Abstract: Much of astrophysical modeling reduces to solving partial differential equations expressing conservation laws. Very recently, proofs of concept have been published that nonlinear PDEs can be solved by harnessing the universal functional approximation capacity of artificial neural networks and large-scale numerical optimization accelerated with GPUs. The neural network approach uses least-squares residual minimization to find approximate global solutions implicitly defined by the PDEs and the boundary conditions over an entire spacetime and parameter space domain. The approach is mesh-free and can thus solve high-dimensional PDEs. We discuss how the approach differs from the standard supervised machine learning. We present experiments carried out in the TensorFlow framework that test the limits of the neural network approach to solving PDEs.