Parametric Taylor series based latent dynamics identification neural networks
Numerical solving parameterised partial differential equations (P-PDEs) is highly practical yet computationally expensive, driving the development of reduced-order models (ROMs). Recently, methods that combine latent space identification techniques with deep learning algorithms (e.g., autoencoders)...
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creator | Lin, Xinlei Xiao, Dunhui |
description | Numerical solving parameterised partial differential equations (P-PDEs) is
highly practical yet computationally expensive, driving the development of
reduced-order models (ROMs). Recently, methods that combine latent space
identification techniques with deep learning algorithms (e.g., autoencoders)
have shown great potential in describing the dynamical system in the lower
dimensional latent space, for example, LaSDI, gLaSDI and GPLaSDI.
In this paper, a new parametric latent identification of nonlinear dynamics
neural networks, P-TLDINets, is introduced, which relies on a novel neural
network structure based on Taylor series expansion and ResNets to learn the
ODEs that govern the reduced space dynamics. During the training process,
Taylor series-based Latent Dynamic Neural Networks (TLDNets) and identified
equations are trained simultaneously to generate a smoother latent space. In
order to facilitate the parameterised study, a $k$-nearest neighbours (KNN)
method based on an inverse distance weighting (IDW) interpolation scheme is
introduced to predict the identified ODE coefficients using local information.
Compared to other latent dynamics identification methods based on autoencoders,
P-TLDINets remain the interpretability of the model. Additionally, it
circumvents the building of explicit autoencoders, avoids dependency on
specific grids, and features a more lightweight structure, which is easy to
train with high generalisation capability and accuracy. Also, it is capable of
using different scales of meshes. P-TLDINets improve training speeds nearly
hundred times compared to GPLaSDI and gLaSDI, maintaining an $L_2$ error below
$2\%$ compared to high-fidelity models. |
doi_str_mv | 10.48550/arxiv.2410.04193 |
format | Article |
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highly practical yet computationally expensive, driving the development of
reduced-order models (ROMs). Recently, methods that combine latent space
identification techniques with deep learning algorithms (e.g., autoencoders)
have shown great potential in describing the dynamical system in the lower
dimensional latent space, for example, LaSDI, gLaSDI and GPLaSDI.
In this paper, a new parametric latent identification of nonlinear dynamics
neural networks, P-TLDINets, is introduced, which relies on a novel neural
network structure based on Taylor series expansion and ResNets to learn the
ODEs that govern the reduced space dynamics. During the training process,
Taylor series-based Latent Dynamic Neural Networks (TLDNets) and identified
equations are trained simultaneously to generate a smoother latent space. In
order to facilitate the parameterised study, a $k$-nearest neighbours (KNN)
method based on an inverse distance weighting (IDW) interpolation scheme is
introduced to predict the identified ODE coefficients using local information.
Compared to other latent dynamics identification methods based on autoencoders,
P-TLDINets remain the interpretability of the model. Additionally, it
circumvents the building of explicit autoencoders, avoids dependency on
specific grids, and features a more lightweight structure, which is easy to
train with high generalisation capability and accuracy. Also, it is capable of
using different scales of meshes. P-TLDINets improve training speeds nearly
hundred times compared to GPLaSDI and gLaSDI, maintaining an $L_2$ error below
$2\%$ compared to high-fidelity models.</description><identifier>DOI: 10.48550/arxiv.2410.04193</identifier><language>eng</language><subject>Computer Science - Learning ; Computer Science - Neural and Evolutionary Computing ; Mathematics - Dynamical Systems</subject><creationdate>2024-10</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2410.04193$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2410.04193$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Lin, Xinlei</creatorcontrib><creatorcontrib>Xiao, Dunhui</creatorcontrib><title>Parametric Taylor series based latent dynamics identification neural networks</title><description>Numerical solving parameterised partial differential equations (P-PDEs) is
highly practical yet computationally expensive, driving the development of
reduced-order models (ROMs). Recently, methods that combine latent space
identification techniques with deep learning algorithms (e.g., autoencoders)
have shown great potential in describing the dynamical system in the lower
dimensional latent space, for example, LaSDI, gLaSDI and GPLaSDI.
In this paper, a new parametric latent identification of nonlinear dynamics
neural networks, P-TLDINets, is introduced, which relies on a novel neural
network structure based on Taylor series expansion and ResNets to learn the
ODEs that govern the reduced space dynamics. During the training process,
Taylor series-based Latent Dynamic Neural Networks (TLDNets) and identified
equations are trained simultaneously to generate a smoother latent space. In
order to facilitate the parameterised study, a $k$-nearest neighbours (KNN)
method based on an inverse distance weighting (IDW) interpolation scheme is
introduced to predict the identified ODE coefficients using local information.
Compared to other latent dynamics identification methods based on autoencoders,
P-TLDINets remain the interpretability of the model. Additionally, it
circumvents the building of explicit autoencoders, avoids dependency on
specific grids, and features a more lightweight structure, which is easy to
train with high generalisation capability and accuracy. Also, it is capable of
using different scales of meshes. P-TLDINets improve training speeds nearly
hundred times compared to GPLaSDI and gLaSDI, maintaining an $L_2$ error below
$2\%$ compared to high-fidelity models.</description><subject>Computer Science - Learning</subject><subject>Computer Science - Neural and Evolutionary Computing</subject><subject>Mathematics - Dynamical Systems</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNqFjr0KwkAQhK-xEPUBrLIvYLyYBLQWxUawSH-suQ0sXi6yd_7k7Y3B3upjhoH5lFpmOi22ZanXKG9-pptiKHSR7fKpOl9QsKUoXEOFvesEAglTgCsGsuAwko9ge48t1wHYDpEbrjFy58HTQ9ANiK9ObmGuJg26QIsfZyo5Hqr9aTUem7twi9Kbr4AZBfL_iw-n5jyE</recordid><startdate>20241005</startdate><enddate>20241005</enddate><creator>Lin, Xinlei</creator><creator>Xiao, Dunhui</creator><scope>AKY</scope><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20241005</creationdate><title>Parametric Taylor series based latent dynamics identification neural networks</title><author>Lin, Xinlei ; Xiao, Dunhui</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-arxiv_primary_2410_041933</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Computer Science - Learning</topic><topic>Computer Science - Neural and Evolutionary Computing</topic><topic>Mathematics - Dynamical Systems</topic><toplevel>online_resources</toplevel><creatorcontrib>Lin, Xinlei</creatorcontrib><creatorcontrib>Xiao, Dunhui</creatorcontrib><collection>arXiv Computer Science</collection><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Lin, Xinlei</au><au>Xiao, Dunhui</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Parametric Taylor series based latent dynamics identification neural networks</atitle><date>2024-10-05</date><risdate>2024</risdate><abstract>Numerical solving parameterised partial differential equations (P-PDEs) is
highly practical yet computationally expensive, driving the development of
reduced-order models (ROMs). Recently, methods that combine latent space
identification techniques with deep learning algorithms (e.g., autoencoders)
have shown great potential in describing the dynamical system in the lower
dimensional latent space, for example, LaSDI, gLaSDI and GPLaSDI.
In this paper, a new parametric latent identification of nonlinear dynamics
neural networks, P-TLDINets, is introduced, which relies on a novel neural
network structure based on Taylor series expansion and ResNets to learn the
ODEs that govern the reduced space dynamics. During the training process,
Taylor series-based Latent Dynamic Neural Networks (TLDNets) and identified
equations are trained simultaneously to generate a smoother latent space. In
order to facilitate the parameterised study, a $k$-nearest neighbours (KNN)
method based on an inverse distance weighting (IDW) interpolation scheme is
introduced to predict the identified ODE coefficients using local information.
Compared to other latent dynamics identification methods based on autoencoders,
P-TLDINets remain the interpretability of the model. Additionally, it
circumvents the building of explicit autoencoders, avoids dependency on
specific grids, and features a more lightweight structure, which is easy to
train with high generalisation capability and accuracy. Also, it is capable of
using different scales of meshes. P-TLDINets improve training speeds nearly
hundred times compared to GPLaSDI and gLaSDI, maintaining an $L_2$ error below
$2\%$ compared to high-fidelity models.</abstract><doi>10.48550/arxiv.2410.04193</doi><oa>free_for_read</oa></addata></record> |
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subjects | Computer Science - Learning Computer Science - Neural and Evolutionary Computing Mathematics - Dynamical Systems |
title | Parametric Taylor series based latent dynamics identification neural networks |
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