⚡ electrical engineering

Reduced-Order Data Assimilation for Thermospheric Density Using Physics-informed SINDyc Models

This paper proposes a computationally efficient reduced-order thermospheric density model using physics-informed SINDy $_c$ -AR that, when coupled with a Kalman filter to assimilate satellite observations, significantly improves density estimation accuracy compared to open-loop predictions, particularly during geomagnetic storms.

Original authors: Sriram Narayanan, Daniele Sicoli, Piyush Mehta

Published 2026-04-28

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CC BY 4.0

Original authors: Sriram Narayanan, Daniele Sicoli, Piyush Mehta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Problem: Predicting the "Invisible Wind" in Space

Imagine you are trying to predict how much a kite will tug on its string. To do this, you need to know how hard the wind is blowing. Now, imagine that instead of a kite, you are managing thousands of satellites orbiting Earth. Instead of wind, you are dealing with the thermosphere—a layer of Earth’s atmosphere that is incredibly thin but acts like a "drag" or a "thick soup" for satellites.

The problem is that this "space wind" is incredibly moody. When the sun gets angry (solar flares) or the Earth’s magnetic field gets shaken (geomagnetic storms), the atmosphere expands and thickens almost instantly.

Currently, scientists have two ways to deal with this:

The "Supercomputer" Method: Massive, incredibly detailed physics models. They are very accurate but so slow and "heavy" that they can't keep up with real-time changes. It’s like trying to predict the weather by simulating every single molecule of air in the world.
The "Rule of Thumb" Method: Simple math formulas that run fast but are "dumb." They don't react to sudden solar storms; they just follow a general pattern. It’s like checking a weather app that only tells you if it’s summer or winter, ignoring the sudden thunderstorm outside your window.

The Solution: The "Smart Sketch" Approach

The researchers in this paper created a third way. They combined a "Smart Sketch" with a "Real-Time Correction" system.

1. The Smart Sketch (Reduced-Order Modeling)

Instead of trying to simulate every single molecule (the Supercomputer method), they used a technique called SINDyc-AR.

Think of this like a master artist who doesn't paint every leaf on a tree, but instead learns the essential shapes and patterns of how a tree moves in the wind. By looking at thousands of hours of high-fidelity simulations, the AI learned the "essence" of the atmosphere. It knows that "If the Sun does this, the atmosphere usually does that." This model is incredibly fast—it’s a lightweight, "smart sketch" that captures the big picture without the heavy math.

2. The Real-Time Correction (Data Assimilation)

Even a smart sketch can be wrong. To fix this, they used a Kalman Filter.

Imagine you are driving a car through heavy fog using only a map. The map (the Smart Sketch) tells you where the road should be, but you can't see the actual pavement. However, every few seconds, your car’s sensors (the Satellites) give you a tiny "ping" about your actual position.

The Kalman Filter is the "brain" that constantly compares the map to the pings. If the map says you are in the middle of the lane, but the sensor says you are drifting toward the curb, the filter instantly nudges your mental map to correct itself. In this paper, the "pings" come from satellites like Swarm and GRACE that measure the actual density they feel as they fly through space.

Why This Matters: The "Multi-Satellite" Superpower

The coolest part of this paper is that this system can listen to multiple satellites at once.

If one satellite is flying over the North Pole and another is over the Equator, the system takes both of their "pings" and uses them to update one single, global "smart sketch." It’s like having several scouts spread across a battlefield, all reporting back to a single general who can then see the entire map with much higher clarity.

The Result

The researchers tested this against the "Supercomputer" and the "Rule of Thumb" methods. They found that:

During calm times: It works great.
During massive solar storms: It is a lifesaver. While the "dumb" models got lost and the "heavy" models were too slow, this system used the satellite "pings" to realize the atmosphere had suddenly thickened and adjusted its predictions in real-time.

In short: They built a way to predict the invisible, changing environment of space that is fast enough to be useful, smart enough to be accurate, and capable of learning from its own mistakes as it flies.

Technical Summary: Reduced-Order Data Assimilation for Thermospheric Density Using Physics-informed SINDyc Models

Problem Statement

Accurate estimation of thermospheric mass density is critical for Low Earth Orbit (LEO) satellite drag prediction, conjunction assessment, and space situational awareness. The thermosphere is a highly dynamic environment that responds nonlinearly to solar extreme ultraviolet (EUV) radiation and geomagnetic activity. Current modeling approaches face a fundamental trade-off:

Physics-based General Circulation Models (GCMs): Highly accurate and physically consistent but computationally prohibitive for real-time applications or rapid uncertainty quantification.
Empirical Models (e.g., NRLMSIS 2.1): Computationally efficient and widely used in operations but lack a time-evolving, physically coupled atmospheric state.

The challenge lies in creating a model that captures the complex, nonlinear physics of the thermosphere at a fraction of the computational cost while remaining capable of being updated in real-time with sparse satellite observations.

Methodology

The authors propose a hybrid framework that couples Reduced-Order Modeling (ROM) with Data Assimilation (DA). The workflow is divided into an offline training phase and an online assimilation phase:

Dimensionality Reduction (PCA): To manage the high-dimensional state space of a GCM, the authors use Principal Component Analysis (PCA) to compress the thermospheric density field into a low-dimensional latent space ( $r=10$ principal components).
System Identification (SINDyc-AR): Instead of a linear model, the authors use Sparse Identification of Nonlinear Dynamics with control (SINDyc-AR). This model is trained on TIE-GCM simulations to identify a sparse set of nonlinear governing equations in the latent space. The "AR" (autoregressive) extension incorporates temporal lags of both the state and external drivers (solar $F_{10.7}$ and geomagnetic $k_p$ indices), allowing the model to capture the atmospheric "memory" of storm events.
Data Assimilation (Extended Kalman Filter): An EKF is implemented to assimilate in situ density measurements from multiple LEO missions (CHAMP, GRACE, GRACE-FO, GOCE, and Swarm).
- Companion-form Lifting: To account for the autoregressive nature of the model, the EKF operates on an augmented state vector that includes lagged latent states, ensuring cross-lag covariances are correctly propagated.
- Log-Density Observation Model: To handle the several orders of magnitude of density variation, measurements are assimilated in $\log_{10}$ space. This results in an affine observation operator with a constant Jacobian, making the EKF update highly stable and well-conditioned.
- Multi-Satellite Extension: The framework is designed to simultaneously assimilate geometrically distinct tracks from multiple satellites into the shared latent space.

Key Contributions

Novel EKF Formulation: Developed an EKF that operates on the companion-form lifting of a SINDyc-AR emulator, utilizing an analytical Jacobian for the nonlinear library to maintain consistency without inflating process noise.
Scale-Invariant Assimilation: Introduced a log-density measurement model that ensures the filter remains well-conditioned across the full dynamic range of the thermosphere.
Multi-Satellite Capability: Demonstrated the ability to calibrate a global latent state using multiple, non-overlapping satellite trajectories.
Open Science: Released a long-term (2000–2025) assimilated thermospheric density dataset to support the research community.

Results

The framework was evaluated against open-loop SINDyc-AR forecasts, empirical models (NRLMSIS 2.1), and the High Accuracy Satellite Drag Model (HASDM).

Single-Satellite Performance: During the extreme 2003 "Halloween" geomagnetic storm, assimilation reduced the Mean Absolute Percentage Error (MAPE) for the withheld GRACE orbit from 98.80% (open-loop) to 22.08%, significantly outperforming empirical models.
Multi-Satellite Synergy: In the November 2009 scenario, adding more satellites improved global coverage. Assimilating CHAMP, GRACE, and GOCE simultaneously reduced the GOCE validation MAPE to 3.47%, demonstrating that multi-orbit coverage effectively calibrates the global latent state.
Model Comparison: While the nonlinear SINDyc-AR generally outperformed the linear DMDc model on withheld orbits within the training regime, the linear DMDc showed better robustness during the 2024 "out-of-training" period, suggesting that nonlinear polynomial terms may struggle with extreme driver extrapolation.

Significance

This work provides a scalable pathway for operational space weather forecasting. By combining the physical insights of GCMs (via SINDyc) with the real-time corrective power of satellite observations (via EKF), the authors have demonstrated a method that is orders of magnitude faster than full GCMs but significantly more responsive to space weather than traditional empirical models. This approach is highly relevant for improving the accuracy of satellite orbit propagation and managing the increasing density of objects in Low Earth Orbit.