EL-KAN: Learning Sparse Conservative Dynamics from Position-Only Trajectories

Learning interpretable particle dynamics from noisy position trajectories requires identifying both which particles interact and the laws governing their interactions. Uneven coverage of interaction distances and unobserved velocities complicate this task, while accurate trajectory fitting alone does not ensure reliable predictions under distribution shifts or interventions. We introduce the Edge-Law Kolmogorov–Arnold Network (EL-KAN), a framework for learning sparse interaction mechanisms in the generated position-only setting, without velocity, force, or graph supervision. EL-KAN represents pairwise interactions through learnable univariate radial potentials whose analytic derivatives produce conservative, equal-and-opposite forces. Weak-form regression transfers temporal differentiation from noisy position observations to known test functions, while sparse graph selection and regularized coefficient refitting identify the interaction structure and governing laws. To address limited radial resolution and discretization bias, EL-KAN combines structured potential candidates, including locally refined radial bases and compact polynomial–exponential dictionaries, with discretization-consistent refitting. Forecasting uses state estimation from an observed trajectory prefix while keeping the learned force field fixed; independent validation selects the potential representation and initialization scheme based on both identification quality and rollout behavior. Experiments on spring, Duffing, and Morse systems under noisy observations and varying sampling resolutions demonstrate effective recovery of sparse interaction structures and nonlinear interaction laws. Compared with the weak-form and anchor controls under the same validation budget, EL-KAN improves long-horizon prediction within and beyond the training distribution and under edge-removal interventions, while retaining explicit, directly editable interaction mechanisms.