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

Simone Brivio, Stefania Fresca, Andrea Manzoni

TL;DR

A major extension of POD-DL-ROMs is considered 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.

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.

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

TL;DR

A major extension of POD-DL-ROMs is considered 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.

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.
Paper Structure (17 sections, 1 theorem, 36 equations, 17 figures, 1 table, 1 algorithm)

This paper contains 17 sections, 1 theorem, 36 equations, 17 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. A general framework: low-rank DL-based architectures
  3. Problem formulation
  4. Neural network architectures and low-rank decompositions
  5. POD-DL-ROMs as low-rank deep learning-based architectures
  6. Mitigating the shortfalls of small data regimes with physics
  7. The computation of the physics-based residual
  8. The PTPI-DL-ROM paradigm
  9. Architecture design and pre-training algorithm
  10. The rationale behind the fine-tuning
  11. Numerical experiments
  12. Eikonal Equation
  13. Advection Diffusion Reaction Equation
  14. Darcy Equations
  15. Navier-Stokes Equations
  16. ...and 2 more sections

Key Result

Lemma 4.1

For any $N \le \min\{N_h,N_sN_t\}$, denote by $\mathbf{V} \in \mathbb{R}^{N_h \times N}$ the POD basis computed through SVD, and by $\mathbf{V}_{\infty} \in \mathbb{R}^{N_h \times N}$ the optimal POD projection matrix. Then, there exists a sampling strategy for $(\boldsymbol{\mu},t) \in \mathcal{P}_

Figures (17)

  • Figure 1: Example of (a) space-discrete (POD+DNN) and (b) space-continuous (DeepONets) low-rank DL-based architectures.
  • Figure 2: The POD-DL-ROM architecture: we highlight the POD projection, the discrete lifting induced by the matrices $\mathbf{V}^T$ and $\mathbf{V}$, and the autoencoder-based architecture.
  • Figure 3: An instance of the time-parameter domains $\mathcal{P}_{sup} \times \mathcal{T}_{sup}, \mathcal{P}_{test} \times \mathcal{T}_{test}$ and $\mathcal{P}_{res} \times \mathcal{T}_{res}$ and their realizations, in dimension $p+1=2$.
  • Figure 4: Computational time required for AD as function of the neural network depth (left) and width (right). We remark that the continuous line represent the average value among $10$ runs, whereas the reported interval shows the minimum and the maximum value registered.
  • Figure 5: Schematic representation of the PTPI-DL-ROM architecture.
  • ...and 12 more figures

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 3.1
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.1
  • proof
  • proof