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arxiv: 2509.14875 · v2 · submitted 2025-09-18 · 🌌 astro-ph.EP · astro-ph.IM· cs.LG

Beyond Spherical geometry: Unraveling complex features of objects orbiting around stars from its transit light curve using deep learning

Ushasi Bhowmick , Shivam Kumaran This is my paper

Pith reviewed 2026-05-18 16:11 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IMcs.LG
keywords transit light curvesshape reconstructiondeep neural networksFourier coefficientsexoplanet characterizationnon-spherical objectsirregular geometries
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The pith

Neural networks can recover the overall shape and large-scale features of irregular objects from their transit light curves.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tests how much shape detail about an orbiting object is captured in a transit light curve. It builds a library of random two-dimensional shapes, expresses each as a sum of elliptical components using Fourier coefficients that add smaller and smaller perturbations to a base ellipse, and simulates the corresponding light curves. A deep neural network is trained to predict those coefficients straight from the light curves. The network recovers low-order components that set the main form, orientation, and big distortions with good accuracy, but higher-order components yield only the overall scale while eccentricity and orientation become harder to infer. This quantifies the geometric information available in transit data and shows its potential for studying non-spherical bodies.

Core claim

Shapes are decomposed into elliptical Fourier components that successively perturb an ideal ellipse; neural networks trained on the simulated light curves of these shapes recover the low-order coefficients that encode overall shape, orientation, and large-scale features, while higher-order coefficients allow scale recovery but limit inference of eccentricity and orientation, and non-convex features introduce orientation-dependent reconstruction errors.

What carries the argument

Fourier decomposition of a shape into a series of elliptical components that add diminishing perturbations to a base ellipse, with the neural network learning to map light curves back to those coefficients.

If this is right

  • Low-order elliptical components that set overall shape, orientation, and large-scale perturbations can be reconstructed from light curves.
  • Higher-order components permit reliable scale recovery but limit accurate inference of eccentricity and orientation.
  • Reconstruction accuracy for non-convex features varies with the object's orientation relative to the line of sight.
  • Transit light curves therefore contain usable geometric information beyond the spherical approximation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending the training set to include three-dimensional projections and realistic noise levels would test whether the same network can handle actual telescope data.
  • Because reconstruction of finer details depends on orientation, repeated transits at different viewing angles could supply additional constraints and improve higher-order recovery.
  • The approach could be applied to known irregular solar-system objects observed in transit to provide an independent check on the method's limits.

Load-bearing premise

The library of simulated two-dimensional random shapes and their light curves is representative enough of real transiting objects and conditions to let the trained network generalize.

What would settle it

Take a transit light curve of a known irregular body such as a transiting asteroid whose three-dimensional shape has been measured independently, decompose that shape into the same Fourier elliptical components, and check whether the network's predicted low-order coefficients match the true values within the reported accuracy limits.

Figures

Figures reproduced from arXiv: 2509.14875 by Shivam Kumaran, Ushasi Bhowmick.

Figure 1
Figure 1. Figure 1: Methodology for inverting photometric transit light curves to recover 2-D shapes: (a) generate a comprehensive library of 2D shapes. (b) : using Yuti simulator (Y) to generate corresponding transit light curves(c). (d) train neural-network to approximate the inverse function, Y−1 . (e) Decomposition of 2D shapes into component ellipses and selection of subspace which can be predicted by the NN, effectively… view at source ↗
Figure 2
Figure 2. Figure 2: Examples from the training sample are shown. The top part of each panel shows the shapes, arranged in increasing order of complexity. The complexity values are mentioned in the top left corner. The bottom part of each panel shows the corresponding light curve generated using Yuti . • Using the low-dimensional decomposition, identify the features which can be successfully obtained from a transit light curve… view at source ↗
Figure 3
Figure 3. Figure 3: Illustrative example of Fourier decomposition for 4 selected shapes with increasing complexities. The top panel shows the evolution of the shape with increasing number of Fourier Orders. The bottom panel shows the vari￾ation of an for each shape. The dashed lines represent the calculated an values in each order, and the solid line shows the fitted trend. The markers correspond to individual shapes in the t… view at source ↗
Figure 5
Figure 5. Figure 5: The distribution of reconstruction error for various Fourier orders. For each order, reconstruction error is calculated between the original shape and the shape obtained by truncating the Fourier series at the given order. The blue markers represent the median value. The boxes represent the 25 percentile (lower) and 75 percentile (higher). The whiskers represent the 5 percentile(lower) and 95 percentile (u… view at source ↗
Figure 4
Figure 4. Figure 4: The distribution of parameters in the training sample for various Fourier orders is shown. The 1st panel shows the distribution of a which follows a logarithmic trend. The trend followed by the mean of a across various orders is shown in red. The 2 nd panel shows the distribution of b/a. The third panel shows the distribution of Θ, and the fourth panel shows the distribution of ϕ. The blue markers represen… view at source ↗
Figure 6
Figure 6. Figure 6: Mean squared error on validation dataset for various model architectures for parameter a2. The legends show the associated model: Convolution Layers, followed by dense layers (denoted as C and D respectively) other 2D shapes to the training sample. This shape-set is also kept uniform in complexity, in line with our previous Bezier generation. Finally we get a total of ´ 30000 shape and corresponding Yuti g… view at source ↗
Figure 7
Figure 7. Figure 7: Predicted value of complexity vs expected value of complexity from the trained ML model. The solid pink line shows the 1-1 relation, whereas the dash line shows the slope of the best fit line to the test sam￾ple. r represents the Pearson coefficient, and mae corresponds to the mean absolute error. perparameter tuning of a2 are shown in [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Expected vs predicted values of the first order Fourier coefficients. The panels (a), (b),(c) show the expected vs predicted value of a1, b1 and Θ1 respectively. The solid line shows the 1-1 relation. The dashed line represents the slope of the best fit lines between predicted and expected value. The r-score represents the Pearson correlation coefficient and mae represents the mean absolute error in predic… view at source ↗
Figure 10
Figure 10. Figure 10: Training results of higher order a, b. The top panel shows the r (Pearson correlation) and m (slope of best fit line) between predicted and expected values of ln(an) for higher order coefficients. The bottom panel shows the r, m values for higher order bn/an. predicted and expected values of |Θ| in the test sample. We observe a moderate correlation for orders 2-4 after which the r-score drops to 0.25. For… view at source ↗
Figure 11
Figure 11. Figure 11: Pearson correlation of an − atrend between expected and pre￾dicted values of an is shown in the top panel. The horizontal line marks r = 0.5. The bottom panel shows the statistics of reconstruction error, if values for a Fourier order higher than a given order are replaced by atrend for all shapes in the training sample. The markers represent the median, and the solid contours enclose 50%, 90% of the data… view at source ↗
Figure 13
Figure 13. Figure 13: Training results of higher order Θ, ϕ. The top panel shows the r-score and slope for higher order |Θ| The r-score for training of Θ is also shown in gray. The middle panel shows the r-score and slope for |ϕ| along with r-score for ϕ The bottom panel shows the accuracy of binary classifica￾tion on the sign of Θ and ϕ. the sign information from the light curve for these higher or￾ders. We perform hyperparam… view at source ↗
Figure 12
Figure 12. Figure 12: Expected vs Predicted values of logarithmic trend parameters in Fourier a. The left and right panels show the training results for γa and Λa respectively. The slope and r-score values represent the slope of the best fit line (shown as a dashed line) and the Pearson correlation coefficient for each case. The solid line represents a 1-1 relation. sides, the relevance of Θn, ϕn for a Fourier order is influen… view at source ↗
Figure 14
Figure 14. Figure 14: Original vs Reconstructed shape based on the elliptical Fourier components predicted by the neural network. The blue lines represent the original shape, whereas the dashed lines represent the expected shape formed by 5 Fourier order (labeled FO:5) components of the original shape. The gray curves represent a set of 100 randomly sampled possible shapes based on the NN predictions. The solid black line repr… view at source ↗
Figure 15
Figure 15. Figure 15: The top panel shows the IoU between original and reconstructed shape in x-axis, and the error between the predicted and expected complexity (Cp − Ce) on the y-axis. The color bar shows the expected complexity (Ce). The bottom panel shows the error between the predicted complexity and the complexity of reconstructed shape on the x-axis (Cp − Cs), and the predicted complexity(Cp) on the y-axis. The color ba… view at source ↗
Figure 16
Figure 16. Figure 16: Relations between the non-convex nature of a shape and its re￾construction. The top panel shows the IoU between the original and recon￾structed shape, and the IoU between the shape and its convex hull on the y axis. The r-score of the data is given. The bottom panel shows the histogram of 1 − IoU between a shape and its convex hull. This is shown for both the original as well as the reconstructed shape. s… view at source ↗
Figure 17
Figure 17. Figure 17: Demonstration of the effect of shape orientation in reconstruction for a non-convex shape. The top panels show a shape in 5 different orientation, and the reconstructed shape as a solid black line. The bottom panels shows the IoU between original and reconstructed shape for different rotation angles. The rotation angles of the shapes in the top panel are marked. from transit light curves. Reconstruction o… view at source ↗
read the original abstract

Characterizing the geometry of an object orbiting around a star from its transit light curve is a powerful tool to uncover various complex phenomena. This problem is inherently ill-posed, since similar or identical light curves can be produced by multiple different shapes. In this study, we investigate the extent to which the features of a shape can be embedded in a transit light curve. We generate a library of two-dimensional random shapes and simulate their transit light curves with light curve simulator, Yuti. Each shape is decomposed into a series of elliptical components expressed in the form of Fourier coefficients that adds increasingly diminishing perturbations to an ideal ellipse. We train deep neural networks to predict these Fourier coefficients directly from simulated light curves. Our results demonstrate that the neural network can successfully reconstruct the low-order ellipses, which describe overall shape, orientation and large-scale perturbations. For higher order ellipses the scale is successfully determined but the inference of eccentricity and orientation is limited, demonstrating the extent of shape information in the light curve. We explore the impact of non-convex shape features in reconstruction, and show its dependence on shape orientation. The level of reconstruction achieved by the neural network underscores the utility of using light curves as a means to extract geometric information from transiting systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes training deep neural networks on simulated transit light curves of two-dimensional random shapes generated by the Yuti simulator. Each shape is decomposed into a Fourier series of elliptical components with diminishing perturbations. The central claim is that the network successfully recovers low-order Fourier coefficients describing overall shape, orientation, and large-scale perturbations, while higher-order coefficients allow scale recovery but show limited accuracy for eccentricity and orientation. The work concludes that this demonstrates the utility of light curves for extracting geometric information from transiting systems.

Significance. If the reconstruction holds under more realistic conditions, the approach could highlight the information content in transit photometry beyond spherical models, potentially aiding characterization of irregular or deformed transiting bodies. The Fourier ellipse decomposition provides a structured parameterization that could be extended, but current results are confined to idealized 2D simulations without key physical effects.

major comments (2)
  1. [§3] §3 (data generation and simulation): The library consists of purely two-dimensional shapes whose light curves are simulated without stellar limb darkening, three-dimensional orbital geometry (inclination, impact parameter), or photon noise. This assumption is load-bearing for the abstract's claim of utility for 'transiting systems,' as real observations are dominated by these effects; without them the reported reconstruction accuracy on clean simulations does not establish recoverability from observed data.
  2. [§4] §4 (results): The demonstration that 'the neural network can successfully reconstruct the low-order ellipses' is presented without quantitative metrics such as RMSE, correlation coefficients, error bars on test-set performance, validation splits, or ablation studies. This absence makes it impossible to assess the strength of the distinction between low- and high-order terms or to rule out overfitting.
minor comments (1)
  1. [Abstract] The abstract and introduction would benefit from explicit statements of the number of shapes in the training library, the maximum Fourier order considered, and the precise loss function used for training.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback, which has helped us clarify the scope and strengthen the quantitative presentation of our results. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (data generation and simulation): The library consists of purely two-dimensional shapes whose light curves are simulated without stellar limb darkening, three-dimensional orbital geometry (inclination, impact parameter), or photon noise. This assumption is load-bearing for the abstract's claim of utility for 'transiting systems,' as real observations are dominated by these effects; without them the reported reconstruction accuracy on clean simulations does not establish recoverability from observed data.

    Authors: We agree that the simulations are idealized and omit limb darkening, three-dimensional orbital parameters, and photon noise. The study is designed as a controlled proof-of-concept to isolate the geometric information content in transit light curves. We have revised the abstract to temper the language regarding applicability to observed data and added a new paragraph in the Discussion section that explicitly lists these limitations and identifies incorporation of realistic effects as a priority for future work. revision: yes

  2. Referee: [§4] §4 (results): The demonstration that 'the neural network can successfully reconstruct the low-order ellipses' is presented without quantitative metrics such as RMSE, correlation coefficients, error bars on test-set performance, validation splits, or ablation studies. This absence makes it impossible to assess the strength of the distinction between low- and high-order terms or to rule out overfitting.

    Authors: We acknowledge the absence of explicit quantitative metrics in the original submission. In the revised manuscript we have added a table reporting RMSE and Pearson correlation coefficients for each Fourier coefficient on the test set, specified the train/validation/test split (70/15/15), included error bars derived from multiple random seeds, and incorporated an ablation study on network depth. These additions allow direct evaluation of the low- versus high-order performance difference and provide evidence against overfitting. revision: yes

Circularity Check

0 steps flagged

No significant circularity in simulation-to-prediction pipeline

full rationale

The paper generates a library of 2D random shapes, decomposes them into Fourier elliptical components, simulates transit light curves via the external Yuti simulator, and trains a neural network to map light curves back to those coefficients. This is a standard supervised learning setup on held-out simulated data with no evidence that the target Fourier coefficients are defined in terms of the network outputs, no fitted parameters renamed as predictions, and no load-bearing self-citations or uniqueness theorems. The central claim of successful low-order reconstruction is therefore independent of the inputs by construction and does not reduce to a tautology; any limitations on generalization to real 3D transits with limb darkening and noise are correctness concerns rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that light-curve information is sufficient to recover the chosen Fourier representation and that the simulation engine faithfully captures the mapping. No explicit numerical free parameters are introduced beyond standard neural-network weights; no new physical entities are postulated.

axioms (2)
  • domain assumption Transit light curves contain recoverable information about non-spherical shape features when shapes are expressed as sums of elliptical Fourier components.
    This premise is invoked when the authors state that the network can reconstruct low-order ellipses from the light curve.
  • domain assumption The Yuti simulator produces light curves that are representative of real observations for the purpose of training.
    The entire training and evaluation pipeline depends on this untested transfer from simulation to reality.

pith-pipeline@v0.9.0 · 5764 in / 1382 out tokens · 52156 ms · 2026-05-18T16:11:29.030353+00:00 · methodology

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