This project will investigate the use of machine learning and neural network methodologies to solve problems involving partial and stochastic differential equations. We will initially consider contaminant dispersal models as an exemplar; in this problem pollutant particles are modelled individually and we are interested in learning the distribution of a large number of such particles. The Fokker-Planck equation for this models is high dimensional, and currently only solvable using Monte Carlo methods. The first part of the project will focus on the efficient approximation of the solution to the forward problem using deep learning methods. A TensorFlow implementation of a deep learning high dimensional PDE solver will be created which incorporates suitable boundary conditions and background flow field. This approach will be analysed analytically where possible and compared to existing methods, such as the MLMC method developed by G. Katsiolides. Once implemented this method of solution will create avenues which can be used to approach the inverse problem of using data to parameterise the model by applying deep learning techniques and/or Bayesian methods; this part of the problem will be explored subsequently. There are a range of applications which could be considered in the later stages of the project, these include, but are not limited to, stochastic PDE models of particle movements, and stochastic optimal control.