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SUMMARY:Enhancing DL-ROMs through mathematical and physical knowledge - St
 efania Fresca\, MOX\, Department of Mathematics\, Politecnico di Milano
DTSTART:20241004T120000Z
DTEND:20241004T130000Z
UID:TALK221893@talks.cam.ac.uk
CONTACT:Ferdia Sherry
DESCRIPTION:Solving differential problems using full order models (FOMs)\,
  such as the finite element method\, usually results in prohibitive comput
 ational costs\, particularly in real-time simulations and multi-query rout
 ines. \nReduced order modeling aims to replace FOMs with reduced order mod
 els (ROMs) characterized by much lower complexity but still able to expres
 s the physical features of the system under investigation.\nWithin this co
 ntext\, deep learning-based reduced order models (DL-ROMs) have emerged as
  a novel and comprehensive approach\, offering efficient and accurate surr
 ogates for solving parametrized time-dependent nonlinear PDEs. \nBy levera
 ging both the mathematical properties and physical knowledge of the system
 \, the accuracy and generalization capabilities of DL-based ROMs can be fu
 rther enhanced. \nBuilding on this motivation\, two main approaches to red
 uced order modeling of parametrized PDEs are introduced: latent dynamics m
 odels (LDMs) and pre-trained physics-informed DL-ROMs (PTPI-DL-ROMs).\n\nL
 DMs represent a novel mathematical framework in which the latent state is 
 constrained to evolve according to an (unknown) ODE. \nA time-continuous s
 etting is employed to derive error and stability estimates for the LDM app
 roximation of the FOM solution. \nThe impact of using an explicit Runge-Ku
 tta scheme in a time-discrete setting is then analyzed\, resulting in the 
 ∆LDM formulation. \nAdditionally\, the learnable setting\, ∆LDMθ \, i
 s explored\, where deep neural networks approximate the discrete LDM compo
 nents\, ensuring a bounded approximation error with respect to the high-fi
 delity solution.\nMoreover\, the framework demonstrates the capability to 
 achieve a time-continuous approximation of the FOM solution in a multi-que
 ry context\, thus being able to compute the LDM approximation at any given
  time instance while retaining a prescribed level of accuracy.\n\nAs the c
 omplexity of PDEs increases\, however\, the computational cost associated 
 with generating synthetic data using high-fidelity solvers for training DL
 -based ROMs also intensifies. \nTo address this challenge\, a significant 
 extension of POD-DL-ROMs is proposed\, integrating the limited labeled dat
 a with the underlying physical laws to achieve reliable approximations in 
 a small data context. \nThe approach relies on a physics-informed loss for
 mulation to compensate for data scarcity\, providing the neural network wi
 th information about the underlying physics. \nBy intertwining the contrib
 utions of data and physics\, PTPI-DL-ROMs incorporate a novel training par
 adigm consisting of an efficient pre-training strategy that enables the op
 timizer to quickly approach the minimum in the loss landscape\, followed b
 y a fine-tuning phase that further enhances prediction accuracy.
LOCATION:MR2 Centre for Mathematical Sciences
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