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

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.