Learning Lie Group Generators from Trajectories

This work investigates the inverse problem of generator recovery in matrix Lie groups from discretized trajectories. Let G be a real matrix Lie group and mathfrakg = Lie(G) its corresponding Lie algebra. A smooth trajectory γ(t) generated by a fixed Lie algebra element xi ∈ mathfrakg follows the exponential flow γ(t) = g0 · exp(t xi). The central task addressed in this work is the reconstruction of such a latent generator xi from a discretized sequence of poses g0, g1, dots, gT subset G, sampled at uniform time intervals. This problem is formulated as a data-driven regression from normalized sequences of discrete Lie algebra increments logleft(gt-1 gt+1right) to the constant generator xi ∈ mathfrakg. A feedforward neural network is trained to learn this mapping across several groups, including SE(2), SE(3), SO(3), and SL(2,ℝ). It demonstrates strong empirical accuracy under both clean and noisy conditions, which validates the viability of data-driven recovery of Lie group generators using shallow neural architectures. This is Lie-RL GitHub Repo https://github.com/Anormalm/LieRL-on-Trajectories. Feel free to make suggestions and collaborations!