MATHICSE Technical Report: A non-intrusive multifidelity method for the reduced order modeling of nonlinear problems

We propose a non-intrusive reduced basis (RB) method for parametrized nonlinear partial differential equations (PDEs) that leverages models of different accuracy. The method extracts parameter locations from a collection of low-fidelity (LF) snapshots for the efficient creation of a high-fidelity (HF) reduced basis and employs multi-fidelity Gaussian process regression (GPR) to approximate the combination coefficients of the reduced basis. LF data is assimilated either via projection onto an LF basis or via an interpolation approach inspired by bifidelity reconstruction. The correlation between HF and LF data is modeled with hyperparameters whose values are automatically determined in the regression step. The proposed methods not only leverage the assimilated LF data to reduce the cost of the offline phase, but also allow for a fast evaluation during the online stage, independent of the computational cost of neither the low- nor the high-fidelity solution. Numerical studies demonstrate the effectiveness of the proposed approach on manufactured examples and problems in nonlinear structural mechanics. Clear benefits of using lower resolution models rather than reduced physics models are observed in both the basis selection and the regression step. An active learning scheme is used for additional snapshot selection at locations with high error. The speed-up in the online evaluation and the high accuracy of extracted quantities of interest makes the multifidelity RB method a powerful tool for outer-loop applications in engineering, as exemplified in uncertainty quantification.

A non-intrusive multifidelity method for the reduced order modeling of nonlinear problems

Mengwu Guo, Jan Sickmann Hesthaven, Mariella Kast

We propose a non-intrusive reduced basis (RB) method for parametrized nonlinear partial differential equations (PDEs) that leverages models of different accuracy. From a collection of low-fidelity (LF) snapshots, parameter locations are extracted for the evaluations of high-fidelity (HF) snapshots to recover a reduced basis. Multi-fidelity Gaussian process regression (GPR) is employed to approximate the combination coefficients of the reduced basis. LF data is assimilated either via projection onto an LF basis or via an interpolation approach inspired by bifidelity reconstruction. The correlation between HF and LF data is modeled with hyperparameters whose values are automatically determined in the regression step. The proposed methods not only leverage the assimilated LF data to reduce the cost of the offline phase, but also allow for a fast evaluation during the online stage, independent of the computational cost of neither the low-nor the high-fidelity solution. Numerical studies demonstrate the effectiveness of the proposed approach on manufactured examples and problems in nonlinear structural mechanics. Clear benefits of using lower resolution models rather than reduced physics models are observed in both the basis selection and the regression step. An active learning scheme is used for additional snapshot selection at locations with high error. The speed-up in the online evaluation and the high accuracy of extracted quantities of interest makes the multifidelity RB method a powerful tool for outer-loop applications in engineering, as exemplified in uncertainty quantification. (C) 2020 Elsevier B.V. All rights reserved.

ELSEVIER SCIENCE SA2020

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

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

2020