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Physics-informed Learning for Orbital Uncertainty Propagation with Error Bounds

Chun-Wei Kong, Morteza Lahijanian, Jay McMahon

arXiv:2609.01881v1physics.comp-pheess.SY

TL;DR

Orbital uncertainty propagation governed by the FP-PDE is difficult in nonlinear, high-dimensional, large space-time settings and lacks rigorous error guarantees. The paper uses a physics-informed space-time PDF model with learned worst-case error bounds, and reports accurate orbital propagation, informative certification, and sub-millisecond evaluation after training.

  • Problem

    Solving the FP-PDE for orbital uncertainty is challenging, while existing approaches lack rigorous a priori error bounds and face high-dimensional or long-horizon limitations.

  • Method

    The paper combines a time-conditioned PINN-GMM for valid space-time PDFs with an error PINN and linear programs for bounding event probabilities.

  • Results

    Across four orbital cases, PINN-GMM keeps maximum g TV below 9.0% and maximum WNE below 7.9%, with a single forward pass taking approximately 0.16 milliseconds.

  • Takeaways & Limitations

    The framework provides accurate hours-scale uncertainty propagation with informative error quantification for orbital benchmarks.

  • Takeaways & Limitations

    The PINN-GMM approximation error and learned error bounds can become problematic at an intermediate time, preventing one formulation from covering the full propagation horizon.

Abstract

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The Fokker-Planck partial differential equation (FP-PDE) governs uncertainty evolution in stochastic dynamical systems. In orbital dynamics, solving the FP-PDE is challenging because of nonlinear motion, high-dimensional states, and large space-time domains. We develop a physics-informed neural network (PINN) approach that approximates the FP-PDE solution as a single space-time probability density, while also quantifying its worst-case approximation error. This approach is, in principle, independent of the choice of state coordinates and neural network architecture. Specifically, to enforce probability density function (PDF) properties into the neural network, we design a Physics-informed Gaussian mixture model (PINN-GMM). Then a companion error PINN learns the dynamics of the approximation error and yields time-dependent bounds that define an ambiguity set of PDFs. This ambiguity set enables rigorous computation of upper and lower bounds on event probabilities through tractable linear programs. Numerical studies on illustrative 1D examples and several 4D--6D orbital test cases demonstrate accurate uncertainty propagation, correct and informative error bounds, and improved reliability over common uncertainty-propagation baseline methods (Gaussian approximation, unscented transform, and Gaussian mixture model). Constructing the PINN-GMM requires offline training, making it costlier than the baseline approximations; once trained, however, a single forward pass returns the density at any time in sub-millisecond time $(0.16~\mathrm{ms}$ in our implementation).

I. Introduction

The paper frames orbital uncertainty propagation as learning a full space-time PDF governed by the FP-PDE, addressing the difficulty and limited guarantees of existing approaches. It introduces PINN-GMM representations, error bounds, and event-probability optimization for higher-dimensional, long-horizon settings.

  • Motivation: The FP-PDE describes exact uncertainty evolution, but solving it for orbital dynamics is difficult because of nonlinear motion, high-dimensional states, and large space-time domains.Finite sampling becomes sparse in high-dimensional domains, especially where probability mass concentrates.
  • Proposed direction: The paper learns the FP-PDE solution as a single space-time PDF rather than propagating uncertainty forward through separate time steps.This formulation can represent arbitrary distributions when the network has sufficient approximation capacity.
  • Prior limitations: Existing uncertainty-propagation methods use restricted parametric families or reduced representations that become limiting in high dimensions and over extended horizons.Reported limitations include state-space integral costs, rapidly growing tensor grids, and modeling restrictions such as no process noise or time-dependent drift and diffusion.
  • Prior limitations: Existing methods generally lack rigorous a priori error bounds for the discrepancy between propagated and true PDFs.This limitation is identified for both traditional approximations and direct FP-PDE representations.
  • Proposed direction: A time-conditioned PINN-GMM enforces PDF properties by construction, while an encoder-decoder variant learns the PDF in a latent space.The representation is intended to yield valid PDFs throughout the continuous time horizon and improve trainability for hours-scale orbital problems.
  • Proposed direction: A companion error PINN produces worst-case error bounds, enabling convex linear programs to compute upper and lower event-probability bounds.The framework also proposes a combined sampling strategy and evaluates orbital cases against traditional uncertainty-propagation methods.

II. Problem

The paper formulates orbital uncertainty propagation through an FP-PDE whose solution is a time-dependent PDF, then seeks a PINN approximation, an error bound, and event-probability bounds. The main challenges are difficult FP-PDE learning over large domains and tractable integration of neural PDF approximations.

  • The FP-PDE describes the evolution of the state uncertainty PDF induced by the orbital stochastic differential equation.
  • In general, the FP-PDE cannot be solved in closed form, motivating a physics-informed neural-network approximation.
  • PINN training uses the FP-PDE operator and initial condition rather than labeled PDF data at later times.
  • Training and error-bound construction remain challenging because orbital propagation involves a large space-time domain.
  • Event-probability bounds require integrating the unknown true PDF over an event set using the learned approximation and error bound.

III. Methods

The methods section develops coordinate choices for reducing domain and dynamical complexity, then introduces a generalized mapped-coordinate framework for representing orbital uncertainty. These choices support space-time density learning and downstream PDF transformations.

  • Coordinate choices: Coordinate selection affects both the PDF domain and the structure of the orbital dynamics.
  • Spherical coordinates: Rotating spherical coordinates can reduce the computation domain, but their density transformation is singular when sin θ = 0.
  • Equinoctial coordinates: Equinoctial coordinates can substantially reduce the PDF computation domain and decouple unperturbed Keplerian dynamics except for a and λ nonlinearities.
  • Equinoctial coordinates: Equinoctial-to-Cartesian density transformation is generally unavailable in closed form because converting mean anomaly to true anomaly requires solving Kepler’s equation.
  • Deviation coordinates: Deviation coordinates localize concentrated uncertainty near a reference trajectory, typically shrinking the domain and simplifying local dynamics.

4. Generalized Mapped Coordinates

The paper generalizes coordinate mappings and uses them with a Gaussian-mixture density representation. PINN-GMM enforces PDF constraints by construction while providing a time-conditioned full density with tractable sampling and marginals.

  • Generalized mapped coordinates: A smooth, invertible map transforms the state into latent coordinates whose dynamics and FP-PDE can be represented in latent space.
  • Generalized mapped coordinates: The resulting encoder–latent-dynamics–decoder architecture supports physics-informed space-time density approximation.
  • PINN-GMM: Gaussian-mixture parameterization enforces nonnegativity, normalization, positive-definite covariances, and active mixture components by construction.
  • PINN-GMM: PINN-GMM represents the density as a time-conditioned Gaussian mixture whose parameters are generated by differentiable neural networks.
  • PINN-GMM: Larger K_GMM captures more complex structure at higher training cost, while no systematic component-selection rule is claimed.
  • PINN-GMM: PINN-GMM supports straightforward sampling and closed-form marginal densities, unlike standard MLP representations.
  • PINN-GMM: Unlike traditional GMM propagation, PINN-GMM directly approximates the full space-time density governed by the FP-PDE.

C. Error Bound Construction and Application

The paper learns a time-dependent error bound for the PINN density, forming an ambiguity set that contains the true PDF under sufficient conditions. Linear programs then compute event-probability bounds from tractable cellwise density bounds.

  • Error-bound construction: The approximation error e(x,t) satisfies an error PDE driven by the residual of the learned PINN density.
  • Error-bound construction: A companion error-PINN produces a time-dependent bound B(t) intended to bound the density approximation error over the computational domain.
  • Ambiguity set: If B(t) satisfies the required error condition, the admissible PDF set contains the true PDF for every time in the propagation interval.
  • Ambiguity set: The first-order bound is uniform in space, making the ambiguity set conservative but compatible with linear-program event-probability bounds.
  • Limitations: Tighter spatially varying bounds require higher-order recursive error learning, which is not practical for the orbital examples.
  • Event-probability bounds: Partitioning the state region into polytope cells and optimizing cell-average densities yields upper and lower event-probability bounds.
  • Event-probability bounds: Event-probability bounds tighten when the error bound decreases or the partition is refined.
  • Event-probability bounds: For PINN-GMM, Gaussian component extrema over compact convex polytopes make the cellwise bounds computable efficiently.

IV. Training Implementation

The training pipeline separately trains the space-time density and error approximations using physics-informed losses, sampling strategies, and an optimizer.

  • Training pipeline: The pipeline trains the space-time PDF ˆp(x, t) and error approximation ˆe(x, t) in separate procedures.The error approximation is used to construct a time-dependent first-order bound B(t).
  • Density training: Density training uses the FP-PDE, an initial-condition batch, a residual batch, and an optimizer.The density algorithm requires ˆpθ, the FP-PDE, a sampling strategy, and an optimizer.
  • Error training: Error training uses the error PDE, including ˆp, together with sampling and optimization to construct B(t).The error network models the true error e := p − ˆp.
  • Optimization loop: The training procedures iterate through optimization steps and return the parameter iterate with the minimum total loss.The pseudocode updates parameters with Backprop(L) and Opt.step(), then returns ˆpθ★.

A. Loss

The method combines PDF-oriented losses with sampling strategies designed to address weak gradient signals and empty regions in high-dimensional domains.

  • Loss: Losses enforce the initial condition and PDE residuals without imposing a boundary-condition loss on ∂X.X is treated as a computational region of interest.
  • Loss: PINN-GMM training uses a Hellinger-type initial-condition loss as a surrogate inspired by the Hellinger distance.The choice is motivated by the suitability of ℓ1-type metrics for density estimation.
  • Loss: The error PINN uses a standard mean-square-error PINN loss for the error PDE.The true error is defined as e := p − ˆp and satisfies the error PDE.
  • Sampling motivation: In high dimensions, uniform samples can fall in low-density or low-residual regions, contributing little to the physics-informed loss.This is associated with the empty-space phenomenon in multivariate settings.
  • Sampling: The combined sampling strategy augments a persistent adaptive pool of informative points with fresh random mini-batches.The pool is periodically enriched with highest-scoring candidates, while fresh samples restore per-iteration domain coverage.
  • Sampling: The adaptive pool scores candidate points using initial-condition discrepancy or differential residual and adds the highest-scoring candidates.Repeated enrichment helps avoid early stagnation and improves stability.

C. Optimization

Optimization uses Adam and parameter-group-specific training for PINN-GMM, while experiments assess approximation, error bounds, event probabilities, and orbital cases.

  • Optimization: Adam updates the neural-network parameters using PyTorch default settings (β1, β2, ε) = (0.9, 0.999, 10^-8).Training saves the iterate with the minimum total loss over at most kmax iterations.
  • Optimization: PINN-GMM optimization separates mean, covariance, and weight network parameters, while fixing the initial-state encoder parameters.A time-dependent encoder is described as a possible extension but is not used.
  • Optimization: Weights and covariances use smaller learning rates, while a global step-decay schedule and gradient clipping improve training stability.The gradient clipping threshold is c_grad = 1, and the smaller learning rate is η(μ_z) = 10^-4.
  • Evaluation: The evaluation compares PINN-GMM with Gaussian approximation, unscented transform, and Gaussian mixture model baselines, plus ablated PINN variants.The ablations isolate the effects of architecture, encoder, and sampling.
  • Experiments: Approximately 7.2-hour horizons demonstrate hours-scale orbital uncertainty propagation with quantified error bounds.The Keplerian cases also compare coordinate choices and include an analytical-reference evaluation.
  • Scope and assumptions: The computational domain must balance information dilution against probability mass approaching its boundary.An overly large X worsens training dilution, whereas an overly small X can make the approximation unreliable.

A. Reference “True” Probability Density

Reference PDFs are constructed from high-fidelity Monte Carlo propagation and evaluated with distributional, pointwise, and likelihood-based measures.

  • Reference construction: High-fidelity Monte Carlo propagates independent initial states sampled from p0(x) through the underlying SDE using Euler–Maruyama.The integration step balances accuracy and computational cost.
  • Reference construction: For orbital dynamics, a GMM is fitted to Monte Carlo samples because direct histogram discretization is inaccurate and intractable in high dimensions.For low-dimensional systems up to 3D, simple grid histograms are considered adequate.
  • Reference construction: The reference GMM is used only as a post-processing estimator, while analytical reference densities are used directly when available.Sample-based measures such as relative divergence do not require a fitted reference density.
  • Reference limitations: KDE references were rejected because bandwidth sensitivity and finite-sample irregularities affected the estimates.A finite-component GMM is smoother and more tractable but introduces parametric fitting error and may miss strongly non-Gaussian structure.
  • Metrics: The evaluation measures total variation, worst-case normalized error, and relative divergence.These quantify global discrepancy, largest normalized pointwise deviation, and likelihood-based agreement, respectively.
  • Total variation: The empirical total-variation estimator samples uniformly from bounded X and converges to TV(p, q; X) as N → ∞.The estimator is used because the defining integral is seldom available in closed form.
  • Relative divergence: Relative divergence equals negative log-likelihood up to a constant for normalized q and is interpreted comparatively because the omitted constant permits negative values.Minimizing it is an empirical surrogate for minimizing generalized KL divergence.

C. 1D Illustrative Example

The 1D studies validate PINN error bounds, examine mixture-component sensitivity, and use the resulting ambiguity sets to bound event probabilities. Results show exact baseline behavior in the linear case and informative, tightening bounds in nonlinear settings.

  • Linear Example: In the linear SDE, GA and UT achieve zero total variation and zero worst-case normalized error, outperforming PINN-MLP on relative divergence.
  • Linear Example: PINN-MLP’s learned bound B(t) upper-bounds the observed worst-case error over the time horizon.
  • Nonlinear Example: For K_GMM = 1, 5, 11, 15, B(t) contains the associated worst normalized error throughout the nonlinear test.Training times were 9.7, 9.9, 12.5, and 13.2 seconds, respectively, illustrating an accuracy–cost trade-off.
  • Nonlinear Example: The K_GMM study does not establish monotone convergence because additional components can provide little benefit and make optimization more difficult.
  • Event-Probability Bounds: Both coarse and refined event-probability discretizations contain the reference probability, while refinement tightens the bounds.The refined computation takes 0.18 seconds per evaluated time point on average.
  • Event-Probability Bounds: GA and GMM point estimates deviate noticeably after t ≈ 2 s, whereas UT remains closer but provides no analogous error-bound interval.The refined linear programs produce densities attaining the lower and upper event-probability bounds.

2. Effect of Coordinate Choice

The orbital studies evaluate PINN-GMM across coordinate systems and increasingly complex perturbed dynamics. Rotating spherical coordinates yield better training and approximations, while learned bounds remain informative across 4D and 6D cases.

  • Perturbed Orbital Cases: In the 6D rotating-spherical case, PINN-GMM total variation remains between 4%–9%, while GA, UT, and GMM errors grow to 50%–54% by the final time.The PINN-GMM error bound contains the worst normalized error.
  • Coordinate Choice: Under the same training budget, Cartesian-coordinate IC and FP-PDE residual losses are about an order of magnitude larger than in rotating spherical coordinates.The Cartesian model’s final-time contours visibly deviate from reference Monte Carlo samples.
  • Coordinate Choice: Rotating spherical coordinates reduce the effective state-domain size and concentrate residual samples where PDF information is present, improving trainability for a fixed sampling budget.
  • Perturbed Orbital Cases: For 4D J2+diffusion, PINN-GMM errors remain relatively small over the full horizon and its bound contains the observed worst normalized error.The transformed density closely follows the Monte Carlo density, capturing drift and probability-mass spreading.
  • Perturbed Orbital Cases: The 4D event-probability computation supplies a certified deviation interval, requiring 21.9 seconds on average; its interval is tighter than in the 6D example.This illustrates that Algorithm 1 can be less conservative in the lower-dimensional 4D setting.
  • Perturbed Orbital Cases: With exact initial-condition matching, the 4D J2+diffusion+low-thrust model starts at the same initial error as GA and UT, then exhibits substantially slower error growth.The learned bound contains the observed worst normalized error over the propagation horizon.

F. Summary of Orbital Experiments

Across four orbital cases, PINN-GMM provides accurate uncertainty propagation with bounded errors and improved robustness, while requiring costly offline training. Its applicability is limited by smooth-dynamics assumptions, impulsive effects, and shorter evaluated horizons.

  • Orbital results: Across four orbital cases, maximum g TV stays below 9.0% and maximum WNE below 7.9%.The proposed method also maintains consistently low and narrow RD ranges.
  • Ablations: Uniform sampling provides poor higher-dimensional coverage, while the encoder improves conditioning and the proposed design trains more reliably than naive variants.Vanilla variants frequently plateau before convergence, and PDF-invalid representations can produce probability estimates far from true values.
  • Computational cost: PINN-GMM requires offline training and is more expensive to construct than GA and UT, but returns density parameters in approximately 0.16 milliseconds per query after training.The paper separates offline construction cost from online evaluation cost.
  • Limitations: The framework assumes a well-posed, continuously differentiable Fokker-Planck equation and does not directly handle impulsive effects.An impulse within the propagation interval cannot be represented by a single Fokker-Planck formulation.
  • Limitations: Approximation errors and learned bounds may grow over longer horizons, while the experiments cover shorter-than-some-study orbital horizons.One-day and multi-day propagation with informative bounds remains future work; CPU runtime also exceeds baseline methods.

Appendix

The appendix defines the orbital coordinates and baseline propagation procedures, then details reference-density construction and experiment configurations. It also reports training behavior for the orbital cases.

  • Orbital representations: Equinoctial elements transform classical Keplerian variables into the state [a, P1, P2, Q1, Q2, λ].The appendix defines P1, P2, Q1, Q2, and λ from eccentricity, inclination, and angular elements.
  • Reference densities: Reference densities without analytical solutions are fitted to Monte Carlo samples using expectation-maximization and are used only for post-processing.The number of mixture components is selected from candidate values {1, 2, 4, 8, 16}.
  • Orbital experiments: The orbital experiments span 4D and 6D Keplerian settings in equinoctial, rotating-spherical, Cartesian, diffusion, and low-thrust configurations.Component counts and training settings vary with density complexity and coordinate representation.
  • Training outcomes: The 6D Cartesian PINN-GMM did not converge after 60000 iterations, so its error PINN was not trained.Its initial-condition and FP-PDE residual losses plateaued at 2.9 × 10^-2 and 2.1 × 10^-2.
  • Training outcomes: For the 4D low-thrust case, time-gated whitened-coordinate parameterization eliminates initial-condition fitting error up to numerical precision.The final FP-PDE residual loss for the density PINN was 6.6 × 10^-3.
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