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

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 timedependent 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.

A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations

“…Our construction is mostly inspired by the recent advancements in nonlinear approximation theory, e.g. [16,17,55], and the increasing use of deep-learning techniques for parametrized PDEs, as in [14,24,39,44].…”

“…This idea has been recently investigated both theoretically, as in [39,55], and practically, e.g. [24,26]. By now, the drawbacks posed by this approach are mainly practical: it is often unclear how the network architecture should be designed and which optimization strategies are better suited for the purpose.…”

“…We then train a third network φ : Θ → R n to learn the reduced map µ → u n µ . In this way, by connecting the architectures of φ and Ψ we obtain the complete model, Φ := Ψ • φ, which we later term as DL-ROM (Deep Learning based Reduced Order Model, in the same spirit of previous works [24,25]).…”

“…Indeed, despite the popularity of autoencoders, e.g. [24,42,47,64], the choice of the latent dimension n is often handled by trial and error. In contrast, we establish precise bounds on n thanks to a rigorous theoretical analysis.…”

Reduced Basis Approximations of Parameterized Dynamical Partial Differential Equations via Neural Networks

“…The error is modeled as a random variable, and recurrent neural networks learn a prediction model in time. Methods also extend to the training portion [8], where the focus is on learning an effective trial space and the mapping over time is still monolithic, as in [5]. In addition, other methods [9], use a multilayered approach for handling time and space (through the use of autoencoders), while still remaining non-intrusive (where the PDE itself is not used in the construction).…”