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CATEGORIES:Cambridge Image Analysis Seminars
SUMMARY:Enhancing DL-ROMs through mathematical and physica
l knowledge - Stefania Fresca\, MOX\, Department o
f Mathematics\, Politecnico di Milano
DTSTART;TZID=Europe/London:20241004T130000
DTEND;TZID=Europe/London:20241004T140000
UID:TALK221893AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/221893
DESCRIPTION:Solving differential problems using full order mod
els (FOMs)\, such as the finite element method\, u
sually results in prohibitive computational costs\
, particularly in real-time simulations and multi-
query routines. \nReduced order modeling aims to r
eplace FOMs with reduced order models (ROMs) chara
cterized by much lower complexity but still able t
o express the physical features of the system unde
r investigation.\nWithin this context\, deep learn
ing-based reduced order models (DL-ROMs) have emer
ged as a novel and comprehensive approach\, offeri
ng efficient and accurate surrogates for solving p
arametrized time-dependent nonlinear PDEs. \nBy le
veraging both the mathematical properties and phys
ical knowledge of the system\, the accuracy and ge
neralization capabilities of DL-based ROMs can be
further enhanced. \nBuilding on this motivation\,
two main approaches to reduced order modeling of p
arametrized PDEs are introduced: latent dynamics m
odels (LDMs) and pre-trained physics-informed DL-R
OMs (PTPI-DL-ROMs).\n\nLDMs represent a novel math
ematical framework in which the latent state is co
nstrained to evolve according to an (unknown) ODE.
\nA time-continuous setting is employed to derive
error and stability estimates for the LDM approxi
mation of the FOM solution. \nThe impact of using
an explicit Runge-Kutta scheme in a time-discrete
setting is then analyzed\, resulting in the ∆LDM f
ormulation. \nAdditionally\, the learnable setting
\, ∆LDMθ \, is explored\, where deep neural networ
ks approximate the discrete LDM components\, ensur
ing a bounded approximation error with respect to
the high-fidelity solution.\nMoreover\, the framew
ork demonstrates the capability to achieve a time-
continuous approximation of the FOM solution in a
multi-query context\, thus being able to compute t
he LDM approximation at any given time instance wh
ile retaining a prescribed level of accuracy.\n\nA
s the complexity of PDEs increases\, however\, the
computational cost associated with generating syn
thetic data using high-fidelity solvers for traini
ng DL-based ROMs also intensifies. \nTo address th
is challenge\, a significant extension of POD-DL-R
OMs is proposed\, integrating the limited labeled
data with the underlying physical laws to achieve
reliable approximations in a small data context. \
nThe approach relies on a physics-informed loss fo
rmulation to compensate for data scarcity\, provid
ing the neural network with information about the
underlying physics. \nBy intertwining the contribu
tions of data and physics\, PTPI-DL-ROMs incorpora
te a novel training paradigm consisting of an effi
cient pre-training strategy that enables the optim
izer to quickly approach the minimum in the loss l
andscape\, followed by a fine-tuning phase that fu
rther enhances prediction accuracy.
LOCATION:MR2 Centre for Mathematical Sciences
CONTACT:Ferdia Sherry
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