Approximating characteristic flows with neural networks for transport-dominated problems

In this talk, we are interested in the numerical discretisation of high-dimensional transport problems, such as kinetic equations. Grid-based methods suffer from the curse of dimensionality: the required number of grid points increases exponentially with the dimension. Particle methods are a popular alternative, but they typically suffer from Monte Carlo noise. 

Neural networks provide a mesh-free alternative. However, solutions of transport problems may develop fine structures over time, which are difficult to represent with a network. 

We present two numerical methods that use neural networks to approximate the characteristic flow of the equation rather than its solution, which is then obtained by composing the initial condition with the approximate flow. 

In the first method, which we call the Neural δf Method, the networks are fitted to the particle trajectories of a particle-in-cell scheme and define the control variate of a δf method, which reduces the Monte Carlo noise. In the second method, which we call the Neural Characteristic Mapping Method, the flows are approximated by physics-informed neural networks (PINNs) and are trained by minimising the residual of the transport equation satisfied by the flow.

This is joint work with Martin Campos Pinto (IPP), Emmanuel Franck (IRMA), Philipp Krah (CEA) and Victor Michel-Dansac (IRMA).