Neural Semi Lagrangian solver for convection-diffusion PDEs
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In this work we consider the approximation of linear and nonlinear convection diffusion PDEs in large dimension. There many classical approaches to solve this PDEs but ion general limited at the dimension two or three. In larger dimension the number of dof needed is too large. Neural network base methods like PINNs [3] are able to deal with larger dimensional problem. However, the training of PINNs is often complex and the accuracy limited. Here we propose an alternative neural network-based method. The ideal is to use sequential in time algorithm [2] (more Classique in finite element or finite volumes) and combine it with semi Lagrangian approaches [1] where the next time step is estimate solving the characteristics. At the end we obtain a simpler algorithm where we solve characteristics on collocation points and project on the neural network space. To assure some accuracy we propose to use a efficient optimizer called natural gradient [4] and adaptive sampling. We will present some validation on parametric transport problems, high-dimension advection diffusion PDEs and on Vlasov Poisson problem.
