American options are among the most liquid derivatives, and the most popular stocks possess hundreds to thousands of options whose prices change constantly during stock exchanges' working hours. Due to the possibility of early exercise, there's no closed formula for the American options, and their fair value has to be computed numerically. The most common approaches, like the binomial tree or the solution of the Black–Scholes PDE with an optimal early-exercise boundary, are time-consuming. Furthermore, the computation of implied volatilities requires iterative pricing of American options, since it is computed by the solution of a nonlinear equation or by a minimizing problem. Consequently, the computation of implied volatilities and sensitivities become problematic for real-time monitoring, which is essential for hedging and risk management. The current work presents the neural networks as an alternative to the standard tools used for American options. The training data is generated by solving the Black–Scholes variational inequality (a parabolic free-boundary problem) by the finite element method. The PDE approach is preferred over the binomial tree due to its higher accuracy and faster convergence, better and stable estimation of the Greeks, and direct access to the early-exercise boundary. Furthermore, one solution of the PDE provides multiple samples for the neural network, i.e., option prices for many strikes. After the generation of the high-fidelity PDE dataset, a neural network is designed to reproduce not only accurate prices but also stable Greeks at microsecond inference latency. The PDE-generated data is appropriately transformed to ensure stable training and robust inference. Boundary and terminal conditions are enforced through the network parameterization to improve stability and physical consistency. The loss function consists of data loss, but it also includes both the PDE residual and the free boundary conditions. It is shown that neural networks can successfully be used to price American options and compute implied volatilities and Greeks. Compared to the PDE solver, the neural network model achieves a relative error below one percent, while the inference latency is on the order of microseconds, making the proposed neural networks an appropriate tool for real-time derivative pricing and risk management.