A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs

Abstract

Conventional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) may incur in severe limitations when dealing with nonlinear time-dependent parametrized PDEs, as these are strongly anchored to the assumption of modal linear superimposition they are based on. For problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method may yield inefficient reduced order models (ROMs) when very high levels of accuracy are required. To overcome this limitation, in this work, we propose a new nonlinear approach to set ROMs 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 full order model (FOM) solutions obtained for different parameter values. We show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs. Moreover, we assess its accuracy and efficiency on 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 efficiently approximate the solution of parametrized PDEs, especially in cases for which a huge number of POD modes would have been necessary to achieve the same degree of accuracy.

Real-Time Simulation of Parameter-Dependent Fluid Flows through Deep Learning-Based Reduced Order Models

“…and is sought in a reduced nonlinear trial manifold Sn N of very small dimension n N; usually, n ≈ n µ + 1-here time is considered as an additional parameter. As for DL-ROMs (see, e.g., [54]), both the reduced dynamics and the reduced nonlinear manifold (or trial manifold) where the ROM solution is sought must be learnt. In particular:…”

POD-Enhanced Deep Learning-Based Reduced Order Models for the Real-Time Simulation of Cardiac Electrophysiology in the Left Atrium

“…Traditional projection-based ROMs built, e.g., through the RB method (Quarteroni et al, 2016), yields inefficient ROMs when dealing with nonlinear timedependent parametrized PDE-ODE system as the one arising from cardiac EP (Fresca et al, 2020). To overcome the limitation of traditional projection-based ROMs, we have recently proposed in Fresca et al (2021) a strategy to construct, in a nonintrusive/data-driven way (indeed neither access or solution to the governing equations are required), DL-based ROMs (DL-ROMs) for nonlinear time-dependent parametrized problems, exploiting deep neural networks (Goodfellow et al, 2016) as a main building block, and a set of FOM snapshots. A first attempt to solve, by means of DL-ROMs, parametrized benchmark test cases in cardiac EP described by the Monodomain equations, has been carried out in Fresca et al (2020).…”

“…where n :[0, T) × R n µ → R n is a differentiable, nonlinear function. As for DL-ROMs (see e.g., Fresca et al, 2021), both the reduced dynamics and the reduced nonlinear manifold where the ROM solution is sought (or trial manifold) must be learnt. In particular,…”

“…Recently, we have introduced a new class of non-intrusivesince just a collection of FOM snapshots is required-nonlinear ROM techniques based on deep learning (DL) algorithms, named DL-ROMs, for the construction of efficient ROMs for parameter-dependent PDEs; in particular, we have focused so far on the Monodomain equations for cardiac EP, both in physiological and pathological scenarios (Fresca et al, 2020), as well as on several other nonlinear time-dependent parametrized problems, see (Fresca et al, 2021). DL-ROMs proved to be computationally efficient during the testing stage, that is, for any new scenario unseen during the training stage, but they might imply overwhelming training costs (and times) when the FOM dimension becomes moderately large.…”

Parameterized neural ordinary differential equations: applications to computational physics problems

“…In [13,14], latent dynamics and nonlinear mappings are modelled as NODEs and autoencoders, respectively; in [4,[71][72][73], autoencoders are used to learn approximate invariant subspaces of the Koopman operator. Relatedly, there have been studies on learning direct mappings via, for example, a neural network, from parameters (including time parameters) to either latent states or approximate solution states [74][75][76][77][78], where the latent states are computed by using autoencoders or linear POD.…”