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### Abstract

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.