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.
Highlights
The solution of a parametrized system of partial differential equations (PDEs) by means of a full-order model (FOM), whenever dealing with real-time or multi-query scenarios, entails prohibitive computational costs if the full order model (FOM) is high-dimensional
In the one-dimensional test cases we aim at assessing the numerical accuracy of the deep learning (DL)-reduced order models (ROMs) approximation, comparing it to the solution provided by a proper orthogonal decomposition (POD)
In the two-dimensional test case we instead focus on computational efficiency, by comparing the computational times of Deep Learning-Based Reduced Order Model (DL-ROM) to the ones entailed by a POD-Galerkin method
Summary
The solution of a parametrized system of partial differential equations (PDEs) by means of a full-order model (FOM), whenever dealing with real-time or multi-query scenarios, entails prohibitive computational costs if the FOM is high-dimensional In the former case, the FOM solution must be computed in a very limited amount of time; in the latter one, the FOM must be solved for a huge number of parameter instances sampled from the parameter space. A widespread family of reduced order modeling techniques relies on the assumption that the reduced order approximation can be expressed by a linear combination of basis functions, built starting from a set of FOM solutions, called snapshots Among these techniques, proper orthogonal decomposition (POD) exploits the singular value decomposition of a suitable snapshot matrix (or the eigen-decomposition of the corresponding snapshot correlation matrix), yielding linear ROMs, that is ROMs employing linear trial spaces, in which the ROM approximation is given by the linear superimposition of POD modes. The solution manifold is approximated through a linear trial manifold, that is, the ROM approximation is sought in a low-dimensional linear trial subspace
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