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 $L_2$ error with POD after the convergent tolerance is reached.

基于投影驱动神经网络（PDNN）的降阶建模通常需要足够的数据集，而物理信息机器学习（PINN）和物理强化神经网络（PRNN）则考虑了降阶系统。然而，这两种方法中使用的基于物理的机器学习技术会导致复杂的神经网络耗费大量的时间、较高的归约基数和大量的残差点。通过对 PDNN、PINN 和 PRNN 的理解，提出了一种基于模型的神经网络（MBNN）来处理没有源项的非线性抛物线偏微分方程（PDE）。与 PINN 相比，采用 PDE 的完全离散方案避免了自动微分技术的昂贵成本。而且，参数空间中残差点的初始条件以适当的比例添加到损失函数中。由于考虑了降阶方程，所提出的MBNN可以在比快照所属时间范围更大的时间范围内预测解。数值结果表明，MBNN 比投影驱动的神经网络取得了更好的性能。一般来说，所提出的低阶约简方法可以得到一致的${\mathrm{大号}}_2$达到收敛容差后 POD 的误差。