A learning method tested on simulated collective motion separated two kinds of influence: the effects agents have on one another and forces acting within an agent from its environment. In synchronization and self-propelled-particle tests, the two versions recovered interaction rules with similar accuracy, while the semi-parametric version produced much better environmental-force estimates across all three benchmarks, including phototaxis. That version was given a prescribed form for the environmental force. The fully non-parametric version learned both types of force directly from trajectory data.
The distinction matters because an interaction kernel is the rule that turns the state of other agents into an influence on one agent. An environmental force represents a separate influence on that agent itself. The central question was whether both could be inferred at the same time and whether the learned rules could predict motion beyond the data used for training.
The clearest gap appeared in force recovery
The Kuramoto synchronization benchmark gave the most detailed comparison. Relative residual error was 9.133 x 10^-3 for the non-parametric method and 9.213 x 10^-3 for the semi-parametric method. Relative interaction-kernel error was nearly the same, at 1.152 x 10^-2 and 1.150 x 10^-2. Environmental-force error, however, fell from 1.992 x 10^-2 with the non-parametric method to 1.713 x 10^-14 with the semi-parametric method. The figures are reported as means with standard deviations across 10 independent training replicates.
The same broad pattern held in the self-propelled-particle benchmark. The two methods recovered the interaction kernel with nearly identical relative error, about 0.12, while the semi-parametric formulation was better at recovering the environmental force and both methods kept trajectory errors small. In the phototaxis test, the semi-parametric environmental-force error was about one order of magnitude lower. Both methods reconstructed and extrapolated trajectories with errors on the order of 10^-6 and 10^-5.
The framework also tried to identify the active rules
The approach is a form of variational learning, meaning it adjusts force functions to reduce the mismatch between predicted and simulated trajectories. Its fully non-parametric estimator minimizes a least-squares objective over multiple trajectory observations. The numerical study kept training, testing, validation and error-evaluation samples separate, then summarized estimator variability across repeated learning trials.
A second part of the study asked the method to choose among candidate dynamical frameworks instead of estimating forces within one fixed description. On synthetic Cucker-Smale data, it selected S2, a second-order alignment-only framework, after admissible candidates tied on validation trajectory error and were separated by model complexity. For phototaxis, it selected S4, the second-order framework combining alignment with environmental forcing, after a validation tie among S4 through S6. In the SPPCS benchmark, S6 was uniquely selected; it included energy-based interaction, alignment and environmental force. In opinion dynamics, the procedure favored F1, a first-order energy-only framework, because it had the lowest complexity among candidates with comparable validation performance.
More trajectories helped more than a denser timeline
Recovery improved when the number of replicate trajectories or the temporal resolution increased. But when the replicate count was just 1, adding more time points had little apparent effect on recovering either the interaction kernel or the environmental force. The result puts the emphasis on seeing varied trajectories, not simply recording each trajectory at finer intervals.
The learned interaction mechanisms also generalized to larger simulated populations and preserved qualitative collective behavior beyond the training interval. In one evaluation, the method reproduced milling, which had been absent during training. This was a test of group-level behavior, so individual agent paths did not need to match the reference trajectories exactly.
A benchmark result is not a real-world validation
The evidence remains limited to synthetic trajectories generated by numerical simulations under the study's specified settings. The preprint therefore shows what the method can recover in the benchmark systems studied, not that it has identified causal interaction laws in a real population. The larger-population result was predictive and qualitative, rather than proof that agent-level predictions remain exact when the system is scaled.
The semi-parametric advantage comes with a built-in dependence on the prescribed environmental-force form, whereas the fully non-parametric formulation offers more flexibility when that form is uncertain. A controlled test also found accurate prediction under 10% relative noise, but that result belongs to the tested noise and derivative-processing settings. The document is an arXiv version 1 preprint dated 25 Aug 2026.
Paper data and sources
Original title: Simultaneous inference of environmental and interaction forces in collective dynamics
Authors: Nipuni de Silva, Ming Zhong, James M. Greene
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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