Non-intrusive reduced order modeling of nonlinear problems using neural networks

We develop a non-intrusive reduced basis (RB) method for parametrized steady-state partial differential equations (PDEs). The method extracts a reduced basis from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD) and employs artificial neural networks (ANNs), particularly multi-layer perceptrons (MLPs), to accurately approxi- mate the coefficients of the reduced model. The search for the optimal number of neurons and the minimum amount of training samples to avoid overfitting is carried out in the offline phase through an automatic routine, relying upon a joint use of the latin hypercube sampling (LHS) and the Levenberg-Marquardt training algorithm. This guarantees a complete offline-online decoupling, leading to an efficient RB method - referred to as POD-NN - suitable also for general nonlinear problems with a non-affine parametric dependence. Numerical studies are presented for the nonlinear Poisson equation and for driven cavity viscous flows, modeled through the steady incompressible Navier-Stokes equations. Both physical and geometrical parametrizations are considered. Several results confirm the accuracy of the POD-NN method and show the substantial speed-up enabled at the online stage as compared to a traditional RB strategy.

Physics-informed machine learning for reduced-order modeling of nonlinear problems

Jan Sickmann Hesthaven, Qian Wang

A reduced basis method based on a physics-informed machine learning framework is developed for efficient reduced-order modeling of parametrized partial differential equations (PDEs). A feedforward neural network is used to approximate the mapping from the time-parameter to the reduced coefficients. During the offline stage, the network is trained by minimizing the weighted sum of the residual loss of the reduced-order equations, and the data loss of the labeled reduced coefficients that are obtained via the projection of high-fidelity snapshots onto the reduced space. Such a network is referred to as physics-reinforced neural network (PRNN). As the number of residual points in time-parameter space can be very large, an accurate network – referred to as physics-informed neural network (PINN) – can be trained by minimizing only the residual loss. However, for complex nonlinear problems, the solution of the reduced-order equation is less accurate than the projection of high-fidelity solution onto the reduced space. Therefore, the PRNN trained with the snapshot data is expected to have higher accuracy than the PINN. Numerical results demonstrate that the PRNN is more accurate than the PINN and a purely data-driven neural network for complex problems. During the reduced basis refinement, the PRNN may obtain higher accuracy than the direct reduced-order model based on a Galerkin projection. The online evaluation of PINN/PRNN is orders of magnitude faster than that of the Galerkin reduced-order model.

2021

Physics-informed machine learning for reduced-order modeling of nonlinear problems

Jan Sickmann Hesthaven, Qian Wang

A physics-informed machine learning framework is developed for the reduced-order modeling of parametrized steady-state partial differential equations (PDEs). During the offline stage, a reduced basis is extracted from a collection of high-fidelity solutions and a reduced-order model is then constructed by a Galerkin projection of the full-order model onto the reduced space. A feedforward neural network is used to approximate the mapping from the physical/geometrical parameters to the reduced coefficients. The network can be trained by minimizing the mean squared residual error of the reduced-order equation on a set of points in parameter space. Such a network is referred to as physics-informed neural network (PINN). As the number of residual points is unlimited, a large data set can be generated to train a PINN to approximate the reduced-order model. However, the accuracy of such a network is often limited. This is improved by using the high-fidelity solutions that are generated to extract the reduced basis. The network is then trained by minimizing the sum of the mean squared residual error of the reduced-order equation and the mean squared error between the network output and the projection coefficients of the high-fidelity solutions. For complex nonlinear problems, the projection of high-fidelity solution onto the reduced space is more accurate than the solution of the reduced-order equation. Therefore, higher accuracy than the PINN for this network - referred to as physics-reinforced neural network (PRNN) - can be expected for complex nonlinear problems. Numerical results demonstrate that the PRNN is more accurate than the PINN and both are more accurate than a purely data-driven neural network for complex problems. During the reduced basis refinement, before reaching its accuracy limit, the PRNN obtains higher accuracy than the direct reduced-order model based on a Galerkin projection.

2020

Model Order Reduction by geometrical parametrization for shape optimization in computational fluid dynamics

Andrea Manzoni, Gianluigi Rozza

Shape Optimization problems governed by partial differential equations result from many applications in computational fluid dynamics; they involve the repetitive evaluation of outputs expressed as functionals of the field variables and usually imply big computational efforts. For this reason looking for computational efficiency in numerical methods and algorithms is mandatory. The interplay between scientific computing and new reduction strategies is crucial in applications of great complexity. In order to achieve an efficient model order reduction, reduced basis methods built upon a high-fidelity ``truth'' finite element approximation -- and combined with suitable geometrical parametrization techniques for efficient shape description -- can be introduced, thus decreasing both the computational effort and the geometrical complexity. Starting from an excursus on classical approaches -- such as local boundary variation and shape boundary parametrization -- we focus on more efficient parametrization techniques which are well suited for a combination with a reduced basis approach, such as the one based on affine mapping (even automatic), nonaffine mapping (coupled with a suitable empirical interpolation technique for better numerical performances) and free-form deformations. We thus describe (and compare) the principal features of these parametrization techniques by showing some applications dealing with shape optimization of parametrized configurations in viscous flows,and discussing computational advantages and efficiency obtained by geometrical and computational model order reduction.

2010

Physics-informed machine learning for reduced-order modeling of nonlinear problems

Jan Sickmann Hesthaven, Qian Wang

2020