Learning phase-space flows using time-discrete implicit Runge-Kutta PINNs
This work provides a computational framework for solving particle motion equations in physics, but it is incremental as it builds on existing PINN methods with a specific adaptation.
The authors tackled solving nonlinear coupled differential equations for particle motion in external fields by adapting implicit Runge-Kutta PINNs to treat coordinates as functions, enabling efficient solutions for time-independent and periodic fields, with successful applications to central force and periodic electric field scenarios.
We present a computational framework for obtaining multidimensional phase-space solutions of systems of non-linear coupled differential equations, using high-order implicit Runge-Kutta Physics- Informed Neural Networks (IRK-PINNs) schemes. Building upon foundational work originally solving differential equations for fields depending on coordinates [J. Comput. Phys. 378, 686 (2019)], we adapt the scheme to a context where the coordinates are treated as functions. This modification enables us to efficiently solve equations of motion for a particle in an external field. Our scheme is particularly useful for explicitly time-independent and periodic fields. We apply this approach to successfully solve the equations of motion for a mass particle placed in a central force field and a charged particle in a periodic electric field.