Recently it has been shown that Physarum-inspired algorithms can solve some network optimization problems. However, it is not yet shown that Physarum-inspired algorithm can solve Node Weighted Steiner Tree Problem (NWSTP). Two new Physarum-inspired algorithms are proposed in this paper to solve NWSTP for the first time. Since all the existing NWSTP benchmark instances have an empty terminal set, new benchmark instances with non-empty terminal sets are generated to cover the shortage of existing benchmark instances. Both proposed algorithms are compared with Genetic Algorithm (GA) and Discrete Particle Swarm Optimization (DPSO) in these benchmark instances. Furthermore, an adapted Dijkstra's algorithm is proposed to provide the optimal solutions for part of these benchmark instances where there are two terminals and the node weights are negative. Simulation results show that our first proposed algorithm can find the optimal solutions for NWSTP with two terminals in graphs with negative node weights, and our second proposed algorithm can find close approximate solutions for NWSTP with multiple terminals in any node weighted graph. Both proposed algorithms provide faster and better NWSTP solutions than GA and DPSO.