A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs (2001.04001v1)

Abstract: Traditional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) suffer from severe limitations when dealing with nonlinear time-dependent parametrized PDEs, because of the fundamental assumption of linear superimposition of modes they are based on. For this reason, in the case of problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method usually yields inefficient reduced order models (ROMs) if one aims at obtaining reduced order approximations sufficiently accurate compared to the high-fidelity, full order model (FOM) solution. To overcome these limitations, in this work, we propose a new nonlinear approach to set reduced order models by exploiting deep learning (DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM, both the nonlinear trial manifold (corresponding to the set of basis functions in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to the projection stage in a linear ROM) are learned in a non-intrusive way by relying on DL algorithms; the latter are trained on a set of FOM solutions obtained for different parameter values. In this paper, we show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs; moreover, we assess its accuracy on test cases featuring different parametrized PDE problems. Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to approximate the solution of parametrized PDEs in situations where a huge number of POD modes would be necessary to achieve the same degree of accuracy.

• The paper introduces a deep learning-based nonlinear ROM to accurately simulate time-dependent parametrized PDEs, outperforming traditional POD methods.
• It leverages convolutional autoencoders and feedforward neural networks to learn nonlinear manifolds and reduced dynamics, significantly reducing computational demands.
• Numerical results on Burgers, transport, and monodomain equations demonstrate enhanced efficiency and stability in handling complex PDE behaviors.

Deep Learning-Based Reduced Order Modeling of Nonlinear Parametrized PDEs

The paper "A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs" addresses critical challenges in efficiently solving nonlinear, time-dependent, parametrized partial differential equations (PDEs) using reduced order models (ROMs). Traditional ROM techniques, such as the reduced basis (RB) method, often rely on linear approximations using proper orthogonal decomposition (POD) bases. These approaches encounter limitations when applied to nonlinear PDEs, particularly for problems with propagating coherent structures common in transport or wave phenomena. The authors propose a novel approach leveraging deep learning (DL) to construct nonlinear ROMs, referred to as DL-ROMs, which promise enhanced accuracy and efficiency.

Key Contributions and Methodology

1. Nonlinear ROM Construction: The paper proposes a deep learning-based framework to establish a nonlinear ROM, termed DL-ROM, for nonlinear, time-dependent parametrized PDEs. This methodology encompasses learning both nonlinear trial manifolds and reduced dynamics through DL models, substantially differing from conventional linear ROM techniques.
2. Deep Learning Models:
   • Nonlinear Trial Manifold Learning: The DL-ROM employs a convolutional autoencoder to autonomously generate a nonlinear trial manifold. This manifold effectively approximates the solution space of the PDEs, overcoming restrictions imposed by linear basis approaches.
   • Reduced Dynamics Learning: For capturing the temporal dynamics of the ROM's intrinsic coordinates, deep feedforward neural networks are tasked with learning the reduced dynamics as a function of time and parameters, operating in a non-intrusive fashion using available FOM solutions.
3. Training and Testing:
   • The training phase involves solving an optimization problem using the ADAM algorithm, focusing on minimizing a loss function that encapsulates reconstruction and dynamic modeling errors.
   • During testing, the well-trained network constructs efficient solutions using new parameter values without requiring full-order simulations.

Numerical Results

The paper provides numerical validation on several PDE problems, including Burgers equation, linear transport equations, and the monodomain equation (arising from cardiac electrophysiology). These tests demonstrate the DL-ROM's capacity to achieve accuracy similar to or surpassing that of high-dimensional linear ROMs while maintaining a lower dimensionality:

• Burgers Equation: The DL-ROM achieved comparable or improved accuracy over POD-based ROMs with significantly fewer basis functions.
• Transport Equations: For systems where traditional ROMs struggled due to oscillations and large gradients, DL-ROMs provided stable and accurate approximations.
• Monodomain Equation: The DL-ROM outperformed local reduced order methods requiring significantly more bases, demonstrating enhanced efficiency and precision.

Implications and Future Work

The implementation of DL-ROMs has profound implications for the sustainability of high-fidelity simulations in real-time applications or scenarios with numerous query instances. By effectively reducing the dimensionality needed to represent complex phenomena, the computational burden is significantly alleviated. Additionally, this framework sets the foundation for further exploration in higher-dimensional spaces or more intricate PDE systems, offering potential expansive utility in computational physics, engineering, and beyond.

The prospects for future paper encompass extending this DL-ROM framework to address PDEs in two or three dimensions and validating real-world applications, seeking to exploit the full potential of DL in reducing computational demands without compromising accuracy.