Reduced Particle in Cell method for the Vlasov-Poisson system using auto-encoder and Hamiltonian neural networks
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We are interested in the Hamiltonian model order reduction of a particle-based Vlasov-Poisson discretization. In such simulations, the plasma distribution is approximated with a large number of computational particles that are evolved using the method of characteristics. Meanwhile, the self-induced electric field is computed on a spatial grid with a discretization of the Poisson equation. This method is named the Particle In Cell (PIC) method. With additional care, we obtain a Hamiltonian PIC, meaning the total energy of the discretized system is preserved. These simulations tend to be computationally intractable, hence we propose a Hamiltonian model order reduction in the number of particles. We combine our previous work [1] on Hamiltonian reduction using deep learning techniques with the key idea of two-step projection-based model order reduction proposed in [2]. In practice, from the full order state space, we operate a first linear symplectic projection onto an intermediate dimensional subspace using a mapping built from the Proper Symplectic Decomposition (PSD). Then, we compose it with a second nonlinear projection onto a low dimensional manifold using an AutoEncoder (AE) neural network. Finally, we propose to learn the resulting reduced order model with a Hamiltonian Neural Network (HNN). The proposed method is validated on several Vlasov-Poisson test cases, such as the linear Landau damping and the two-stream instability.
