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Fluid-Spatiotemporal Stochastic Geometry for Information Flow in Non-Stationary Fields: Kinematics and Field Reconstruction

Wen-Yu Dong, Weiwei Jiang, Song Zhao, Qi Bi, Sheng Chen

arXiv:2610.23277v1cs.NI

TL;DR

The paper addresses how to recover directional macroscopic transport from aggregate density evolution when density alone does not identify the full vector field. It selects a unique weighted minimum-energy representative under stated conditions, then derives kinematic quantities and evaluates reconstruction against controlled tests and held-out traffic crossings.

  • Problem

    Aggregate density evolution constrains flux divergence and normal boundary flux but can leave density-invisible circulation undetermined, so directional macroscopic transport is not uniquely recoverable from density alone.

    Source: §I · p. 1

  • Method

    The paper formulates a weighted inverse boundary value problem and selects an instantaneous minimum-energy velocity representative from fields consistent with estimated density evolution, source, and boundary data.

    Source: §I · p. 1, B. Weighted Minimum-Energy Selection… · p. 6

  • Results

    Under stated positivity and compatibility conditions, the selected representative is unique; experiments examine reconstruction consistency and sensitivity, and pNEUMA traces validate internal transport against held-out crossing measurements.

    Source: §VII · p. 16

  • Takeaways & Limitations

    The reconstructed field supports information-flux, divergence, and centroid quantities as a kinematic description of macroscopic demand transport.

    Source: §VII · p. 16

    The selected velocity is a macroscopic representative, not individual user velocities, and its pointwise-in-time minimum-energy selection does not establish a global Wasserstein geodesic for the observed density path.

    Source: B. Weighted Minimum-Energy Selection… · p. 6

Abstract

from arXiv · show

Classical stochastic geometry primarily addresses forward analysis by prescribing spatial point-process laws and mobility models. It does not directly address the inverse problem of recovering directional macroscopic transport from aggregate density evolution in non-stationary spatial networks. This paper develops the kinematic foundation of fluid-spatiotemporal stochastic geometry (F-STSG), representing an evolving mobile population by an intensity field under an assumed continuum limit while retaining the network infrastructure as a discrete spatial measure. We formulate macroscopic transport reconstruction as an inverse boundary value problem. Under a strictly positive bounded intensity and compatible source and boundary data, an instantaneous weighted minimum-energy criterion selects a unique velocity representative from the admissible class. The reconstruction reduces to a weighted elliptic Neumann problem and separates density-identifiable transport from weighted-solenoidal circulation that is not identifiable from density evolution alone. We further derive the information flux, interface transport, velocity-divergence relation, and centroid dynamics. Numerical experiments examine reconstruction consistency and sensitivity, while the public New Era of Urban Traffic Monitoring with Aerial Footage (pNEUMA) dataset provides held-out validation of internal transport. A separate unmanned aerial vehicle (UAV) study evaluates the centroid as a causal downstream control reference under observation delay, motion constraints, and execution perturbations.

I. INTRODUCTION

The paper addresses the inverse problem of recovering macroscopic transport from aggregate density evolution, then develops F-STSG as a weighted reconstruction framework that distinguishes identifiable transport from density-invisible circulation. It positions this approach alongside related transport, stochastic-geometry, continuum, and field-estimation work, and derives further kinematic quantities and evaluations.

I. INTRODUCTION

  • The inverse problem begins with observed intensity evolution and specified source and boundary data, unlike forward models that prescribe mobility or velocity-density relations.The selected transport is a macroscopic representative, not an estimate of individual trajectories or microscopic utilities.Source: §I · p. 1, 3) Continuum and mean-field models · p. 2

1) Dynamic optimal transport:

  • Dynamic optimal transport uses a continuity constraint and kinetic-energy objective over a density–velocity path; this paper instead selects a weighted minimum-energy velocity at each observation time.The selection does not require the observed density path to be a finite-horizon Wasserstein geodesic.Source: 1) Dynamic optimal transport · p. 2

2) Mobility-aware stochastic geometry:

  • Mobility-aware stochastic geometry models prescribed displacement or time-varying intensity, whereas F-STSG reconstructs a density-identifiable transport representative from estimated density evolution and source and boundary information.Demand-aware UAV deployment methods provide application-level references and baselines for the later control study.Source: 2) Mobility-aware stochastic geometry · p. 2

3) Continuum and mean-field models:

  • Continuum and mean-field models commonly prescribe a velocity-density relation, cost, or agent objective and solve a forward evolution, equilibrium, or control problem.F-STSG instead treats density evolution, source, and boundary exchange as inputs to a fixed-time inverse selection.Source: 3) Continuum and mean-field models · p. 2

4) Spatiotemporal estimation and forecasting:

  • Spatiotemporal graph and data-driven methods can estimate or forecast density fields for conservation-constrained reconstruction, with source and boundary terms supplied by measurements or separate models.Euclidean geometry is not necessarily retained as an explicit state variable across the broader graph and data-driven class.Source: 4) Spatiotemporal estimation and… · p. 2

B. Main Contributions

  • F-STSG reconstructs a unique minimum-energy velocity representative under stated positivity, compatibility, and boundary assumptions, separating density-identifiable transport from invisible circulation.The framework also derives network-level kinematic quantities and evaluates reconstruction and centroid-based control.Source: B. Main Contributions · p. 3
  • For weakly varying positive backgrounds, a first-order perturbation, Green’s-function representation, and spectral decomposition extend the analysis beyond radial fields.The modal form describes how heterogeneity modifies reconstructed transport through the background operator’s eigenstructure.Source: B. Main Contributions · p. 3
  • The paper derives network-level interface transport, population balance, divergence, material-derivative relations, and centroid dynamics from the reconstructed field.Controlled experiments examine convergence and sensitivity, pNEUMA traces validate internal crossing rates against held-out measurements, and a separate UAV study evaluates centroid-based control.Source: B. Main Contributions · p. 3

II. SYSTEM MODEL

F-STSG represents the network through a continuous mobile-population intensity field coupled to discrete infrastructure. Its continuum model derives from a scaling limit and supports a balance law with source and boundary exchange, while the approximation’s reliability depends on local observation support.

II. SYSTEM MODEL

  • F-STSG couples a continuous intensity field for the mobile population with a discrete random measure for network infrastructure, allowing transport to be interpreted relative to cell boundaries.The core kinematic theory applies to general simple, stationary ergodic point processes, although numerical validation assumes a Poisson point process for tractability.Source: §II · p. 3, A. Network Skeleton: Discrete Random… · p. 4

A. Network Skeleton: Discrete Random Measures

  • The infrastructure measure represents base-station locations, and users associate with the station providing maximum biased received power under a weighted Voronoi tessellation.The association rule uses tier-specific transmit power, bias weight, and path-loss exponent.Source: A. Network Skeleton: Discrete Random… · p. 4

B. Continuum Limit and Macroscopic Intensity Field

  • Under population scaling, normalized empirical measures are assumed to converge weakly in probability to a deterministic density, defining the macroscopic intensity field λ(x, t).The field follows a balance law with velocity, local source or sink, and possible boundary exchange; the main analytical development uses a closed, source-free special case.Source: B. Continuum Limit and Macroscopic… · p. 4
  • Continuum reconstruction can remain poor where local effective observation counts are small, even when total population is large.Sparse rural-IoT settings may require coarser bins, stronger regularization, a parametric density model, or a hybrid discrete–continuous description.Source: B. Continuum Limit and Macroscopic… · p. 4

C. Observation of the Macroscopic Intensity Field

The reconstruction pipeline estimates positive intensity and its time derivative from aggregated observations, then uses source and boundary information to recover transport. The paper also characterizes shape-changing intensity evolution, couples demand to active infrastructure, and frames the inverse reconstruction as minimum-energy transport selection.

C. Observation of the Macroscopic Intensity Field

  • Aggregate samples, occupancy counts, or load telemetry can estimate intensity, while neighboring regularized estimates provide its time derivative; boundary and source terms must be observed, modeled, or set to zero.Validation of reconstructed transport requires trajectory or boundary-crossing measurements withheld from reconstruction, and the estimated intensity must remain positive on the modeled support.Source: C. Observation of the Macroscopic… · p. 5

D. Topological Dynamics: Non-Separable Intensity Fields

  • A separable intensity field preserves spatial shape, so nontrivial redistribution in a closed source-free model with constant total mass requires non-separable evolution.For the power-law example, a time-varying decay exponent changes spatial shape and requires nontrivial macroscopic transport in the closed source-free model.Source: D. Topological Dynamics: Non-Separable… · p. 5

E. Dynamic State Process: Fluid-Infrastructure Coupling

  • F-STSG represents active infrastructure as a thinned discrete measure whose activation may depend on local intensity, while the reconstructed continuum transport is interpreted relative to cell boundaries and active resources.Users associate with the active base station providing maximum biased received power.Source: E. Dynamic State Process:… · p. 5

III. FIELD RECONSTRUCTION: THE INVERSE BOUNDARY VALUE PROBLEM

  • The inverse problem reconstructs the minimum-energy macroscopic velocity consistent with estimated density evolution and specified source and boundary data.The continuum field is the aggregate state, while latent dynamics also include local generation or depletion and boundary exchange.Source: §III · p. 5

A. Conservation Laws and Underdetermined System

The conservation law constrains density evolution through flux divergence and boundary exchange, but does not uniquely determine velocity. A pointwise weighted minimum-energy criterion selects a unique representative under stated positivity and compatibility assumptions, without recovering density-invisible circulation.

A. Conservation Laws and Underdetermined System

  • Density evolution supplies one scalar constraint on the vector velocity, leaving weighted-solenoidal components invisible when source and boundary data are known.Unknown source or boundary exchange is additionally confounded with transport.Source: A. Conservation Laws and… · p. 6

B. Weighted Minimum-Energy Selection and Its Optimal-Transport Connection

  • The minimum-energy objective has the kinetic-energy form used in dynamic optimal transport, but the reconstruction is performed independently at each observation time.The paper does not assert that the observed density path is a global Wasserstein geodesic.Source: B. Weighted Minimum-Energy Selection… · p. 6
  • The minimum-energy construction selects the unique irrotational admissible velocity under positive bounded intensity and the stated domain and compatibility conditions.The selected field is a pointwise-in-time minimizer, not a claim that the full density path is a Wasserstein geodesic.Source: B. Weighted Minimum-Energy Selection… · pp. 6–7
  • The selected field is a macroscopic transport representative; density-invisible weighted-solenoidal circulation is not recovered from density evolution alone.Such circulation may affect microscopic mobility or channel performance, which requires additional trajectory or channel information for quantitative evaluation.Source: B. Weighted Minimum-Energy Selection… · pp. 6–7

C. Existence and Uniqueness of Scalar Potential

Under positive, bounded density and compatible data, the weighted Neumann problem has a unique selected velocity field. The density and balance constraints identify transport only up to compatible circulation, while radial symmetry yields a closed-form velocity.

C. Existence and Uniqueness of Scalar Potential

  • The result assumes a bounded connected Lipschitz domain, strictly positive bounded intensity, compatibility, and prescribed boundary data.The theorem applies separately at each time satisfying these conditions.Source: C. Existence and Uniqueness of Scalar… · p. 7
  • A weak solution for the scalar potential exists and is unique up to an additive constant, making the selected velocity field unique.The zero-mean gauge fixes the potential representative; its gradient is unchanged by adding a constant.Source: C. Existence and Uniqueness of Scalar… · p. 7
  • Nonzero outer-boundary flux represents crossings of the observation boundary, while internal interfaces are handled separately by flux integrals.Only crossings of the outer boundary enter the prescribed boundary-flux data.Source: C. Existence and Uniqueness of Scalar… · p. 7
  • The selected velocity depends continuously on density, forcing, and boundary-flux data, with perturbations entering through their respective errors.Unresolved source variations may alias into the forcing error; if the source is unknown, transport and source activity are not identifiable from density alone.Source: C. Existence and Uniqueness of Scalar… · p. 7
  • For a sufficiently regular radially symmetric field on a disk with no source or boundary flux, radial velocity is the negative enclosed accumulation rate divided by local density.Net accumulation implies inward velocity, while net depletion implies outward velocity.Source: C. Existence and Uniqueness of Scalar… · pp. 7–8
  • The theorem does not cover vanishing-density regions, unknown sources, unrestricted microscopic velocity components, or temporal regularity without additional assumptions.The result establishes pointwise-in-time well-posedness, not continuity or differentiability of the reconstructed field over time.Source: C. Existence and Uniqueness of Scalar… · p. 7

D. Analytical Properties and Physical Interpretation

The selected field supports balance-consistent kinematic quantities, but does not itself guarantee forecasting or control performance. Reconstruction proceeds from aggregated observations to velocity and flux for a separate controller, with computational cost and latency depending on the update pipeline.

D. Analytical Properties and Physical Interpretation

  • The reconstructed field is balance-consistent and depends continuously on coefficient, forcing, and boundary data, but does not alone guarantee forecasting or control performance.Prediction and control also depend on observation error, latency, prediction, and the downstream controller.Source: D. Analytical Properties and Physical… · p. 8

E. Reconstruction Workflow and Computational Requirements

  • At each update, the workflow estimates aggregate density and its time derivative, checks compatibility, solves the weighted elliptic problem, and returns velocity and flux to a separate controller.The observations are spatially aggregated rather than individual trajectories.Source: E. Reconstruction Workflow and… · p. 8
  • The inverse solve scales as O(NitNnz) operations, while memory scales as O(Nnz + (Lt + 1)Mg); a fixed local stencil gives Nnz = O(Mg).Iteration count still depends on grid resolution, conditioning, preconditioning, and stopping tolerance.Source: E. Reconstruction Workflow and… · p. 8
  • For a 20 × 20 pNEUMA grid, inverse reconstruction averaged 6.733 ms per update, with a 95th-percentile latency of 8.715 ms.The benchmark covers 2,095 timed executions over 419 real field updates, from prepared aggregate fields through recovery of the potential, velocity, and flux; it excludes telemetry, preprocessing, field estimation, and downstream control.Source: E. Reconstruction Workflow and… · p. 8
  • The measured workstation reconstruction time does not establish end-to-end real-time feasibility, which depends on the full pipeline and update interval.Aggregation, estimation, temporal differentiation, and inversion must together finish within the update interval.Source: E. Reconstruction Workflow and… · p. 8

IV. PERTURBATION ANALYSIS AND GREEN’S FUNCTION FORMALISM

For small perturbations around a positive background field, the paper linearizes the weighted elliptic reconstruction and expresses its response through a Neumann Green’s function. Outside that regime, reconstruction uses the full weighted elliptic problem.

IV. PERTURBATION ANALYSIS AND GREEN’S FUNCTION FORMALISM

  • When the positive background field varies weakly, with ϵ ≪1, a first-order approximation is derived; otherwise, the full weighted elliptic problem is solved.The approximation is explicitly restricted to the weak-variation regime.Source: §IV · p. 9

A. Linearized Field Equation

  • The intensity is decomposed into a specified background state and a mass-conserving first-order perturbation.The background pair satisfies the governing equation at zeroth order.Source: A. Linearized Field Equation · p. 9
  • Substitution into the governing PDE and retention of O(ϵ) terms yields a linearized equation for the potential perturbation.Rearranging isolates the weighted elliptic operator based on the background intensity.Source: A. Linearized Field Equation · p. 9
  • The effective forcing is Seff = ∂tδλ + ∇·(δλv0), the first-order mass imbalance induced by the density perturbation.The background velocity is v0 = −∇ϕ0, and the response operator is L0 = ∇·(λ0∇).Source: A. Linearized Field Equation · p. 9

B. Green’s Function Integral Representation

  • A zero-mean gauge defines the Neumann Green’s function for the background operator despite the global mass-conservation constraint.The construction uses homogeneous Neumann boundary conditions.Source: B. Green’s Function Integral… · p. 9
  • Within the linearization regime, the Green’s function maps compatible first-order effective forcing to the first-order velocity perturbation.Outside the small-perturbation and positive-background assumptions, the full weighted elliptic problem must be solved directly.Source: B. Green’s Function Integral… · p. 9

C. Spectral Decomposition of the Weighted Operator

The weighted elliptic operator’s spectral expansion attenuates higher-eigenvalue spatial modes in the potential response. The resulting information flux represents modeled-intensity transport across interfaces, while wireless-performance interpretation requires a separate model.

C. Spectral Decomposition of the Weighted Operator

  • The Neumann Green’s-function expansion excludes the constant eigenmode associated with global mass conservation.The background operator is self-adjoint under the stated Neumann boundary condition, with a zero eigenvalue for the constant eigenfunction.Source: C. Spectral Decomposition of the… · p. 9
  • Each spatial eigenmode contributes to the potential perturbation with a factor 1/νn, so larger-νn modes contribute less; this is not temporal decay or evidence of noise robustness.Finite-count and estimator sensitivity are evaluated separately.Source: C. Spectral Decomposition of the… · p. 10

V. KINEMATIC ANALYTICAL TOOLBOX

  • The kinematic identities require spatial and temporal differentiability assumptions beyond those needed for weak existence and uniqueness.Prediction additionally requires causal estimation and an explicit model of subsequent field evolution.Source: §V · p. 10

A. Information Flux Vector

  • The information flux is the product of local intensity and selected macroscopic velocity, describing transport of the modeled intensity measure.Its signed integral over an oriented interface gives instantaneous transport in the chosen normal direction.Source: A. Information Flux Vector · p. 10
  • For a fixed regional partition, internal interface fluxes represent exchange between neighboring regions, while outer-boundary crossings are counted separately.The regional balance also includes outward flux through the portion of each region on the domain boundary.Source: A. Information Flux Vector · p. 10
  • Coupling transport to area spectral efficiency requires a specified wireless-performance model; circulation effects on channel or interference dynamics are not identifiable from density alone.If intensity denotes demand rather than active-link density, an activity or scheduling map is also required.Source: A. Information Flux Vector · p. 10

B. Topological Compression and Material Derivative

The section relates velocity divergence to material density change and derives how the intensity centroid moves. In a source-free closed system, centroid velocity is the intensity-weighted mean velocity; sources and boundary exchange can modify it.

B. Topological Compression and Material Derivative

  • For a conservative system, congestion divergence equals the negative logarithmic rate of intensity change along a moving fluid element; D < 0 marks compression, while density growth may also reflect S > 0.The material derivative captures the rate of change experienced by a moving fluid element.Source: B. Topological Compression and… · pp. 10–11

C. Global Centroid Dynamics

  • In a source-free closed system, centroid velocity reduces to the intensity-weighted mean velocity; more generally, source activity and boundary exchange contribute through their first moments.Source: C. Global Centroid Dynamics · p. 11

VI. NUMERICAL AND EMPIRICAL VALIDATION

The experiments serve distinct roles, including finite-population convergence checks. Under the matched generator and estimator, density-estimation error follows an approximately Npop^-1/2 trend over the tested range, but operational continuum adequacy remains dataset-dependent.

VI. NUMERICAL AND EMPIRICAL VALIDATION

  • The validation experiments distinguish matched consistency checks, controlled sensitivity tests, held-out pNEUMA transport validation, and a UAV control use case.Source: §VI · p. 11

A. Finite-Population Convergence

  • At the largest simulated population (Npop ≈ 10^7), the selected estimator’s error was below 10^-3 with a Poisson generator; finite-range results were consistent with expected asymptotic convergence under the matched generator and smoothing rule.Snapshot variability decreased with population size.Source: A. Finite-Population Convergence · p. 12
  • These finite-range results do not validate operational mobility traces; continuum adequacy depends on aggregation scale, reporting interval, effective observation count, and estimator bias.Source: A. Finite-Population Convergence · p. 12

B. Matched Minimum-Energy Reconstruction

Matched tests check identifiability, controlled perturbations examine sensitivity, and pNEUMA traces provide held-out validation of internal transport. On pNEUMA, reconstructed crossing rates outperform a centroid-translation baseline on correlation and error metrics, while sensitivity tests identify observation noise and update counts as important constraints.

B. Matched Minimum-Energy Reconstruction

C. Sensitivity to Observation, Estimation, and Model Perturbations

D. Real-Trace Validation With pNEUMA Mobility Data

  • Across three pNEUMA sessions and interfaces, F-STSG scored Pearson 0.511 and Spearman 0.515 versus −0.028 and −0.040; MAE was 0.766 versus 0.877 vehicles/s and RMSE 0.965 versus 1.076 vehicles/s.The averages compare F-STSG with the centroid-translation baseline.Source: Table IV · p. 14, D. Real-Trace Validation With pNEUMA… · p. 14

E. Validation and Temporal Interpretation of Kinematic Metrics

A matched periodic example checks the divergence and centroid identities and illustrates their timing relative to center density. The paper also uses the intensity-weighted centroid as a delayed-observation control reference, with motion constraints and execution-layer effects treated separately.

E. Validation and Temporal Interpretation of Kinematic Metrics

  • In the selected periodic radial example, peak compression and inward transport preceded the center-density maximum at 12 h, and both indicators crossed zero at that maximum.The exact lead times are specific to this example.Source: E. Validation and Temporal… · p. 14

1) Centroid Reference and Controller:

2) Evaluation Setup and Results:

The evaluation tests causal centroid control under observation delay and execution perturbations, alongside reconstruction validation using pNEUMA traces. C3 improves service distance over C2, while abrupt demand changes and execution nonidealities expose practical limits.

2) Evaluation Setup and Results:

  • C3 extrapolates the centroid using a least-squares linear fit to up to five available estimates, while C1 repeats the latest density peak and C2 repeats the latest centroid.C4 uses the future true centroid only as a noncausal reference; each run samples 50,000 counts on a 25 × 25 grid.Source: 2) Evaluation Setup and Results · p. 15
  • C3 achieves a 12.80% mean paired service-distance gain over C2 and a 16.47% gain over C1 across 90 runs, with mean tracking lag of 3.33 versus 7.00 steps.The gain over C2 accompanies a 9.59% increase in pooled control effort; nominal direct execution shows no solver failure or listed constraint violation.Source: 2) Evaluation Setup and Results · p. 16
  • As observation delay rises from zero to eight steps, the mean paired C3-over-C2 gain increases from 1.29% to 21.94%, alongside higher control effort.In abrupt migration, the gain falls to 5.81%, indicating that linear centroid extrapolation is less effective when demand changes abruptly.Source: Figure 8 · p. 15, 2) Evaluation Setup and Results · p. 16

3) Execution-Layer Stress Test:

  • The execution stress test models finite response time, speed and acceleration saturation, and bounded correlated wind, rather than full quadrotor dynamics.The tested cases include a combined condition with all three effects, and paired controllers share the same wind realization.Source: 3) Execution-Layer Stress Test · p. 16
  • Under combined execution nonidealities, C3 retains a 12.68% paired service gain over C2, but obstacle violations occur in 13 of 90 runs.In abrupt migration, tracking failures affect 14.56% of updates and the C3-over-C2 gain is 4.44%; the surrogate does not validate six-degree-of-freedom flight dynamics or hardware.Source: 3) Execution-Layer Stress Test · p. 16, Table V · p. 16

VII. CONCLUSION

  • The paper formulates transport reconstruction as a weighted inverse boundary value problem, selecting a unique minimum-energy velocity representative under positivity and compatibility conditions.It derives information flux, divergence, and intensity-weighted centroid; pNEUMA traces validate internal transport against held-out crossing measurements, while UAV control is evaluated separately.Source: §VII · p. 16

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