="https://ui.adsabs.harvard.edu/abs/1999AJ....117..629W/abstract" Weinberg, 1999</a>,
<a href="https://ui.adsabs.harvard.edu/abs/2022MNRAS.510.6201P/abstract" Peterxen, Weinberg & Katz, 2022></a>).</p>
<p> The collaboration has applied <b> EXP </b> to various simulations to: </p>
<ul>
<li> Follow the evolution of galactic bars (<a”https://ui.adsabs.harvard.edu/abs/2021MNRAS.501.5408W/abstract” Weinberg & Petersen 2020</a>),
as well as its interaction with a dark matter halo (<a “https://ui.adsabs.harvard.edu/abs/2025arXiv251009751H/abstract” Hunt et al, 2025></a>)</li>
<li> Distinguish intrinsic halo instabilities from evolution driven by disk/halo coupling in the simulation of an isolated galaxy
(https://ui.adsabs.harvard.edu/abs/2023MNRAS.521.1757J/abstract Johnson, Petersen et al 2023);</li>
<li> Isolate the signatures of multiple interacting satellites in a simulated galactic disk (Petersen et al 2025, in prep);</li>
<li> Connect features found in phase-space local patches of a simulated disk into global structures (Tavangar et al 2025, in prep);</li>
<li> Characterize the morphological transformation of the SMC and LMC as they orbit our Milky Way (Rathore et al 2025, in prep)</li>
</ul>
</article>
<article class="thumb">
<a href="images/fulls/plain.png" class="image"><img src="images/thumbs/8.png" alt="" /></a>
<h2> <b> EXP</b>: applications to observations</h2>
<p>Two dimensional basis function expansions can also be performed on observational data. As shown in
<a href=”https://ui.adsabs.harvard.edu/abs/2025MNRAS.539..661G/abstract”>Ganapathy et al 2025</a>, 2D
expansions on image data can be used to describe the light (stellar) distribution in a galaxy, and
provide a language for succinctly, quantitatively summarizing the morphological features. A
Fourier-Laguerre basis is a natural choice for expanding imaging data, capturing both the angular
(Fourier) and radial (Laguerre) information. Expansions using this basis can be used to quantify
lopsidedness in galaxies
(e.g. <a href=”https://ui.adsabs.harvard.edu/abs/2025MNRAS.539..661G/abstract”>Ganapathy et al 2025</a>),
measure galaxy inclination (e.g. Martinez et al, in prep), and identify morphological features like bars.
These expansions are also how we map an image of a
(<a href=”https://carriefilion.github.io/#Sonification”>into a sound </a>) via sonification. Similarly,
we can perform expansions of integral field spectrograph data, which allow for analyses of both velocity
and chemical information. </p>
</article>
<article class="thumb">
<a href="images/fulls/plain.png" class="image"><img src="images/thumbs/9.png" alt="" /></a>
<h2> Looking Forward</h2>
<ul>
<li>Time series analyses more generally </li>
<ul>
<li>DMD</li>
<li>Advanced mSSA techniques</li>
</ul>
<li>Other fields in galaxies</li>
<ul>
<li>Velocity</li>
<li>Chemistry</li>
<li>Ages</li>
</ul>
<li>Other applications beyond galaxies</li>
<ul>
<li>Accretion disks</li>
<li>Planetary disks</li>
<li>Astrophysical Plasmas</li>
<li>Proto-stellar disks</li>
</ul>
</ul>
</article>
<article class="thumb">
<a href="images/fulls/plain.png" class="image"><img src="images/thumbs/7.png" alt="" /></a>
<h2>How to get started</h2>
<p>We have built and compiled a variety of resources to help you get started with <b> Exp </b> and basis function expansions!</p>
<p> Check out our <a href="https://github.com/EXP-code">GitHub page</a> and accompanying
<a href="https://exp-docs.readthedocs.io/en/latest/topics/multistep.html"readthedocs></a> for how to install EXP.</p>
<p> If you want to experiment with EXP, try out the <a href="https://github.com/EXP-code/EXP-container">Docker image</a> and
<a href="https://exp-docs.readthedocs.io/en/latest/topics/multistep.html"readthedocs></a> documentation. If you want to run pyEXP and EXP examples,
be sure to clone the respective repositories to wherever you are working with the Docker image </p>
<p>If you want to want to learn more about mSSA, check out this <a href="https://michael-petersen.github.io/papers/mssa/MSSA-Tutorial-Slideshow.slides.html#/">webpage</a> and these papers:
<a href="https://ui.adsabs.harvard.edu/abs/2021MNRAS.501.5408W/abstract">1</a> and
<a href="https://ui.adsabs.harvard.edu/abs/2023MNRAS.521.1757J/abstract">2</a> (see also the
<a href = "https://exp-docs.readthedocs.io/en/latest/topics/ssa.html"><b> EXP </b> readthedocs</a> for more information) </p>
</dl>
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<h2>Get to know us</h2>
<p>The <b> EXP </b> collaboration is developing the next generation of tools for galactic dynamics to tackle disequilibrium dynamics.
Our team combines expertise in (i) analytic models, (ii) numerical simulations and (iii) data analysis.
We use basis function expansions as a common language that unites all three of these realms of expertise.
Looking forward, our proposed tools have the potential
to enhance dynamical discovery within astrophysics more generally and on any scale. </p>
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<h3>Contact us on ...</h3>
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