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Physics-Guided Flow Matching: A Generative AI Framework for Global Total Electron Content Forecasting

Si-Tong Yang, Feng Ye, Xian Lu, Dong Lin, Benjamin C. Martinez

arXiv:2610.37159v1cs.LGphysics.space-phstat.ML

TL;DR

Global TEC forecasting matters for space-weather applications, yet flow matching has been insufficiently explored for this task. PGFM guides flow matching with ionospheric PDE dynamics and adaptively fuses physical and data-driven estimates. It improves spatial and temporal accuracy over representative benchmarks, with learned weights showing spatially varying physical contributions.

  • Problem

    Global TEC forecasting is important for space-weather applications, while flow matching's application to this task remains insufficiently explored.

    Source: §1 · p. 2

  • Method

    PGFM constructs a PDE-guided physical flow and combines a physical estimate of TEC tendency with a data-driven velocity estimate through spatially adaptive fusion.

    Source: §1 · p. 4, §3.2.2 · p. 10

  • Results

    PGFM improves spatial and temporal forecasting accuracy over representative benchmarks, and its learned weights show that physical contributions vary spatially.

    Source: §6 · p. 19

  • Takeaways & Limitations

    The results demonstrate that PDE-based physical guidance can be incorporated into generative global TEC forecasting with location-varying physical contributions.

    Source: §6 · p. 19

    The TEC balance approximation fixes the lower boundary and neglects correction terms from temporal and horizontal variation of the upper boundary, assuming they are small relative to retained terms.

    Source: §3.2.1 · p. 8

Abstract

from arXiv · show

Accurate forecasting of global total electron content (TEC) is important for space weather monitoring and the reliable operation of satellite communication and navigation systems. We propose a physics-guided generative artificial intelligence (AI) framework for day-ahead global TEC forecasting. The framework builds on flow matching, which learns to gradually transform samples from an initial distribution into new samples that share the statistical characteristics of the target data. The proposed model incorporates ionospheric dynamics described by a partial differential equation (PDE) into this transformation process, guiding the changes between initial and target samples using physical knowledge. The model further combines the physical dynamics with statistical patterns captured from data to learn this transformation, adaptively adjusting their relative contributions at each location. The model is trained and evaluated using one year of global TEC data generated by the Thermosphere Ionosphere Electrodynamics General Circulation Model (TIEGCM). The results demonstrate that our model improves both spatial and temporal forecasting accuracy over representative benchmarks, including statistical, deep learning, and generative AI methods at all lead times and across latitudes. The average root mean square error (RMSE) is 39.2% lower than that of ConvLSTM and 23.8% lower than that of the same framework without physical guidance. The learned contribution weights vary systematically with latitude and across geographic locations, showing how the physical contribution changes across the forecast domain. These findings demonstrate that explicit physical guidance improves generative forecasts of global TEC and highlight the value of incorporating physical dynamics into generative AI modeling.

1 Introduction

Global TEC forecasting matters for space-weather applications, but flow matching remains insufficiently explored for this task. PGFM uses PDE-guided flow and spatially adaptive fusion of physical and data-driven estimates.

1 Introduction

  • Ionospheric electron-density changes can degrade satellite communications and GNSS performance, motivating reliable forecasting for space-weather applications.TEC is widely used to characterize ionospheric variability and its effects on radio propagation.Source: §1 · p. 2
  • Coupled physics-based models can require substantial computational resources and real-time solar-wind observations that may be impractical for operational forecasting.Source: §1 · p. 3
  • Flow matching learns a velocity field for continuous transformations between source and target data distributions, and global TEC forecasting remains insufficiently explored.Its continuous transformations make it a promising framework for TEC maps that evolve continuously under ionospheric processes.Source: §1 · pp. 3–4
  • PGFM constructs a physically aligned flow from PDE-generated intermediate states and learns its associated velocity field while retaining generative forecasting flexibility.This contrasts with generic interpolation or statistically motivated flow constructions.Source: §1 · p. 4
  • The model combines physical dynamics estimated from historical information with a data-driven velocity field through jointly learned spatial weights.The relative contributions can vary across locations.Source: §1 · p. 4

2 Data Description and Problem Formulation

The study uses one year of TIEGCM simulations to construct global TEC maps and formulate multistep forecasts conditioned on past TEC, physical drivers, and ionospheric dynamics.

2.1 Data Description

  • TIEGCM version 3.0 simulates the coupled thermosphere-ionosphere system, and its lower levels are nudged with hourly SD-WACCM-X winds, temperatures, and geopotential heights.The model covers approximately 97–600 km altitude.Source: §2.1 · pp. 4–5
  • TEC is defined as vertically integrated electron density for each ionospheric column, converted to TEC units and collected into a global map.The integration domain begins at approximately 97 km; its upper boundary can vary with location and time.Source: §2.1 · p. 5
  • One year of TIEGCM data is stored every 15 minutes on a 72 × 144 global grid, with 2.5° × 2.5° spatial resolution.Source: §2.1 · p. 5
  • Each TEC field is a 72 × 144 map, with 72 latitude points and 144 longitude points.Source: §2.1 · p. 5
  • The dataset includes solar radio flux F10.7, geomagnetic index Kp, and variables describing O+ production, chemical loss, electrodynamic drift, and neutral-wind transport.The transport variables include horizontal and vertical E × B drift.Source: §2.1 · p. 5

2.2 Forecasting Problem Formulation

  • Forecasting is formulated as multistep spatiotemporal prediction conditioned on past TEC maps, available solar and geomagnetic drivers, and physical-process equations.Source: §2.2 · p. 5
  • Figure 2 presents a representative global TEC snapshot alongside solar and geomagnetic drivers and horizontal and vertical E × B drift fields.The TEC and drift snapshots are from 17 February 2023 at 22:00 UTC.Source: Figure 2 · p. 6

3 Methodology

The method constructs a flow-matching probability path that follows PDE-modeled ionospheric evolution, then learns a forecast velocity by combining physical and data-driven estimates. It integrates that velocity to make hourly TEC forecasts autoregressively.

3.1 Flow Matching

  • Flow matching learns a time-dependent velocity field that transports samples along a probability path from a prior distribution toward the target data distribution.The method specifies endpoint coupling and a conditional path between paired samples; training aims for intermediate states to follow the designed distributions.Source: §3.1 · p. 7, §3.2.1 · p. 7

3.2.1 Physics-Guided Probability Path

  • The physical path is built from the TEC evolution equation, combining photochemical production and loss with plasma transport.The TEC balance is obtained by vertically integrating the electron-density evolution equation.Source: §3.2.1 · p. 8
  • The PDE-guided path uses simulated ionospheric evolution as its conditional mean, with stochastic variance around that trajectory for intermediate training states.The variance is largest at the midpoint and decreases toward the endpoints; the TEC balance assumes upper-boundary correction terms are small relative to retained terms.Source: §3.2.1 · pp. 8–9

3.2.2 Adaptive “Velocity” Fusion with Physical Dynamics

  • A learned spatial weighting map fuses physical and data-driven velocities, varying their relative contributions by location and flow coordinate.The physical estimator predicts net TEC tendency from ion-dynamic history and physical drivers; the data-driven estimator learns from historical TEC and drivers.Source: §3.2.2 · pp. 10–11

3.2.3 Forecast

  • The model forecasts one hour at a time, appending each predicted TEC map to the history window and repeating until the 24-hour horizon.At each forecast cycle, the flow coordinate resets to τ = 0.Source: §3.2.3 · pp. 11–12

4 Experiments

Experiments use TIEGCM-generated global TEC data to assess PGFM in a one-day autoregressive forecasting setting. The design compares it with statistical, deep-learning, and flow-matching benchmarks and evaluates forecast-map errors.

4.1 Experiment Design

  • The study uses one year of TIEGCM global TEC data, training one-hour transitions autoregressively to a maximum 24-hour forecast horizon.The monthly split yields 305 training days, 24 validation days, and 36 test days.Source: §4.1 · p. 12
  • PGFM is compared with statistical, deep-learning, and vanilla flow-matching benchmarks; an oracle with true future physical dynamics is diagnostic only.The oracle is excluded from operational rankings because future physical information is unavailable in real-time forecasting.Source: §4.1 · pp. 12–13

4.2 Evaluation Metrics

  • Forecast maps are evaluated against the TIEGCM reference at the valid time using RMSE, MAE, and nMAE over valid spatial grid cells.For lead time h, the reference map is the TIEGCM map at t + h.Source: §4.2 · p. 13

5 Results and Discussion

PGFM reproduces large-scale global TEC patterns, with errors that increase at longer lead times and vary by latitude. Across the 24-hour horizon it has the lowest errors among deployable models, while its learned physical–data-driven fusion weights vary across the forecast domain.

5.1 Global TEC Forecasts and Spatial Error Patterns

  • PGFM reproduces principal large-scale TEC patterns, but errors grow with lead time and become especially concentrated at low latitudes by 12 hours.Absolute-error profiles peak at low latitudes, where PGFM has its largest reductions; nMAE rises toward high latitudes and more strongly toward the southern polar region.Source: §5.1 · pp. 14–15

5.2 Overall Forecasting Performance

  • PGFM has the lowest errors among deployable models across reported metrics and lead times; averaged over 24 hours, RMSE is 1.892 TECU, 39.2% below ConvLSTM and 23.8% below FMB.PGFM also reduces average MAE and nMAE relative to both comparators.Source: §5.2 · p. 15, Table 1 · p. 16
  • Forecast errors rise with lead time, but PGFM remains the lowest-error deployable model throughout the 24-hour horizon, while the strongest comparator shifts from FMB at 1–12 hours to ConvLSTM at 24 hours.At 24 hours, PGFM’s RMSE is 3.266 TECU versus 3.439 TECU for ConvLSTM.Source: §5.2 · p. 15, Figure 7 · p. 17

5.3 Analysis of the Learned “Velocity”

  • The learned fusion weight λ is generally larger at low and lower-middle latitudes and decreases toward middle and high latitudes, indicating spatially adaptive contributions from the physical branch.The southern polar region shows more varied net O+ rates; 15-minute TIEGCM sampling may limit information about temporal variability in polar physical-process rates.Source: §5.3 · p. 17, Figure 8 · p. 18
  • The fused PGFM “velocity” better recovers the target structures’ spatial extent than FMB, while the data-driven and physical branches capture different aspects of the target pattern.Here, “velocity” is a change in flow-matching state space, not physical plasma drift velocity.Source: §5.3 · pp. 17–18, Figure 9 · p. 19

6 Conclusion

The framework combines PDE-guided flow matching with spatial fusion of physical and data-driven velocity estimates. On one year of TIEGCM data, it improves spatial and temporal forecasting accuracy over representative benchmarks.

6 Conclusion

  • The model improves spatial and temporal forecasting accuracy over representative forecasting benchmarks in evaluation using TIEGCM-generated data.The learned weighting fields show that the physical contribution varies spatially.Source: §6 · p. 19
  • The probability path follows ionospheric PDE evolution, while the O+ continuity equation supplies a physical velocity estimate fused with a data-driven estimate.The fusion mechanism adapts spatially.Source: §6 · p. 19
  • Analysis of branch velocities shows how adaptive fusion changes the learned transport field and aligns it with the target structure.Source: §6 · p. 19
  • Future work will investigate latent-space formulations to reduce computational and memory costs and consider extension to other systems governed by known PDEs.Source: §6 · p. 19

Appendix A Flow Matching Background

Flow matching learns a vector field that transports samples from a prior distribution toward a target distribution. It uses tractable conditional paths to train this field without explicitly evaluating unavailable marginal densities and velocities.

Appendix A Flow Matching Background

  • Flow matching learns a time-dependent vector field that transports samples from a prior distribution toward a target distribution.The learned field is integrated from flow coordinate 0 to 1 to transport prior samples toward the target.Source: Appendix A · pp. 25–26
  • Training specifies a probability-density path between prior and target distributions, whose velocity field governs probability-mass transport along that path.Source: Appendix A · pp. 25–26
  • Because marginal densities and their velocity fields are generally unavailable in closed form, flow matching constructs tractable conditional paths between coupled endpoint samples.A general Gaussian conditional path combines the endpoints and Gaussian noise.Source: Appendix A · p. 26
  • The neural vector field is trained against a conditional target velocity, avoiding explicit evaluation of the marginal density and velocity.Source: Appendix A · p. 26

Appendix B Network Architecture

The architecture estimates velocity using separate data-driven and physical components, then combines them into a final predicted velocity.

Appendix B Network Architecture

  • The architecture has two components: a data-driven velocity estimator and a physical velocity estimator.Source: Appendix B · p. 26
  • The data-driven and physical estimators provide distinct velocity estimates within the overall architecture.Source: Appendix B · p. 26
  • The two components are combined to produce the final predicted velocity.Source: Appendix B · p. 26

B1 Data-Driven“velocity”Estimator

The data-driven velocity estimator uses a two-dimensional U-Net with residual blocks and encoded TEC and physical-driver histories. It standardizes inputs and produces a single-channel standardized velocity.

B1 Data-Driven“velocity”Estimator

B2 Physical“velocity”Estimator

A ConvGRU estimates the physical branch’s net TEC tendency from historical ionospheric dynamics and driver covariates, producing stepwise maps over the forecast cycle.

B2 Physical“velocity”Estimator

  • Historical inputs combine four ionospheric process terms with scalar and spatial physical-driver covariates, ending at forecast initialization.The process terms are O+ production, chemical loss, electrodynamic drift, and neutral-wind transport; future dynamics are predicted rather than supplied as conditioning inputs.Source: B2 Physical“velocity”Estimator · p. 29
  • The two-layer ConvGRU encoder summarizes physical history, and a corresponding two-layer decoder recurrently produces future-dynamics features.Decoder layers start from the corresponding encoder’s final hidden states; the encoder processes inputs chronologically.Source: B2 Physical“velocity”Estimator · p. 29
  • Each decoder step yields one physical net TEC-tendency map in TECU per second, and the sequence spans the forecast cycle.The output head maps the second-layer hidden state to a single-channel map representing the combined physical-process contribution.Source: B2 Physical“velocity”Estimator · p. 29

B3 Adaptive“velocity”Fusion

The physical and data-driven branches are aligned at each flow coordinate, then combined through a learnable spatial weighting map trained with the flow-matching objective.

B3 Adaptive“velocity”Fusion

  • The K physical predictions are assigned to uniform subintervals of flow time and converted into normalized flow-space velocity using the forecast interval and TEC standard deviation.For a one-hour transition with tendencies in TECU per second, the forecast interval is 3600 seconds.Source: B3 Adaptive“velocity”Fusion · pp. 29–30
  • At each location and flow coordinate, λ weights the physical estimate while 1−λ weights the data-driven estimate.The spatial weighting map has values between 0 and 1.Source: B3 Adaptive“velocity”Fusion · p. 30
  • Flow-matching training learns the fusion weights to reduce the discrepancy between the fused estimate and the target velocity.Source: B3 Adaptive“velocity”Fusion · p. 30

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