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MULTIFACETED MATHEMATICS FOR PREDICTIVE DIGITAL TWINS
The overarching goal of the M2dt Center is to catalyze research and education on
multifaceted mathematical foundations of predictive digital twins for complex
systems.
A digital twin is a set of virtual information constructs that mimics the
structure, context, and behavior of a natural, engineered, or social system (or
system-of-systems), is dynamically updated with data from its physical twin, has
a predictive capability, and informs decisions that realize value. The
bidirectional interaction between the virtual and the physical is central to the
digital twin.
Data streams from the physical system are assimilated into the virtual
representation to reduce uncertainties and improve predictions, which in turn
are used as a basis for controlling the physical system, optimizing data
acquisition, and providing decision support. To support critical
high-consequence decisions, the DT must be trustworthy through a combination of
explainability and uncertainty quantification. Digital twins are key to
unlocking the full potential of decades of investment in computational modeling
– moving beyond forward simulation to underpin intelligent automation in complex
systems by supporting data-driven decision making.
The tight, dynamic interplay between computational models and physical systems,
the rapid time scales needed for decision making and control, the heterogeneity
of data sources that must be assimilated, the coupled nature of target systems,
and the need for robustness to data and model uncertainties to support
high-consequence decisions – these present frontier mathematical and
computational challenges for DTs and place them at the forefront of applied math
and computational science research today.
M2dt addresses these challenges via an integrated and multifaceted research
program that unites the complementary perspectives of physics-based
computational science (PDE solvers, data assimilation, reduced-order models,
optimal control, uncertainty quantification) with statistical/data sciences
(machine learning, graphical models, dimensionality reduction, optimal
experimental design, optimization) to advance the state of the art in these
areas and at their interfaces. In particular, we are pursuing an integrated and
multifaceted program of research on foundations of digital twins in:
1. new mathematical and statistical frameworks and computational algorithms,
and new representations blending graphical and PDE models, to enable
model-based inference from data and dynamic decision making for complex
systems;
2. new dimensionality reduction methods exploiting nonlinear manifolds;
3. new data-driven, machine learning-based, structure-preserving reduced-order
models and surrogates of high fidelity models that address high dimensional
state and parameter spaces;
4. new algorithms orchestrating use of hierarchies of models in digital twins,
given real-time and heterogeneous computing constraints on latencies,
bandwidth, response times, and data availability, on heterogeneous compute
platforms from edge devices to exascale systems; and
5. new decomposition algorithms to tackle the challenges of data assimilation
and optimal control for complex systems.
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<h1>Multifaceted Mathematics for Predictive Digital Twins</h1>
<p>The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex systems.</p>
<p>A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a set of virtual information constructs that mimics the structure, context, and behavior of a natural, engineered, or social system (or system-of-systems), is dynamically updated with data from its physical twin, has a predictive capability, and informs decisions that realize value. The bidirectional interaction between the virtual and the physical is central to the digital twin.</p>
<p>Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used as a basis for controlling the physical system, optimizing data acquisition, and providing decision support. To support critical high-consequence decisions, the DT must be trustworthy through a combination of explainability and uncertainty quantification. Digital twins are key to unlocking the full potential of decades of investment in computational modeling – moving beyond forward simulation to underpin intelligent automation in complex systems by supporting data-driven decision making.</p>
<p>The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity of data sources that must be assimilated, the coupled nature of target systems, and the need for robustness to data and model uncertainties to support high-consequence decisions – these present frontier mathematical and computational challenges for DTs and place them at the forefront of applied math and computational science research today.</p>
<p>M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational science (PDE solvers, data assimilation, reduced-order models, optimal control, uncertainty quantification) with statistical/data sciences (machine learning, graphical models, dimensionality reduction, optimal experimental design, optimization) to advance the state of the art in these areas and at their interfaces. In particular, we are pursuing an integrated and multifaceted program of research on foundations of digital twins in:</p>
<ol>
<li>new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based inference from data and dynamic decision making for complex systems;</li>
<li>new dimensionality reduction methods exploiting nonlinear manifolds;</li>
<li>new data-driven, machine learning-based, structure-preserving reduced-order models and surrogates of high fidelity models that address high dimensional state and parameter spaces;</li>
<li>new algorithms orchestrating use of hierarchies of models in digital twins, given real-time and heterogeneous computing constraints on latencies, bandwidth, response times, and data availability, on heterogeneous compute platforms from edge devices to exascale systems; and</li>
<li>new decomposition algorithms to tackle the challenges of data assimilation and optimal control for complex systems.</li>
</ol>
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"level" => "debug"
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"slack" => array:6 [▶
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"emoji" => ":boom:"
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2 => "/var/www/html/m2dt/resources/blueprints"
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7 => "/var/www/html/m2dt/resources/sites.yaml"
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9 => "/var/www/html/m2dt/public/assets"
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"php_memory_limit" => "-1"
"php_max_execution_time" => "0"
"ajax_timeout" => "600000"
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"groups" => "/var/www/html/m2dt/resources/users/groups.yaml"
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# Multifaceted Mathematics for Predictive Digital Twins\n
\n
The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex systems.\n ◀The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex s ▶
\n
A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a set of virtual information constructs that mimics the structure, context, and behavior of a natural, engineered, or social system (or system-of-systems), is dynamically updated with data from its physical twin, has a predictive capability, and informs decisions that realize value. The bidirectional interaction between the virtual and the physical is central to the digital twin.\n ◀A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a ▶
\n
Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used as a basis for controlling the physical system, optimizing data acquisition, and providing decision support. To support critical high-consequence decisions, the DT must be trustworthy through a combination of explainability and uncertainty quantification. Digital twins are key to unlocking the full potential of decades of investment in computational modeling – moving beyond forward simulation to underpin intelligent automation in complex systems by supporting data-driven decision making.\n ◀ Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used a ▶
\n
The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity of data sources that must be assimilated, the coupled nature of target systems, and the need for robustness to data and model uncertainties to support high-consequence decisions – these present frontier mathematical and computational challenges for DTs and place them at the forefront of applied math and computational science research today. \n ◀The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity ▶
\n
M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational science (PDE solvers, data assimilation, reduced-order models, optimal control, uncertainty quantification) with statistical/data sciences (machine learning, graphical models, dimensionality reduction, optimal experimental design, optimization) to advance the state of the art in these areas and at their interfaces. In particular, we are pursuing an integrated and multifaceted program of research on foundations of digital twins in:\n ◀M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational sc ▶
1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based inference from data and dynamic decision making for complex systems; \n ◀1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based in ▶
2. new dimensionality reduction methods exploiting nonlinear manifolds;\n
3. new data-driven, machine learning-based, structure-preserving reduced-order models and surrogates of high fidelity models that address high dimensional state and parameter spaces; \n ◀3. new data-driven, machine learning-based, structure-preserving reduced-order models and surrogates of high fidelity models that address high dimensional state ▶
4. new algorithms orchestrating use of hierarchies of models in digital twins, given real-time and heterogeneous computing constraints on latencies, bandwidth, response times, and data availability, on heterogeneous compute platforms from edge devices to exascale systems; and \n ◀4. new algorithms orchestrating use of hierarchies of models in digital twins, given real-time and heterogeneous computing constraints on latencies, bandwidth, ▶
5. new decomposition algorithms to tackle the challenges of data assimilation and optimal control for complex systems.
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# Multifaceted Mathematics for Predictive Digital Twins\n
\n
The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex systems.\n ◀The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex s ▶
\n
A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a set of virtual information constructs that mimics the structure, context, and behavior of a natural, engineered, or social system (or system-of-systems), is dynamically updated with data from its physical twin, has a predictive capability, and informs decisions that realize value. The bidirectional interaction between the virtual and the physical is central to the digital twin.\n ◀A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a ▶
\n
Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used as a basis for controlling the physical system, optimizing data acquisition, and providing decision support. To support critical high-consequence decisions, the DT must be trustworthy through a combination of explainability and uncertainty quantification. Digital twins are key to unlocking the full potential of decades of investment in computational modeling – moving beyond forward simulation to underpin intelligent automation in complex systems by supporting data-driven decision making.\n ◀ Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used a ▶
\n
The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity of data sources that must be assimilated, the coupled nature of target systems, and the need for robustness to data and model uncertainties to support high-consequence decisions – these present frontier mathematical and computational challenges for DTs and place them at the forefront of applied math and computational science research today. \n ◀The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity ▶
\n
M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational science (PDE solvers, data assimilation, reduced-order models, optimal control, uncertainty quantification) with statistical/data sciences (machine learning, graphical models, dimensionality reduction, optimal experimental design, optimization) to advance the state of the art in these areas and at their interfaces. In particular, we are pursuing an integrated and multifaceted program of research on foundations of digital twins in:\n ◀M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational sc ▶
1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based inference from data and dynamic decision making for complex systems; \n ◀1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based in ▶
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3. new data-driven, machine learning-based, structure-preserving reduced-order models and surrogates of high fidelity models that address high dimensional state and parameter spaces; \n ◀3. new data-driven, machine learning-based, structure-preserving reduced-order models and surrogates of high fidelity models that address high dimensional state ▶
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# Multifaceted Mathematics for Predictive Digital Twins\n
\n
The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex systems.\n ◀The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex s ▶
\n
A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a set of virtual information constructs that mimics the structure, context, and behavior of a natural, engineered, or social system (or system-of-systems), is dynamically updated with data from its physical twin, has a predictive capability, and informs decisions that realize value. The bidirectional interaction between the virtual and the physical is central to the digital twin.\n ◀A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a ▶
\n
Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used as a basis for controlling the physical system, optimizing data acquisition, and providing decision support. To support critical high-consequence decisions, the DT must be trustworthy through a combination of explainability and uncertainty quantification. Digital twins are key to unlocking the full potential of decades of investment in computational modeling – moving beyond forward simulation to underpin intelligent automation in complex systems by supporting data-driven decision making.\n ◀ Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used a ▶
\n
The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity of data sources that must be assimilated, the coupled nature of target systems, and the need for robustness to data and model uncertainties to support high-consequence decisions – these present frontier mathematical and computational challenges for DTs and place them at the forefront of applied math and computational science research today. \n ◀The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity ▶
\n
M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational science (PDE solvers, data assimilation, reduced-order models, optimal control, uncertainty quantification) with statistical/data sciences (machine learning, graphical models, dimensionality reduction, optimal experimental design, optimization) to advance the state of the art in these areas and at their interfaces. In particular, we are pursuing an integrated and multifaceted program of research on foundations of digital twins in:\n ◀M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational sc ▶
1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based inference from data and dynamic decision making for complex systems; \n ◀1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based in ▶
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4. new algorithms orchestrating use of hierarchies of models in digital twins, given real-time and heterogeneous computing constraints on latencies, bandwidth, response times, and data availability, on heterogeneous compute platforms from edge devices to exascale systems; and \n ◀4. new algorithms orchestrating use of hierarchies of models in digital twins, given real-time and heterogeneous computing constraints on latencies, bandwidth, ▶
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}
#augmentable: Statamic\Entries\Entry {#2368 ▶}
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}
cp_url
"https://m2dt.oden.utexas.edu/cp"
csrf_field
0 of 0 Illuminate\Support\HtmlString {#633 ▼
#html: "<input type="hidden" name="_token" value="gccGUPK2NIR73Ka4gjAej0EexlTzLkzyexYw2g7f" autocomplete="off">"
}
csrf_token
"gccGUPK2NIR73Ka4gjAej0EexlTzLkzyexYw2g7f"
current_date
0 of 0 Illuminate\Support\Carbon @1731252635 {#2319 ▼
#endOfTime: false
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#constructedObjectId: "000000000000090f0000000000000000"
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#dumpProperties: array:3 [▶
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]
#dumpLocale: null
#dumpDateProperties: null
date: 2024-11-10 15:30:35.307521 UTC (+00:00)
}
current_full_url
"https://m2dt.oden.utexas.edu"
current_layout
"layout"
current_template
"home"
current_uri
"/"
current_url
"https://m2dt.oden.utexas.edu"
current_user
null
date
0 of 0 Statamic\Fields\Value {#2416 ▼
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this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "date"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
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}
#raw: null
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edit_url
0 of 0 Statamic\Fields\Value {#2424 ▼
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this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "edit_url"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
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}
entry_id
0 of 0 Statamic\Fields\Value {#2425 ▼
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this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "entry_id"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
#handle: "entry_id"
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}
environment
"PRODUCTION"
get
[]
get_post
[]
homepage
"/"
id
0 of 0 Statamic\Fields\Value {#2427 ▼
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this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "id"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
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}
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is_entry
0 of 0 Statamic\Fields\Value {#2429 ▼
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this: Statamic\Structures\AugmentedPage {#2328 …}
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file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
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is_homepage
true
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0 of 0 Statamic\Fields\Value {#2431 ▼
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use: {▶
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locale
0 of 0 Statamic\Fields\Value {#2433 ▼
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this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "locale"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
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}
#raw: null
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logged_in
false
logged_out
true
mount
0 of 0 Statamic\Fields\Value {#2435 ▼
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class: "Statamic\Structures\AugmentedPage"
this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "mount"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
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now
0 of 0 Illuminate\Support\Carbon @1731252635 {#2319 ▼
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]
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date: 2024-11-10 15:30:35.307521 UTC (+00:00)
}
old
[]
order
0 of 0 Statamic\Fields\Value {#2437 ▼
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class: "Statamic\Structures\AugmentedPage"
this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "order"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
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}
origin_id
0 of 0 Statamic\Fields\Value {#2439 ▼
-resolver: Closure(Value $value) {#2438 ▶
class: "Statamic\Structures\AugmentedPage"
this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "origin_id"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
#handle: "origin_id"
#fieldtype: null
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}
page
0 of 0 Statamic\Structures\Page {#2304 ▼
#tree: Statamic\Structures\CollectionTree {#2257 ▶
#handle: "pages"
#locale: "default"
#tree: array:4 [▶
0 => array:1 [▶
"entry" => "home"
]
1 => array:1 [▶
"entry" => "26875ded-0ca7-433f-b4ab-621373d19cda"
]
2 => array:1 [▶
"entry" => "9bc7ba4e-012d-4797-bbfa-6b2c46f326dd"
]
3 => array:1 [▶
"entry" => "cc78691a-e5e5-4081-8853-501e55d959fe"
]
]
#cachedFlattenedPages: Illuminate\Support\Collection {#2354 ▶
#items: array:5 [▶
0 => Statamic\Structures\Page {#2304}
1 => Statamic\Structures\Page {#2351 ▶
#tree: Statamic\Structures\CollectionTree {#2257}
#reference: "26875ded-0ca7-433f-b4ab-621373d19cda"
#route: "{parent_uri}/{slug}"
#parent: Statamic\Structures\Page {#2304}
#children: []
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#id: "26875ded-0ca7-433f-b4ab-621373d19cda"
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-structure: null
#supplements: Illuminate\Support\Collection {#2348 ▶
#items: []
#escapeWhenCastingToString: false
}
#selectedQueryColumns: null
}
2 => Statamic\Structures\Page {#2353 ▶
#tree: Statamic\Structures\CollectionTree {#2257}
#reference: "9bc7ba4e-012d-4797-bbfa-6b2c46f326dd"
#route: "{parent_uri}/{slug}"
#parent: Statamic\Structures\Page {#2304}
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#id: "9bc7ba4e-012d-4797-bbfa-6b2c46f326dd"
#url: null
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#supplements: Illuminate\Support\Collection {#2346 ▶
#items: []
#escapeWhenCastingToString: false
}
#selectedQueryColumns: null
}
3 => Statamic\Structures\Page {#2347 ▶
#tree: Statamic\Structures\CollectionTree {#2257}
#reference: "cc78691a-e5e5-4081-8853-501e55d959fe"
#route: "{parent_uri}/{slug}"
#parent: Statamic\Structures\Page {#2304}
#children: []
#isRoot: false
#id: "cc78691a-e5e5-4081-8853-501e55d959fe"
#url: null
#title: null
#depth: 1
#data: []
#augmentationReferenceKey: null
#setAugmentationReferenceKey: false
-absoluteUrl: null
-absoluteUrlWithoutRedirect: null
-blueprint: null
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-structure: null
#supplements: Illuminate\Support\Collection {#2311 ▶
#items: []
#escapeWhenCastingToString: false
}
#selectedQueryColumns: null
}
4 => Statamic\Structures\Page {#2370 ▶
#tree: Statamic\Structures\CollectionTree {#2257}
#reference: "b2f92961-6a8d-460c-902d-55515da9b03a"
#route: "{parent_uri}/{slug}"
#parent: Statamic\Structures\Page {#2304}
#children: []
#isRoot: false
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#url: null
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#supplements: Illuminate\Support\Collection {#2369 ▶
#items: []
#escapeWhenCastingToString: false
}
#selectedQueryColumns: null
}
]
#escapeWhenCastingToString: false
}
#withEntries: false
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#original: array:1 [▶
"tree" => array:4 [▶
0 => array:1 [▶
"entry" => "home"
]
1 => array:1 [▶
"entry" => "26875ded-0ca7-433f-b4ab-621373d19cda"
]
2 => array:1 [▶
"entry" => "9bc7ba4e-012d-4797-bbfa-6b2c46f326dd"
]
3 => array:1 [▶
"entry" => "cc78691a-e5e5-4081-8853-501e55d959fe"
]
]
]
-structureCache: Statamic\Structures\CollectionStructure {#2345 ▶
#title: null
#handle: "pages"
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}
}
#reference: "home"
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-structure: Statamic\Structures\CollectionStructure {#2345 ▶}
#supplements: Illuminate\Support\Collection {#2303 ▶
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parent
0 of 0 Statamic\Fields\Value {#2441 ▼
-resolver: Closure(Value $value) {#2440 ▶
class: "Statamic\Structures\AugmentedPage"
this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "parent"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
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permalink
0 of 0 Statamic\Fields\Value {#2443 ▼
-resolver: Closure(Value $value) {#2442 ▶
class: "Statamic\Structures\AugmentedPage"
this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "permalink"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
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post
[]
private
0 of 0 Statamic\Fields\Value {#2445 ▼
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class: "Statamic\Structures\AugmentedPage"
this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "private"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
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}
published
0 of 0 Statamic\Fields\Value {#2447 ▼
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class: "Statamic\Structures\AugmentedPage"
this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "published"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
#handle: "published"
#fieldtype: null
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#shallow: false
}
response_code
200
site
0 of 0 Statamic\Sites\Site {#538 ▼
#handle: "default"
#config: array:3 [▶
"name" => "m2dt"
"url" => "/"
"locale" => "en_US"
]
#rawConfig: array:3 [▶
"name" => "{{ config:app:name }}"
"url" => "/"
"locale" => "en_US"
]
}
sites
0 of 0 Illuminate\Support\Collection {#2320 ▼
#items: array:1 [▶
0 => Statamic\Sites\Site {#538 ▶
#handle: "default"
#config: array:3 [▶
"name" => "m2dt"
"url" => "/"
"locale" => "en_US"
]
#rawConfig: array:3 [▶
"name" => "{{ config:app:name }}"
"url" => "/"
"locale" => "en_US"
]
}
]
#escapeWhenCastingToString: false
}
slug
0 of 0 Statamic\Fields\Value {#2449 ▼
-resolver: Closure(Value $value) {#2448 ▶
class: "Statamic\Structures\AugmentedPage"
this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "slug"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
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#fieldtype: null
#augmentable: null
#shallow: false
}
status
0 of 0 Statamic\Fields\Value {#2451 ▼
-resolver: Closure(Value $value) {#2450 ▶
class: "Statamic\Structures\AugmentedPage"
this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "status"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
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#raw: null
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}
template
0 of 0 Statamic\Fields\Value {#2453 ▼
-resolver: Closure(Value $value) {#2452 ▶
class: "Statamic\Structures\AugmentedPage"
this: Statamic\Structures\AugmentedPage {#2328 …}
use: {▶
$key: "template"
$fields: Illuminate\Support\Collection {#2330 …}
}
file: "/var/www/html/m2dt/vendor/statamic/cms/src/Data/AbstractAugmented.php"
line: "50 to 59"
}
#raw: null
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#fieldtype: null
#augmentable: null
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}
title
0 of 0 Statamic\Fields\Value {#2455 ▼
-resolver: null
#raw: "Home"
#handle: "title"
#fieldtype: Statamic\Fieldtypes\Text {#3468 ▶
#field: Statamic\Fields\Field {#3459 ▶
#handle: "title"
#prefix: null
#config: array:2 [▶
"type" => "text"
"required" => true
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#parent: Statamic\Entries\Entry {#2368 ▶
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-computedCallbackCache: Illuminate\Support\Collection {#2289 ▶
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-siteCache: Statamic\Sites\Site {#538 ▶
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"url" => "/"
"locale" => "en_US"
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#withComputedData: true
#data: Illuminate\Support\Collection {#2356 ▶
#items: array:7 [▶
"blueprint" => "pages"
"title" => "Home"
"template" => "home"
"author" => "68edd57b-eaab-4c5d-9bee-cee149367db0"
"updated_by" => "68edd57b-eaab-4c5d-9bee-cee149367db0"
"updated_at" => 1730563136
"content" => """
# Multifaceted Mathematics for Predictive Digital Twins\n
\n
The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex systems.\n ◀The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex s ▶
\n
A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a set of virtual information constructs that mimics the structure, context, and behavior of a natural, engineered, or social system (or system-of-systems), is dynamically updated with data from its physical twin, has a predictive capability, and informs decisions that realize value. The bidirectional interaction between the virtual and the physical is central to the digital twin.\n ◀A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a ▶
\n
Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used as a basis for controlling the physical system, optimizing data acquisition, and providing decision support. To support critical high-consequence decisions, the DT must be trustworthy through a combination of explainability and uncertainty quantification. Digital twins are key to unlocking the full potential of decades of investment in computational modeling – moving beyond forward simulation to underpin intelligent automation in complex systems by supporting data-driven decision making.\n ◀ Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used a ▶
\n
The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity of data sources that must be assimilated, the coupled nature of target systems, and the need for robustness to data and model uncertainties to support high-consequence decisions – these present frontier mathematical and computational challenges for DTs and place them at the forefront of applied math and computational science research today. \n ◀The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity ▶
\n
M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational science (PDE solvers, data assimilation, reduced-order models, optimal control, uncertainty quantification) with statistical/data sciences (machine learning, graphical models, dimensionality reduction, optimal experimental design, optimization) to advance the state of the art in these areas and at their interfaces. In particular, we are pursuing an integrated and multifaceted program of research on foundations of digital twins in:\n ◀M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational sc ▶
1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based inference from data and dynamic decision making for complex systems; \n ◀1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based in ▶
2. new dimensionality reduction methods exploiting nonlinear manifolds;\n
3. new data-driven, machine learning-based, structure-preserving reduced-order models and surrogates of high fidelity models that address high dimensional state and parameter spaces; \n ◀3. new data-driven, machine learning-based, structure-preserving reduced-order models and surrogates of high fidelity models that address high dimensional state ▶
4. new algorithms orchestrating use of hierarchies of models in digital twins, given real-time and heterogeneous computing constraints on latencies, bandwidth, response times, and data availability, on heterogeneous compute platforms from edge devices to exascale systems; and \n ◀4. new algorithms orchestrating use of hierarchies of models in digital twins, given real-time and heterogeneous computing constraints on latencies, bandwidth, ▶
5. new decomposition algorithms to tackle the challenges of data assimilation and optimal control for complex systems.
"""
]
#escapeWhenCastingToString: false
}
#supplements: Illuminate\Support\Collection {#2365 ▶
#items: []
#escapeWhenCastingToString: false
}
#initialPath: "/var/www/html/m2dt/content/collections/pages/home.md"
#published: true
#selectedQueryColumns: null
#selectedQueryRelations: []
#original: array:13 [▶
"collection" => "pages"
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"slug" => "home"
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"path" => "/var/www/html/m2dt/content/collections/pages/home.md"
"blueprint" => "pages"
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"author" => "68edd57b-eaab-4c5d-9bee-cee149367db0"
"updated_by" => "68edd57b-eaab-4c5d-9bee-cee149367db0"
"content" => """
# Multifaceted Mathematics for Predictive Digital Twins\n
\n
The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex systems.\n ◀The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex s ▶
\n
A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a set of virtual information constructs that mimics the structure, context, and behavior of a natural, engineered, or social system (or system-of-systems), is dynamically updated with data from its physical twin, has a predictive capability, and informs decisions that realize value. The bidirectional interaction between the virtual and the physical is central to the digital twin.\n ◀A <a href="https://www.nationalacademies.org/our-work/foundational-research-gaps-and-future-directions-for-digital-twins" target="_blank">digital twin</a> is a ▶
\n
Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used as a basis for controlling the physical system, optimizing data acquisition, and providing decision support. To support critical high-consequence decisions, the DT must be trustworthy through a combination of explainability and uncertainty quantification. Digital twins are key to unlocking the full potential of decades of investment in computational modeling – moving beyond forward simulation to underpin intelligent automation in complex systems by supporting data-driven decision making.\n ◀ Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used a ▶
\n
The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity of data sources that must be assimilated, the coupled nature of target systems, and the need for robustness to data and model uncertainties to support high-consequence decisions – these present frontier mathematical and computational challenges for DTs and place them at the forefront of applied math and computational science research today. \n ◀The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity ▶
\n
M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational science (PDE solvers, data assimilation, reduced-order models, optimal control, uncertainty quantification) with statistical/data sciences (machine learning, graphical models, dimensionality reduction, optimal experimental design, optimization) to advance the state of the art in these areas and at their interfaces. In particular, we are pursuing an integrated and multifaceted program of research on foundations of digital twins in:\n ◀M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational sc ▶
1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based inference from data and dynamic decision making for complex systems; \n ◀1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based in ▶
2. new dimensionality reduction methods exploiting nonlinear manifolds;\n
3. new data-driven, machine learning-based, structure-preserving reduced-order models and surrogates of high fidelity models that address high dimensional state and parameter spaces; \n ◀3. new data-driven, machine learning-based, structure-preserving reduced-order models and surrogates of high fidelity models that address high dimensional state ▶
4. new algorithms orchestrating use of hierarchies of models in digital twins, given real-time and heterogeneous computing constraints on latencies, bandwidth, response times, and data availability, on heterogeneous compute platforms from edge devices to exascale systems; and \n ◀4. new algorithms orchestrating use of hierarchies of models in digital twins, given real-time and heterogeneous computing constraints on latencies, bandwidth, ▶
5. new decomposition algorithms to tackle the challenges of data assimilation and optimal control for complex systems.
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