Physics Informed Neural Networks for Inverse Advection-Diffusion Problems

In this study, we will address the problem of localising a source of pollutant given a sparse set of noisy data. By sparse we mean spatially separated time-series measurements in space. To this end, we will be adopting a machine learning algorithm, the Physics Informed Neural Network (PINN). PINNs are neural networks which aim to be possible alternatives to conventional forward and inverse PDE solvers. PINNs are quite generalisable as the same method can be applied to a wide array of problems. Furthermore, PINNs are not restricted to computational meshes, which gives us the freedom to use sparse and incomplete data-sets. Consequently, we wished to leverage the generality and power of this novel method on a transport equation which models pollutant transport, the advection-diffusion equation. We will be investigating how we can localise the source by solving the advection-diffusion equation in an inverse manner. It is the inverse problem which will be the main goal of the study, as forward numerical solvers of linear PDEs are already robust, fast and accurate. However, the forward problem will also be looked at as a preliminary to the construction of the PINN. We compute the reference solutions of the transport equation using traditional numerical methods, which gives us training data to be fed to the PINN. Then we employ the PINNs on different test cases involving time-dependent transport equations in one- and two-dimensional spatial domains. Our results show that the PINN is indeed able to solve for these problems and localise sources with high accuracy in certain situations