PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs

• URL: http://arxiv.org/abs/2405.08558v1
• Date: Tue, 14 May 2024 12:46:12 GMT
• Title: PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs
• Authors: Simone Brivio, Stefania Fresca, Andrea Manzoni,
• Abstract summary: In this paper, we consider a major extension of POD-DL-ROMs by making them physics-informed. We first complement POD-DL-ROMs with a trunk net architecture, endowing them with the ability to compute the problem's solution at every point in the spatial domain. In particular, we take advantage of the few available data to develop a low-cost pre-training procedure.
• Score: 0.6827423171182154
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The coupling of Proper Orthogonal Decomposition (POD) and deep learning-based ROMs (DL-ROMs) has proved to be a successful strategy to construct non-intrusive, highly accurate, surrogates for the real time solution of parametric nonlinear time-dependent PDEs. Inexpensive to evaluate, POD-DL-ROMs are also relatively fast to train, thanks to their limited complexity. However, POD-DL-ROMs account for the physical laws governing the problem at hand only through the training data, that are usually obtained through a full order model (FOM) relying on a high-fidelity discretization of the underlying equations. Moreover, the accuracy of POD-DL-ROMs strongly depends on the amount of available data. In this paper, we consider a major extension of POD-DL-ROMs by enforcing the fulfillment of the governing physical laws in the training process -- that is, by making them physics-informed -- to compensate for possible scarce and/or unavailable data and improve the overall reliability. To do that, we first complement POD-DL-ROMs with a trunk net architecture, endowing them with the ability to compute the problem's solution at every point in the spatial domain, and ultimately enabling a seamless computation of the physics-based loss by means of the strong continuous formulation. Then, we introduce an efficient training strategy that limits the notorious computational burden entailed by a physics-informed training phase. In particular, we take advantage of the few available data to develop a low-cost pre-training procedure; then, we fine-tune the architecture in order to further improve the prediction reliability. Accuracy and efficiency of the resulting pre-trained physics-informed DL-ROMs (PTPI-DL-ROMs) are then assessed on a set of test cases ranging from non-affinely parametrized advection-diffusion-reaction equations, to nonlinear problems like the Navier-Stokes equations for fluid flows.

Related papers

• Physics Informed Deep Learning for Strain Gradient Continuum Plasticity [0.0]
  We use a space-time discretization based on physics informed deep learning to approximate solutions of rate-dependent strain gradient plasticity models. Taking inspiration from physics informed neural networks, we modify the loss function of a PIDL model in several novel ways. We show how PIDL methods could address the computational challenges posed by strain plasticity models.
  arXiv  Detail & Related papers  (2024-08-13T06:02:05Z)
• Training Deep Surrogate Models with Large Scale Online Learning [48.7576911714538]
  Deep learning algorithms have emerged as a viable alternative for obtaining fast solutions for PDEs. Models are usually trained on synthetic data generated by solvers, stored on disk and read back for training. It proposes an open source online training framework for deep surrogate models.
  arXiv  Detail & Related papers  (2023-06-28T12:02:27Z)
• PriorBand: Practical Hyperparameter Optimization in the Age of Deep Learning [49.92394599459274]
  We propose PriorBand, an HPO algorithm tailored to Deep Learning (DL) pipelines. We show its robustness across a range of DL benchmarks and show its gains under informative expert input and against poor expert beliefs.
  arXiv  Detail & Related papers  (2023-06-21T16:26:14Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models [0.0]
  Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional ROMs. This work introduces the $mu t$-POD-LSTM-ROM framework for efficient numerical approximation of parametrized PDEs.
  arXiv  Detail & Related papers  (2022-01-25T10:15:17Z)
• Dynamic Network-Assisted D2D-Aided Coded Distributed Learning [59.29409589861241]
  We propose a novel device-to-device (D2D)-aided coded federated learning method (D2D-CFL) for load balancing across devices. We derive an optimal compression rate for achieving minimum processing time and establish its connection with the convergence time. Our proposed method is beneficial for real-time collaborative applications, where the users continuously generate training data.
  arXiv  Detail & Related papers  (2021-11-26T18:44:59Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models [0.2538209532048866]
  Reduced order models (ROMs) provide reliable approximations to parameter-dependent fluid dynamics problems in rapid times. Deep learning (DL)-based ROMs overcome all these limitations by learning in a non-intrusive way both the nonlinear trial manifold and the reduced dynamics. The resulting POD-DL-ROMs are shown to provide accurate results in almost real-time for the flow around a cylinder benchmark, the fluid-structure interaction between an elastic beam attached to a fixed, rigid block and a laminar incompressible flow, and the blood flow in a cerebral aneurysm.
  arXiv  Detail & Related papers  (2021-06-10T13:07:33Z)
• Model-Constrained Deep Learning Approaches for Inverse Problems [0.0]
  Deep Learning (DL) is purely data-driven and does not require physics. DL methods in their original forms are not capable of respecting the underlying mathematical models. We present and provide intuitions for our formulations for general nonlinear problems.
  arXiv  Detail & Related papers  (2021-05-25T16:12:39Z)
• POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition [0.0]
  Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional reduced order models (ROMs) In this paper we propose a possible way to avoid an expensive training stage of DL-ROMs, by (i) performing a prior dimensionality reduction through POD, and (ii) relying on a multi-fidelity pretraining stage. The proposed POD-DL-ROM is tested on several (both scalar and vector, linear and nonlinear) time-dependent parametrized PDEs.
  arXiv  Detail & Related papers  (2021-01-28T07:34:15Z)
• DiffPD: Differentiable Projective Dynamics with Contact [65.88720481593118]
  We present DiffPD, an efficient differentiable soft-body simulator with implicit time integration. We evaluate the performance of DiffPD and observe a speedup of 4-19 times compared to the standard Newton's method in various applications.
  arXiv  Detail & Related papers  (2021-01-15T00:13:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.