Differentiable simulations for particle tracking in accelerators: analysis, benchmarking and optimization

Optimization of beamlines and lattices is a common problem in accelerator physics, which is usually solved with semi-analytical methods and numerical optimization routines. However, these are usually of the gradient-free or finite-differences type, whose computational cost grows quickly with the number of optimization parameters. On the other hand, the cost of gradient-based optimization can scale well with the number of parameters, but only if the computation of the gradient is itself efficient (e.g. not via finite differences, which are inefficient). Recently, there has been an emergence of so-called "differentiable" codes that efficiently provide gradients. Nevertheless, analysis and benchmarking comparisons of these techniques have largely been absent from the literature. In this work, we develop our own differentiable code, via auto-differentiation. We analyze and benchmark differentiability against finite differences in two test cases, a space-charge FODO cell from the literature and a realistic future beamline at CERN. The analyses and benchmarking are done both theoretically and empirically. Finally, we embed the gradient provided by the differentiable code in gradient-based optimization routines and compare with gradient-free methods. This work offers the first such analysis and benchmarking and thus contributes towards the development of more efficient and performant particle accelerators.