The motion of particles is influenced by various physical effects. One of the most challenging problems in particle dynamics is forcing inference, which requires determining the unknown forcing function from measured data, such as particle trajectories or flow observations. If the forcing function can be determined accurately, it reveals the physical effects that dominate the particles' motion.

In this talk, we formulate the forcing inference problem as an optimization problem. The cost function measures the difference between the measured and simulated particle distributions. The constraints are expressed by both the particle dynamic equations and characteristic ODEs. To update the parameters representing the forcing function, we use a gradient-based method. During this process, we derive the gradient of the cost function using the adjoint method to avoid the heavy computation involved in directly calculating derivatives. This involves constructing the Lagrangian function and deriving the corresponding adjoint equations. Numerical experiments verify the effectiveness of the proposed adjoint-based method.