Off the Normal Path: Learning Spatial Density Models of Node Mobility
Wanxin GaoIoanis NikolaidisJanelle Harms
Aug 2026
Machine LearningData Science
Abstract
We consider the problem of learning models of spatial density functions, representing the steady-state density of mobile nodes moving on a two-dimensional terrain. Deriving such models can assist in network design and optimization problems, e.g., by accelerating the computation of the density function during a parameter sweep. We address the question of applicability of off-the-shelf mixture density network models and of, two varieties of, normalizing flows for the description of mobile node density over a disk. We introduce the use of M\"obius distributions to retain symmetric spatial relations. Our results indicate that mixtures of M\"obius distributions provide interpretable, parsimonious models for the studied steady state density distributions, that match or outperform the alternatives.
This article investigates several physics-informed and hybrid machine learning strategies that incorporate physics knowledge in experimental data-driven deep-learning models for predicting the bond quality and porosity of fused filament fabrication (FFF) parts. Three types of strategies are explored to incorporate physics constraints and multi-physics FFF simulation results into a deep neural network (DNN), thus ensuring consistency with physical laws: (1) incorporate physics constraints within the loss function of the DNN, (2) use physics model outputs as additional inputs to the DNN model, and (3) pre-train a DNN model with physics model input-output and then update it with experimental data. These strategies help to enforce a physically consistent relationship between bond quality and tensile strength, thus making porosity predictions physically meaningful. Eight different combinations of the above strategies are investigated. The results show how the combination of multiple strategies produces accurate machine learning models even with limited experimental data.
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