A non-intrusive neural network model order reduction algorithm for parameterized parabolic PDEs

Abstract

Reduced-order modeling based on projection-driven neural network (PDNN) generally needs sufficient data set while physics-informed machine learning (PINN) and physics-reinforced neural network (PRNN) take the reduced order systems into consideration. However, the physics-informed machine learning technique used in these two methods gives rise to expensive time consumption for complex neural network, higher reduced basis and a large amount of residual points. With understanding of PDNN, PINN and PRNN, a model-based neural network (MBNN) is proposed to cope with nonlinear parabolic partial differential equations (PDEs) without source terms. Compared with PINN, the fully discrete scheme of PDEs is adopted to avoid expensive cost for automatic differentiation technique. Moreover, initial conditions at residual points in parameter space are added to loss function with a proper proportion. Since the reduced order equation is taken into account, the proposed MBNN can predict solutions in a larger time range than the time range to which the snapshot belongs. Numerical results show that MBNN achieves better performance than the projection-driven neural network. Generally, the proposed method for lower order reduction can get a consistent L2 error with POD after the convergent tolerance is reached.