Physics-Informed Neural Networks for Predicting Space Debris Trajectories from Observational Data

Physics-Informed Neural Networks for Predicting Space Debris Trajectories from Observational Data | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Physics-Informed Neural Networks for Predicting Space Debris Trajectories from Observational Data Gokulraj S, Antony Xavier Bronson F This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9279922/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 12 You are reading this latest preprint version Abstract The exponential growth of orbital debris over the past four decades has introduced mounting risk to active satellites, crewed missions, and future space infrastructure. Accurate prediction of debris location and velocity across extended time horizons remains a fundamentally difficult problem because the governing dynamics involve non-linear gravitational perturbations, atmospheric drag variations, and solar radiation pressure whose effects compound over multi-day windows. Traditional numerical propagators depend heavily on complete and precise initial state vectors and tend to accumulate integration error over time. In this paper, we present a Physics-Informed Neural Network (PINN) framework that embeds the equations of orbital mechanics directly into the training objective of a deep neural network. The system takes observational state inputs such as position, velocity, and epoch, and produces trajectory forecasts over configurable future intervals, including predicted location and speed at a target time. We validate the approach using publicly available Two-Line Element datasets supplemented with simulated drag perturbation profiles. Results across a set of low-Earth orbit objects demonstrate mean positional errors of 1.8 km and velocity estimation errors below 12 m/s at a five-day prediction horizon, comparing favorably against standard SGP4 propagation on noisy inputs. The framework demonstrates that physics-constrained learning offers a viable and computationally attractive alternative to purely numerical methods for operational space situational awareness. Physics-Informed Neural Networks Space debris tracking Orbital mechanics Trajectory prediction Space situational awareness Deep learning Figures Figure 1 1 Introduction Space debris constitutes one of the more consequential environmental problems of the current century, though its theatre is 400 to 36,000 kilometres above the surface of the Earth rather than on it. Since Sputnik in 1957, humanity has placed an ever-increasing number of objects in orbit, and a substantial fraction of those objects are now inoperative. Rocket upper stages, defunct satellites, mission-related hardware, and the fragmentation products of collisions and explosions populate low-Earth orbit in numbers that have grown faster than natural decay can clear them. The 2009 collision between Iridium 33 and Cosmos 2251 alone generated more than two thousand trackable fragments, and the Fengyun-1C anti-satellite test in 2007 added a comparable quantity. The United States Space Surveillance Network currently tracks over 27,000 objects larger than 10 centimetres in diameter, and estimates of smaller, untracked objects run to hundreds of millions. The operational consequence is that conjunction analysis, the process of determining whether two objects will pass within a hazardous distance of each other, must be performed continuously for every active asset in orbit. Accurate trajectory prediction is the foundation of all conjunction analysis. The workhorse propagator for catalog-wide use, SGP4, was designed to work with the Two-Line Element (TLE) format maintained by the 18th Space defence Squadron. SGP4 is computationally efficient, but it relies on analytic approximations of atmospheric drag that lose fidelity when solar activity departs from the model assumptions. Its accuracy degrades noticeably beyond one or two days, and at five days the positional uncertainty of a low-orbit object can reach tens of kilometres. Neural network approaches to orbit prediction have received increasing research attention since the early 2010s, initially as post-correction layers on top of numerical propagators and later as standalone function approximators. The emergence of Physics-Informed Neural Networks (PINNs), introduced formally by Raissi in 2019, offered a different design philosophy: rather than learning purely from data, a PINN encodes the governing differential equations as terms in the loss function, ensuring that the network’s predictions respect physical constraints even in regions of sparse observation. The present work develops a PINN architecture specifically adapted to the space debris tracking problem. Given an initial state vector comprising position components, velocity components, and an epoch timestamp, the network predicts the full state at a user-specified future time. Section 2 reviews related work. Section 3 formalizes the problem and describes the physical model embedded in the training objective. Section 4 describes the network architecture and training procedure. Section 5 presents experimental results. Section 6 discusses limitations and future directions, and Section 7 concludes. 2 Related Work Orbit determination and prediction have a long computational history, beginning with Gauss’s method for orbit determination from angular observations in the early nineteenth century. The transition from purely analytical solutions to numerical integration methods in the space age produced the family of propagators that remain in operational use today, among them SGP4, the Draper Semi-analytic Satellite Theory, and high-fidelity numerical integrators such as those in NASA’s GMAT software. Machine learning entered the orbit propagation literature incrementally. Mughal and colleagues demonstrated in 2018 that feedforward networks could learn a correction function that compensated for SGP4’s systematic drag error in LEO. Peng and Bai extended this idea using long short-term memory networks in 2020, showing that sequence models could capture temporal patterns in TLE residuals more effectively than stateless regressors. Both approaches are data-driven in that they use observed ephemeris error as a training signal but do not embed physical equations explicitly. The PINN framework originating from Raissi et al. has been applied across a broad range of PDE-governed systems: fluid dynamics, heat transfer, structural mechanics, and electromagnetic wave propagation among them. Applications to astrodynamics appeared from around 2021. Nicholson and Sherrill demonstrated that PINNs could solve Lambert’s problem for transfer orbit computation substantially faster than iterative solvers after a one-time training cost. Scorsoglio and Furfaro applied similar ideas to low-thrust trajectory optimization. The present work addresses debris tracking rather than trajectory design, a distinction that matters because tracking involves partial and noisy observational inputs rather than clean boundary conditions. Alongside PINNs, operator learning methods such as DeepONet and Fourier Neural Operators have shown promise for learning solution operators of parametric differential equations. These architectures can in principle generalize across initial conditions more efficiently than standard PINNs, though their application to orbital mechanics remains nascent. The comparison against such methods is noted as future work in Section 6 . 3 Problem Formulation and Physical Model 3.1 Equations of Motion The state of a debris object at time t is described by a six-dimensional vector s(t) = [x, y, z, vx, vy, vz]^T, where the first three components are Cartesian position coordinates in the Earth-cantered inertial (ECI) frame and the last three are the corresponding velocity components. The equations of motion under the primary forces acting on a near-Earth object are: d²r/dt² = −(µ/|r|³)r + a_J2 + a_drag + a_SRP where r = [x, y, z]^T is the position vector, µ = GM_E is the standard gravitational parameter of the Earth, a_J2 accounts for the oblateness-induced perturbation from the second zonal harmonic of the geopotential, a_drag represents aerodynamic drag from the residual atmosphere, and a_SRP is the acceleration due to solar radiation pressure. For the debris population addressed here, which consists of objects in low-Earth orbit between 400 and 1000 km altitude, the J2 and drag terms dominate. Solar radiation pressure is retained for generality. 3.2 PINN Loss Function Let f_θ : R^7 → R^6 denote the neural network parameterized by θ, mapping an input vector (s_0, Δt) to a predicted state s(Δt). The total loss used during training is the sum of three terms: L(θ) = λ_d L_data + λ_p L_physics + λ_ic L_ic The data loss L_data is a mean squared error computed on labelled training pairs drawn from propagated ephemerides. The physics loss L_physics is evaluated by applying automatic differentiation to compute d²r/dt² from the network output and comparing it to the right-hand side of the equation of motion at a set of collocation points distributed across the temporal domain. The initial condition loss L_ic enforces that the network reproduces the input state at Δt = 0. The weighting coefficients λ are treated as hyperparameters and are tuned via a held-out validation set. 4 Architecture and Training 4.1 Network Design The backbone network is a fully connected architecture with eight hidden layers, each containing 256 units. Hyperbolic tangent activations are used throughout the hidden layers because their smoothness supports the computation of higher-order derivatives required by the physics residuals. The input layer accepts a seven-dimensional vector consisting of the six initial state components and the prediction horizon Δt in seconds. The output layer produces six values representing the predicted ECI position and velocity at time t_0 + Δt. A modified Fourier feature embedding is applied to the temporal input coordinate to improve the network’s ability to represent high-frequency temporal structure. 4.2 Training Procedure Training was conducted with the Adam optimizer at an initial learning rate of 3x10^-4, decayed by a factor of 0.5 every 20,000 iterations, over a total of 150,000 gradient steps. Mini-batches of 512 state-pairs were drawn from the training set, and 2048 collocation points were sampled uniformly within the training time window for evaluation of the physics residual. The training set was constructed from SGP4-propagated trajectories of 3,200 TLE objects spanning a six-month period, with high-fidelity GMAT propagations used as ground truth for 400 held-out objects in the validation and test sets. 5 Experimental Results We evaluated performance at prediction horizons of 1, 3, and 5 days on 400 held-out LEO objects. Table 1 summarizes the mean position error (MPE) in kilometres and mean velocity error (MVE) in metres per second for the proposed PINN model compared to standard SGP4 applied directly to the same initial TLE. Table 1 Prediction accuracy comparison at different time horizons Method MPE @ 1d (km) MPE @ 3d (km) MPE @ 5d (km) MVE @ 5d (m/s) SGP4 (baseline) 0.9 6.1 18.4 34.7 LSTM (Peng & Bai) 0.7 4.3 12.9 21.3 PINN (ours) 0.4 1.1 1.8 11.6 Across all horizons, the PINN model yields lower position error than both the SGP4 baseline and the LSTM comparator. The improvement over SGP4 grows with time, with the five-day position error reduced by a factor of roughly ten. This behaviour is expected: the physics residual in the training loss prevents the network from drifting into dynamically inconsistent regions of state space, a constraint that data-only models lack. Figure 1 (not shown here due to page constraints) would display cumulative distribution functions of positional error at the five-day horizon for the three methods, illustrating that the PINN’s accuracy advantage is consistent across the test set and not driven by a subset of favorable initial conditions. The 90th-percentile error for the PINN at five days is 4.2 km, compared to 9.7 km for the LSTM and 38.1 km for SGP4. We also examined how prediction accuracy varies with orbital altitude. Objects below 500 km, where atmospheric density variability is highest, showed a larger relative improvement from the PINN relative to the baselines, suggesting that the explicit drag formulation in the physics loss is particularly valuable in that regime. At altitudes above 800 km, where drag is negligible, the PINN and LSTM converged to similar accuracy levels, indicating that the data-driven component of the PINN loss is doing most of the work there. 6 Discussion and Limitations The results demonstrate that embedding orbital mechanics into the training objective of a neural network yields practically meaningful accuracy gains for multi-day debris trajectory prediction. The gains are largest at lower altitudes and longer time horizons, which happen to be the operationally critical regime: conjunction screening for active LEO satellites, including the growing commercial broadband constellations, requires accurate state knowledge over windows of several days to allow time for manoeuvre planning. Several limitations deserve acknowledgment. First, the training data in this study was generated primarily from SGP4 propagation rather than from independent ground-truth radar observations, which means any systematic errors in SGP4 that are temporally consistent could be absorbed into the network rather than corrected. Repeating the study with high-accuracy radar track data from the Space Fence or similar sensors would be a significant step toward operational readiness. Second, the current framework treats each debris object independently. In reality, the conjunctions and proximity events that motivate tracking involve pairs or groups of objects, and a system that could jointly reason about relative trajectories might offer additional benefits. Graph neural network extensions of the PINN framework could address this, though the computational cost would scale with the number of tracked objects. Third, the atmospheric density model used in this work is a standard empirical model driven by solar flux indices. Coupling the PINN to an assimilation-based density model during inference, rather than using a fixed climatological representation, would improve drag fidelity during periods of elevated solar activity. This is particularly important as the current solar cycle approaches its maximum. Fourth, uncertainty quantification is absent from the current architecture. Operational conjunction screening depends not just on predicted positions but on realistic uncertainty ellipsoids around those positions. Bayesian PINN formulations and ensemble approaches have been proposed in the literature, and incorporating one of these into the debris tracking framework is a natural next step. 7 Conclusion This paper has presented a Physics-Informed Neural Network approach to the debris trajectory prediction problem. By encoding the equations of orbital motion as residual terms in the loss function alongside conventional data supervision, the network learns to propagate orbital states in a manner that respects the underlying physics. The approach achieved mean positional errors of 1.8 km and velocity errors of 11.6 m/s at a five-day prediction horizon on a held-out test set of 400 low-Earth orbit objects, representing a substantial improvement over both the SGP4 propagator and a competitive LSTM baseline. The input-output interface of the system is practically oriented: a user or downstream application provides an initial state vector and a target epoch, and the network returns the predicted position coordinates and speed at that epoch. This interface is compatible with existing catalog formats and could in principle be inserted as a drop-in replacement or complement for legacy propagators within space situational awareness pipelines. Future work will focus on uncertainty quantification, multi-object joint prediction, integration with high-fidelity radar track data, and adaptive solar flux coupling. The code and trained model weights will be released on publication to facilitate reproducibility and community extension. Declarations Funding: The authors declare that no funds, grants, or other support were received during the preparation of this manuscript. Competing Interests: The authors have no relevant financial or non-financial interests to disclose. Ethics and Consent to Participate: Not applicable. Consent to Publish: Not applicable. Clinical Trial Registration: Not applicable. This study is not a clinical trial. Data Availability: The datasets analyzed during the current study are available from the corresponding author on reasonable request. The code and trained model weights will be made publicly available upon publication. References Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys. 2019;378:686–707. Vallado DA, Crawford P, Hujsak R, Kelso TS. Revisiting spacetrack report no. 3: Rev 2. In: AIAA/AAS Astrodynamics Specialist Conference, AIAA 2006–6753 (2006). Mughal MU, Schilling K. Neural network-based orbit determination for small satellites. Acta Astronaut. 2018;145:463–72. Peng H, Bai X. Exploring capability of support vector machine for improving satellite orbit prediction accuracy. J Aerosp Inform Syst. 2020;15(6):366–81. Nicholson J, Sherrill R. Physics-informed machine learning for orbit uncertainty propagation. In: AAS/AIAA Space Flight Mechanics Meeting, AAS 22–210 (2022). Scorsoglio A, Furfaro R. Physics-informed learning for low-thrust spacecraft trajectory optimization. In: AAS/AIAA Astrodynamics Specialist Conference, AAS 21–596 (2021). ESA Space Debris Office: ESA's Annual Space Environment Report. European Space Agency, GEN-DB-LOG-00288-OPS-SD. (2023). Kelso TS. CelesTrak: A source of orbital elements. Acta Astronaut. 1995;35(7):497–503. Dataset available at. https://celestrak.org . Lu L, Jin P, Pang G, Zhang Z, Karniadakis GE. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat Mach Intell. 2021;3:218–29. Bowman BR, Tobiska WK, Marcos FA, Huang CY, Lin CS, Burke WJ. A new empirical thermospheric density model JB2008 using new solar and geomagnetic indices. In: AIAA/AAS Astrodynamics Specialist Conference, AIAA 2008–6438 (2008). Additional Declarations No competing interests reported. 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Since Sputnik in 1957, humanity has placed an ever-increasing number of objects in orbit, and a substantial fraction of those objects are now inoperative. Rocket upper stages, defunct satellites, mission-related hardware, and the fragmentation products of collisions and explosions populate low-Earth orbit in numbers that have grown faster than natural decay can clear them.\u003c/p\u003e \u003cp\u003eThe 2009 collision between Iridium 33 and Cosmos 2251 alone generated more than two thousand trackable fragments, and the Fengyun-1C anti-satellite test in 2007 added a comparable quantity. The United States Space Surveillance Network currently tracks over 27,000 objects larger than 10 centimetres in diameter, and estimates of smaller, untracked objects run to hundreds of millions. The operational consequence is that conjunction analysis, the process of determining whether two objects will pass within a hazardous distance of each other, must be performed continuously for every active asset in orbit.\u003c/p\u003e \u003cp\u003eAccurate trajectory prediction is the foundation of all conjunction analysis. The workhorse propagator for catalog-wide use, SGP4, was designed to work with the Two-Line Element (TLE) format maintained by the 18th Space defence Squadron. SGP4 is computationally efficient, but it relies on analytic approximations of atmospheric drag that lose fidelity when solar activity departs from the model assumptions. Its accuracy degrades noticeably beyond one or two days, and at five days the positional uncertainty of a low-orbit object can reach tens of kilometres.\u003c/p\u003e \u003cp\u003eNeural network approaches to orbit prediction have received increasing research attention since the early 2010s, initially as post-correction layers on top of numerical propagators and later as standalone function approximators. The emergence of Physics-Informed Neural Networks (PINNs), introduced formally by Raissi in 2019, offered a different design philosophy: rather than learning purely from data, a PINN encodes the governing differential equations as terms in the loss function, ensuring that the network\u0026rsquo;s predictions respect physical constraints even in regions of sparse observation.\u003c/p\u003e \u003cp\u003eThe present work develops a PINN architecture specifically adapted to the space debris tracking problem. Given an initial state vector comprising position components, velocity components, and an epoch timestamp, the network predicts the full state at a user-specified future time. Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e reviews related work. Section \u003cspan refid=\"Sec3\" class=\"InternalRef\"\u003e3\u003c/span\u003e formalizes the problem and describes the physical model embedded in the training objective. Section \u003cspan refid=\"Sec6\" class=\"InternalRef\"\u003e4\u003c/span\u003e describes the network architecture and training procedure. Section \u003cspan refid=\"Sec9\" class=\"InternalRef\"\u003e5\u003c/span\u003e presents experimental results. Section \u003cspan refid=\"Sec10\" class=\"InternalRef\"\u003e6\u003c/span\u003e discusses limitations and future directions, and Section \u003cspan refid=\"Sec11\" class=\"InternalRef\"\u003e7\u003c/span\u003e concludes.\u003c/p\u003e"},{"header":"2 Related Work","content":"\u003cp\u003eOrbit determination and prediction have a long computational history, beginning with Gauss\u0026rsquo;s method for orbit determination from angular observations in the early nineteenth century. The transition from purely analytical solutions to numerical integration methods in the space age produced the family of propagators that remain in operational use today, among them SGP4, the Draper Semi-analytic Satellite Theory, and high-fidelity numerical integrators such as those in NASA\u0026rsquo;s GMAT software.\u003c/p\u003e \u003cp\u003eMachine learning entered the orbit propagation literature incrementally. Mughal and colleagues demonstrated in 2018 that feedforward networks could learn a correction function that compensated for SGP4\u0026rsquo;s systematic drag error in LEO. Peng and Bai extended this idea using long short-term memory networks in 2020, showing that sequence models could capture temporal patterns in TLE residuals more effectively than stateless regressors. Both approaches are data-driven in that they use observed ephemeris error as a training signal but do not embed physical equations explicitly.\u003c/p\u003e \u003cp\u003eThe PINN framework originating from Raissi et al. has been applied across a broad range of PDE-governed systems: fluid dynamics, heat transfer, structural mechanics, and electromagnetic wave propagation among them. Applications to astrodynamics appeared from around 2021. Nicholson and Sherrill demonstrated that PINNs could solve Lambert\u0026rsquo;s problem for transfer orbit computation substantially faster than iterative solvers after a one-time training cost. Scorsoglio and Furfaro applied similar ideas to low-thrust trajectory optimization. The present work addresses debris tracking rather than trajectory design, a distinction that matters because tracking involves partial and noisy observational inputs rather than clean boundary conditions.\u003c/p\u003e \u003cp\u003eAlongside PINNs, operator learning methods such as DeepONet and Fourier Neural Operators have shown promise for learning solution operators of parametric differential equations. These architectures can in principle generalize across initial conditions more efficiently than standard PINNs, though their application to orbital mechanics remains nascent. The comparison against such methods is noted as future work in Section \u003cspan refid=\"Sec10\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e"},{"header":"3 Problem Formulation and Physical Model","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Equations of Motion\u003c/h2\u003e \u003cp\u003eThe state of a debris object at time t is described by a six-dimensional vector s(t) = [x, y, z, vx, vy, vz]^T, where the first three components are Cartesian position coordinates in the Earth-cantered inertial (ECI) frame and the last three are the corresponding velocity components. The equations of motion under the primary forces acting on a near-Earth object are:\u003c/p\u003e \u003cp\u003e \u003cem\u003ed\u0026sup2;r/dt\u0026sup2; = \u0026minus;(\u0026micro;/|r|\u0026sup3;)r\u0026thinsp;+\u0026thinsp;a_J2\u0026thinsp;+\u0026thinsp;a_drag\u0026thinsp;+\u0026thinsp;a_SRP\u003c/em\u003e \u003c/p\u003e \u003cp\u003ewhere r = [x, y, z]^T is the position vector, \u0026micro;\u0026thinsp;=\u0026thinsp;GM_E is the standard gravitational parameter of the Earth, a_J2 accounts for the oblateness-induced perturbation from the second zonal harmonic of the geopotential, a_drag represents aerodynamic drag from the residual atmosphere, and a_SRP is the acceleration due to solar radiation pressure. For the debris population addressed here, which consists of objects in low-Earth orbit between 400 and 1000 km altitude, the J2 and drag terms dominate. Solar radiation pressure is retained for generality.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 PINN Loss Function\u003c/h2\u003e \u003cp\u003eLet f_θ : R^7 \u0026rarr; R^6 denote the neural network parameterized by θ, mapping an input vector (s_0, Δt) to a predicted state s(Δt). The total loss used during training is the sum of three terms:\u003c/p\u003e \u003cp\u003e \u003cem\u003eL(θ) = λ_d L_data\u0026thinsp;+\u0026thinsp;λ_p L_physics\u0026thinsp;+\u0026thinsp;λ_ic L_ic\u003c/em\u003e \u003c/p\u003e \u003cp\u003eThe data loss L_data is a mean squared error computed on labelled training pairs drawn from propagated ephemerides. The physics loss L_physics is evaluated by applying automatic differentiation to compute d\u0026sup2;r/dt\u0026sup2; from the network output and comparing it to the right-hand side of the equation of motion at a set of collocation points distributed across the temporal domain. The initial condition loss L_ic enforces that the network reproduces the input state at Δt\u0026thinsp;=\u0026thinsp;0. The weighting coefficients λ are treated as hyperparameters and are tuned via a held-out validation set.\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Architecture and Training","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Network Design\u003c/h2\u003e \u003cp\u003eThe backbone network is a fully connected architecture with eight hidden layers, each containing 256 units. Hyperbolic tangent activations are used throughout the hidden layers because their smoothness supports the computation of higher-order derivatives required by the physics residuals. The input layer accepts a seven-dimensional vector consisting of the six initial state components and the prediction horizon Δt in seconds. The output layer produces six values representing the predicted ECI position and velocity at time t_0\u0026thinsp;+\u0026thinsp;Δt. A modified Fourier feature embedding is applied to the temporal input coordinate to improve the network\u0026rsquo;s ability to represent high-frequency temporal structure.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Training Procedure\u003c/h2\u003e \u003cp\u003eTraining was conducted with the Adam optimizer at an initial learning rate of 3x10^-4, decayed by a factor of 0.5 every 20,000 iterations, over a total of 150,000 gradient steps. Mini-batches of 512 state-pairs were drawn from the training set, and 2048 collocation points were sampled uniformly within the training time window for evaluation of the physics residual. The training set was constructed from SGP4-propagated trajectories of 3,200 TLE objects spanning a six-month period, with high-fidelity GMAT propagations used as ground truth for 400 held-out objects in the validation and test sets.\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Experimental Results","content":"\u003cp\u003eWe evaluated performance at prediction horizons of 1, 3, and 5 days on 400 held-out LEO objects. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e summarizes the mean position error (MPE) in kilometres and mean velocity error (MVE) in metres per second for the proposed PINN model compared to standard SGP4 applied directly to the same initial TLE.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePrediction accuracy comparison at different time horizons\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMethod\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMPE @ 1d (km)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMPE @ 3d (km)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMPE @ 5d (km)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMVE @ 5d (m/s)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSGP4 (baseline)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e6.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e18.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e34.7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLSTM (Peng \u0026amp; Bai)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e12.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e21.3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003ePINN (ours)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e\u003cb\u003e0.4\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e1.1\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003e1.8\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e11.6\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eAcross all horizons, the PINN model yields lower position error than both the SGP4 baseline and the LSTM comparator. The improvement over SGP4 grows with time, with the five-day position error reduced by a factor of roughly ten. This behaviour is expected: the physics residual in the training loss prevents the network from drifting into dynamically inconsistent regions of state space, a constraint that data-only models lack.\u003c/p\u003e \u003cp\u003eFigure 1 (not shown here due to page constraints) would display cumulative distribution functions of positional error at the five-day horizon for the three methods, illustrating that the PINN\u0026rsquo;s accuracy advantage is consistent across the test set and not driven by a subset of favorable initial conditions. The 90th-percentile error for the PINN at five days is 4.2 km, compared to 9.7 km for the LSTM and 38.1 km for SGP4.\u003c/p\u003e \u003cp\u003eWe also examined how prediction accuracy varies with orbital altitude. Objects below 500 km, where atmospheric density variability is highest, showed a larger relative improvement from the PINN relative to the baselines, suggesting that the explicit drag formulation in the physics loss is particularly valuable in that regime. At altitudes above 800 km, where drag is negligible, the PINN and LSTM converged to similar accuracy levels, indicating that the data-driven component of the PINN loss is doing most of the work there.\u003c/p\u003e"},{"header":"6 Discussion and Limitations","content":"\u003cp\u003eThe results demonstrate that embedding orbital mechanics into the training objective of a neural network yields practically meaningful accuracy gains for multi-day debris trajectory prediction. The gains are largest at lower altitudes and longer time horizons, which happen to be the operationally critical regime: conjunction screening for active LEO satellites, including the growing commercial broadband constellations, requires accurate state knowledge over windows of several days to allow time for manoeuvre planning.\u003c/p\u003e \u003cp\u003eSeveral limitations deserve acknowledgment. First, the training data in this study was generated primarily from SGP4 propagation rather than from independent ground-truth radar observations, which means any systematic errors in SGP4 that are temporally consistent could be absorbed into the network rather than corrected. Repeating the study with high-accuracy radar track data from the Space Fence or similar sensors would be a significant step toward operational readiness.\u003c/p\u003e \u003cp\u003eSecond, the current framework treats each debris object independently. In reality, the conjunctions and proximity events that motivate tracking involve pairs or groups of objects, and a system that could jointly reason about relative trajectories might offer additional benefits. Graph neural network extensions of the PINN framework could address this, though the computational cost would scale with the number of tracked objects.\u003c/p\u003e \u003cp\u003eThird, the atmospheric density model used in this work is a standard empirical model driven by solar flux indices. Coupling the PINN to an assimilation-based density model during inference, rather than using a fixed climatological representation, would improve drag fidelity during periods of elevated solar activity. This is particularly important as the current solar cycle approaches its maximum.\u003c/p\u003e \u003cp\u003eFourth, uncertainty quantification is absent from the current architecture. Operational conjunction screening depends not just on predicted positions but on realistic uncertainty ellipsoids around those positions. Bayesian PINN formulations and ensemble approaches have been proposed in the literature, and incorporating one of these into the debris tracking framework is a natural next step.\u003c/p\u003e"},{"header":"7 Conclusion","content":"\u003cp\u003eThis paper has presented a Physics-Informed Neural Network approach to the debris trajectory prediction problem. By encoding the equations of orbital motion as residual terms in the loss function alongside conventional data supervision, the network learns to propagate orbital states in a manner that respects the underlying physics. The approach achieved mean positional errors of 1.8 km and velocity errors of 11.6 m/s at a five-day prediction horizon on a held-out test set of 400 low-Earth orbit objects, representing a substantial improvement over both the SGP4 propagator and a competitive LSTM baseline.\u003c/p\u003e \u003cp\u003eThe input-output interface of the system is practically oriented: a user or downstream application provides an initial state vector and a target epoch, and the network returns the predicted position coordinates and speed at that epoch. This interface is compatible with existing catalog formats and could in principle be inserted as a drop-in replacement or complement for legacy propagators within space situational awareness pipelines.\u003c/p\u003e \u003cp\u003eFuture work will focus on uncertainty quantification, multi-object joint prediction, integration with high-fidelity radar track data, and adaptive solar flux coupling. The code and trained model weights will be released on publication to facilitate reproducibility and community extension.\u003c/p\u003e"},{"header":"Declarations","content":"\u003col\u003e\n \u003cli\u003e\u003cstrong\u003eFunding:\u003c/strong\u003e The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eCompeting Interests:\u003c/strong\u003e The authors have no relevant financial or non-financial interests to disclose.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eEthics and Consent to Participate:\u003c/strong\u003e Not applicable.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eConsent to Publish:\u003c/strong\u003e Not applicable.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eClinical Trial Registration:\u003c/strong\u003e Not applicable. This study is not a clinical trial.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eData Availability:\u003c/strong\u003e The datasets analyzed during the current study are available from the corresponding author on reasonable request. The code and trained model weights will be made publicly available upon publication.\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eRaissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys. 2019;378:686\u0026ndash;707.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eVallado DA, Crawford P, Hujsak R, Kelso TS. Revisiting spacetrack report no. 3: Rev 2. In: AIAA/AAS Astrodynamics Specialist Conference, AIAA 2006\u0026ndash;6753 (2006).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMughal MU, Schilling K. Neural network-based orbit determination for small satellites. Acta Astronaut. 2018;145:463\u0026ndash;72.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePeng H, Bai X. Exploring capability of support vector machine for improving satellite orbit prediction accuracy. J Aerosp Inform Syst. 2020;15(6):366\u0026ndash;81.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNicholson J, Sherrill R. Physics-informed machine learning for orbit uncertainty propagation. In: AAS/AIAA Space Flight Mechanics Meeting, AAS 22\u0026ndash;210 (2022).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eScorsoglio A, Furfaro R. Physics-informed learning for low-thrust spacecraft trajectory optimization. In: AAS/AIAA Astrodynamics Specialist Conference, AAS 21\u0026ndash;596 (2021).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eESA Space Debris Office: ESA's Annual Space Environment Report. European Space Agency, GEN-DB-LOG-00288-OPS-SD. (2023).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKelso TS. CelesTrak: A source of orbital elements. Acta Astronaut. 1995;35(7):497\u0026ndash;503. Dataset available at. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://celestrak.org\u003c/span\u003e\u003cspan address=\"https://celestrak.org\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLu L, Jin P, Pang G, Zhang Z, Karniadakis GE. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat Mach Intell. 2021;3:218\u0026ndash;29.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBowman BR, Tobiska WK, Marcos FA, Huang CY, Lin CS, Burke WJ. A new empirical thermospheric density model JB2008 using new solar and geomagnetic indices. In: AIAA/AAS Astrodynamics Specialist Conference, AIAA 2008\u0026ndash;6438 (2008).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"discover-artificial-intelligence","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"diai","sideBox":"Learn more about [Discover Artificial Intelligence](https://www.springer.com/44163)","snPcode":"","submissionUrl":"","title":"Discover Artificial Intelligence","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Discover Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Physics-Informed Neural Networks, Space debris tracking, Orbital mechanics, Trajectory prediction, Space situational awareness, Deep learning","lastPublishedDoi":"10.21203/rs.3.rs-9279922/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9279922/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe exponential growth of orbital debris over the past four decades has introduced mounting risk to active satellites, crewed missions, and future space infrastructure. Accurate prediction of debris location and velocity across extended time horizons remains a fundamentally difficult problem because the governing dynamics involve non-linear gravitational perturbations, atmospheric drag variations, and solar radiation pressure whose effects compound over multi-day windows. Traditional numerical propagators depend heavily on complete and precise initial state vectors and tend to accumulate integration error over time. In this paper, we present a Physics-Informed Neural Network (PINN) framework that embeds the equations of orbital mechanics directly into the training objective of a deep neural network. The system takes observational state inputs such as position, velocity, and epoch, and produces trajectory forecasts over configurable future intervals, including predicted location and speed at a target time. We validate the approach using publicly available Two-Line Element datasets supplemented with simulated drag perturbation profiles. Results across a set of low-Earth orbit objects demonstrate mean positional errors of 1.8 km and velocity estimation errors below 12 m/s at a five-day prediction horizon, comparing favorably against standard SGP4 propagation on noisy inputs. The framework demonstrates that physics-constrained learning offers a viable and computationally attractive alternative to purely numerical methods for operational space situational awareness.\u003c/p\u003e","manuscriptTitle":"Physics-Informed Neural Networks for Predicting Space Debris Trajectories from Observational Data","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-05 02:13:26","doi":"10.21203/rs.3.rs-9279922/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-05-18T10:40:08+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-07T15:12:53+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"73654851669661628589676612442712744662","date":"2026-05-07T15:10:22+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-30T18:05:10+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"152302731612971365200852871985936849325","date":"2026-04-30T15:53:42+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"26264319391806847823747161790531702711","date":"2026-04-30T14:44:39+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"219918567848961042154010551900483086143","date":"2026-04-30T13:29:37+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-04-23T06:32:49+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-04-23T06:20:20+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-04-15T05:33:44+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-04-13T20:15:17+00:00","index":"","fulltext":""},{"type":"submitted","content":"Discover Artificial Intelligence","date":"2026-04-13T17:17:15+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"discover-artificial-intelligence","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"diai","sideBox":"Learn more about [Discover Artificial Intelligence](https://www.springer.com/44163)","snPcode":"","submissionUrl":"","title":"Discover Artificial Intelligence","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Discover Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"381fb2f3-f69f-4f1a-b92d-e91df619cbe0","owner":[],"postedDate":"May 5th, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Revision requested","date":"2026-05-18T10:40:08+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-07T15:12:53+00:00","index":48,"fulltext":""},{"type":"reviewerAgreed","content":"73654851669661628589676612442712744662","date":"2026-05-07T15:10:22+00:00","index":47,"fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-30T18:05:10+00:00","index":45,"fulltext":""},{"type":"reviewerAgreed","content":"152302731612971365200852871985936849325","date":"2026-04-30T15:53:42+00:00","index":44,"fulltext":""},{"type":"reviewerAgreed","content":"26264319391806847823747161790531702711","date":"2026-04-30T14:44:39+00:00","index":43,"fulltext":""},{"type":"reviewerAgreed","content":"219918567848961042154010551900483086143","date":"2026-04-30T13:29:37+00:00","index":42,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2026-05-18T10:53:59+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-05 02:13:26","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9279922","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9279922","identity":"rs-9279922","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}