Speaker
Description
We present a novel method to generate a symplectic transfer map for any beamline element defined by its magnetic vector potential, even when it is known only on a 3D grid. The method uses a neural network (NN) as an ansatz to solve the Hamilton-Jacobi (HJ) equation for the unknown generating function of the second kind. This generating function is chosen to connect the solution of a simpler system with an exact analytical solution (e.g., an ideal hard-edge quadrupole) to the system with a more complex field configuration (e.g., a quadrupole with an Enge fringe field profile). This design dramatically reduces the learning burden on the NN. The learned generating function defines the trajectories and the element's symplectic transfer map via implicit equations for the particle's position and explicit equations for its momenta. The implicit equations are typically solved to machine precision in just a few iterations using Newton’s method combined with automatic differentiation capability of the NN. The method's accuracy can be conveniently estimated by how well the NN solution satisfies the original Hamiltonian. We validate the method with 1D and 2D examples for drift space, hard-edge quadrupole, and quadrupole with Enge fringe field profile.
Funding Agency
Work supported by the U.S. Department of Energy under Contract No. DE-SC0012704, and the Field Work Proposal 2025-BNL-PS040
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