Simulating Partial Differential Equations with Neural Networks

In this paper, we present a novel approach for simulating solutions of partial differential equations using neural networks. We consider a time-stepping method similar to the finite-volume method, where the flux terms are computed using neural networks. To train the neural network, we collect ’sensor’ data on small subsets of the computational domain. Thus, our neural network learns the local behavior of the solution rather than the global one. This leads to a much more versatile method that can simulate the solution to equations whose initial conditions are not in the same form as the initial conditions we train with. Also, using sensor data from a small portion of the domain is much more realistic than methods where a neural network is trained using data over a large domain.