Cosmo-PINN: A Physics-Informed Neural Network for Cosmological Reconstruction

arXiv:2605.30139 · astro-ph.CO, gr-qc · Submitted 2026-05-28 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: I'm Vera.

Jocelyn: And I'm Jocelyn, here with Subrahmanyan.

Vera: Today's paper: "Cosmo-PINN: A Physics-Informed Neural Network for Cosmological Reconstruction".

Jocelyn: The gist The Cosmo-PINN framework introduces a Physics-Informed Neural Network for reconstructing cosmological theory by embedding physical laws as hard constraints in the loss function to ensure physically consistent solutions,…

Vera: First, who's behind it and why it matters.

Title and authors: Jocelyn: Now let's talk about what the paper calls a data loss function. It’s based on late-time cosmological observations like Baryon Acoustic Oscillations and Type Ia Supernovae >

Vera: They use Cosmic Chronometers, CC data to measure the Hubble parameter within the redshift range zero point zero nine to one point nine six five >

Jocelyn: The loss function for CC is defined as chi squared cc equals sum over i of HPINN i minus Hobs i over sigma i squared >

Subrahmanyan: That’s a standard least squares setup for the Hubble parameter measurement, looking at the residuals against observational uncertainty >

Vera: Then you’ve got Baryon Acoustic Oscillations data from DESI DR2, and they define the likelihood as chi squared bao equals dPINN BAO minus dobs BAO tobs BAO inverse C negative one dPINN BAO minus dobs BAO >

Jocelyn: That’s a matrix operation involving the distance ratio and volume-averaged distance ratio from those BAO measurements >

Subrahmanyan: And then there are the Type Ia Supernovae, they use three compilations: PantheonPlus, Union3 point 0, and DES-Dovekie, providing measurements of the distance modulus muobs as a function of redshift zi > <ref:2605.30139#pg1>

Vera: The data loss component is LDATA which is lambda CC lcc plus lambda BAO lbao plus lambda SN lsn >

Jocelyn: So the total training loss is just this weighted sum of all those observational fits, with lambdas being the weights you tune >

Subrahmanyan: And that’s what connects the theoretical reconstruction part to the actual measurements they are trying to match >

Vera: It shows how they balance matching what we see with satisfying what we know about gravity and energy density >

Jocelyn: It’s a very holistic setup for training, combining everything from expansion history measurements to distance scale ratios >

The paper's summary: Subrahmanyan: So the paper suggests a few key improvements to this Cosmo-PINN framework itself that they want to highlight >

Vera: They focus on making it more robust, specifically by testing convergence with different sets of initial conditions to see if the PINN converges to a consistent solution >

Jocelyn: They also studied the stability of the Chebyshev polynomial order, training with N equals two, three, five, six and eight to see what happens >

Subrahmanyan: And they found that between N equals three and eight, the obtained results are consistent across those different polynomial orders >

Vera: That’s good because it suggests their choice of representation isn't overly sensitive to the exact polynomial degree used for reconstruction >

Jocelyn: Then there’s this part about posterior uncertainties. They use Hamiltonian Monte Carlo, HMC sampling with the NUTS sampler to get those uncertainties >

Subrahmanyan: That lets them propagate posterior samples through the network to obtain credible intervals for the reconstructed functions >

Vera: That gives us a way to quantify how certain they are about their results, which is crucial when you’re looking at cosmological parameters like H0 or omega m0 >

Jocelyn: It also tests whether the PINN converges to a consistent solution using different initial conditions, so that the solutions provided by Cosmo-PINN are consistent and insensitive to the choice of initial conditions >

Subrahmanyan: And they’ve introduced parameter priors LIC into the total loss which anchor those parameters near Planck two thousand eighteen values, preventing degeneracies > <ref:2605.30139#pg1>

Vera: That’s a nice touch because it helps keep those parameters from wandering off to weird values during the reconstruction process >

The paper's improvements: Jocelyn: So to wrap up this paper on Cosmo-PINN, they are showing that this framework provides a stable and physically consistent way to reconstruct cosmological observables and the underlying theory >

Vera: It really proves that embedding physical laws within the loss function is necessary for obtaining physically consistent solutions >

Subrahmanyan: It shows that purely data-driven reconstruction can lead to ghost scenarios where omega DE(z) can have values omega DE(z) less than zero during the matter dominated era, which led to an ill-defined behaviour for wDE(z) at the transition redshift >

Jocelyn: And they compare it directly against a purely data-driven neural network with the same architecture and show small oscillations appear in that pure ANN reconstruction >

Vera: That comparison really highlights why imposing physical constraints within the loss function is necessary for obtaining physically consistent solutions >

Subrahmanyan: It also shows that without those constraints, a purely data-driven reconstruction supports a ghost scenario where omega DE(z) can have values omega DE(z) less than zero during the matter dominated era, which led to an ill-defined behaviour for wDE(z) at the transition redshift >

Jocelyn: Overall, Cosmo-PINN is a stable and physically consistent framework for reconstructing cosmological observables and of the underlying cosmological theory >

Conclusion: Vera: So we’ve been looking at Cosmo-PINN, which is this Physics-Informed Neural Network for Cosmological Reconstruction >

Jocelyn: Right, so they use these physical laws as hard constraints in the loss function to make sure the reconstructed cosmological solutions are actually consistent with physics and not just statistical fits >

Subrahmanyan: That’s the core idea, using the field equations as a hard constraint to ensure consistency throughout the training domain >

Vera: They’re training this AI on late-time data from Baryon Acoustic Oscillations, Type Ia Supernovae, and Cosmic Chronometers >

Jocelyn: And they use those data losses—CC loss, BAO likelihood, SN loss—to train the network to match what we actually observe >

Subrahmanyan: The results show that this framework gives us a stable way to reconstruct the equation of state parameter wDE(z) without restricting the functional form a priori >

Vera: They found two main scenarios: one where wDE(z) crosses that phantom divide line, and another where it’s forced to be constant at minus one for the quintessence case >

Jocelyn: The comparison with a purely data-driven network really showed those small oscillations that you don't get when you include all the physical constraints >

Subrahmanyan: That pure ANN scenario can support a ghost scenario where omega DE(z) can be less than zero during the matter dominated era, which means it’s physically impossible without the PINN loss function >

Vera: It really shows that imposing those physical laws in there is necessary for getting solutions that make sense cosmologically >

Jocelyn: So for anyone listening who only cares about what we see, this means the AI isn't just guessing based on patterns; it’s being forced to obey gravity and energy conservation >

Subrahmanyan: And for us theoreticians, it gives us a tool to test specific models without getting stuck in the degeneracies that pure data fitting usually causes >

Vera: It’s a powerful way to anchor those parameters near Planck values using those soft priors LIC, which helps keep things grounded >

Jocelyn: We’re still looking at what these reconstructions tell us about the nature of dark energy itself, whether it's tracking something or just staying constant >

Subrahmanyan: It’s a good step toward extracting physical theory properties directly from observational data without having to assume a specific model structure first >

Vera: That’s Cosmo-PINN for now. We’ll keep watching these results as we look at the next set of survey data coming in >

Andronikos Paliathanasis

Institute of Systems Science, Durban University of Technology · Centre for Space Research, North-West University · Departamento de Matemáticas, Universidad Católica del Norte · National Institute for Theoretical and Computational Sciences (NITheCS)

astro-ph.CO, gr-qc

Submitted: 2026-05-28

Updated: 2026-10-14

Comments: 23 pages, 17 figures, updated architecture, updated robustness tests, new reconstruction in Appendix A

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 79/100

The gist: The gist The Cosmo-PINN framework introduces a Physics-Informed Neural Network for reconstructing cosmological theory by embedding physical laws as hard constraints in the loss function to ensure

Key concepts

Cosmo-PINN Framework: This framework merges standard Artificial Neural Networks with Physics-Informed Neural Network (PINN) principles. It enforces cosmological field equations as hard constraints within the loss function during training, ensuring that every reconstructed point adheres to both observational data and fundamental physical laws simultaneously.
Physics-Informed Constraints: Instead of just fitting data points, this method embeds the governing cosmological field equations directly into the network's loss function. This forces the network to learn a solution that satisfies physical laws at every step of training, preventing unphysical solutions common in purely data-driven models.
Equation of State Parameter wDE(z): This parameter describes how dark energy behaves over cosmic time. It dictates the relationship between its pressure and its energy density. The paper reconstructs this function to see if it crosses critical boundaries, such as the phantom divide where wDE = -1.
Data Loss Function (LDATA): This component of the total loss function combines various observational data sets—Cosmic Chronometers, Baryon Acoustic Oscillations, and Supernovae. It quantifies how well the network's predictions match real-world measurements across different cosmological probes.

Terminology

Loss Function — 2× in this paper · explained in 2 episodes
Physical Constraints — 1× in this paper · explained in 7 episodes
Quintessence — 1× in this paper · explained in 2 episodes

Summary

The gist The Cosmo-PINN framework introduces a Physics-Informed Neural Network for reconstructing cosmological theory by embedding physical laws as hard constraints in the loss function to ensure physically consistent solutions, which is important because it overcomes limitations of purely data-driven approaches where recovered solutions are not necessarily physically consistent

Cosmo-PINN Framework

The Cosmo-PINN framework combines Artificial Neural Network (ANN) reconstructions with the physics enforcement of Physics-Informed Neural Networks (PINNs) by imposing cosmological field equations as hard constraints in the loss function, ensuring that reconstructed observables are consistent with observational data at each point and with the underlying physical law throughout the training domain

Architecture The Cosmo-PINN architecture is designed to reconstruct the Hubble function H (z) and free functions of the theory, such as the equation of state parameter wDE (z), and the energy density ρDE (z), while simultaneously inferring cosmological parameters H0, omegam0, and rdrag
Physical Law The full set of physical quantities is trained under the requirement of physical consistency by embedding the cosmological field equations directly within the loss function as a hard constraint
Input Mapping The input to the network is the redshift scalar z ∈ [0, zmax], which is mapped onto the unit interval via a normalized transformation x = z/zmax, such that x ∈ [0, 1]
Function Representation The equation of state parameter wDE (x) is employed using a Chebyshev polynomial expansion wDE(x) = Σ n=0 c n Tn(2x − 1), with trainable coefficients c n
Conservation Law Enforcement The PDE loss component imposes the physical law, defined as LPDE = λPDE Σ (R 2/2)(xi)
Initial Conditions The reduced dark energy density ρ¯DE(z) is expressed as ρ¯DE(z) = ¯ρDE0 exp [x NDE(x)], where the exponential ansatz enforces the initial condition ¯ρDE0 = omegaDE0, and the requirement ¯ρDE > 0 at every point
Parameter Priors The cosmological parameters H0, omegam0, and rdrag are introduced into the total loss via expressions LIC which act as soft priors that anchor the learnable cosmological parameters near the Planck 2018 values
Smoothness Penalties The Lbounds component imposes strong boundary conditions such as wDE is within the range wDE ∈ [−1, 1] in the quintessence limit using a sigmoid function
Weak Smoothness Penalties LSP Weak smoothness penalties are introduced by introducing penalties on the first and second derivatives of the trained functions to avoid unphysical oscillations related to overfitting

Data Loss Function

The analysis is based on late-time cosmological observations, employing datasets from Baryon Acoustic Oscillations (BAO), Type Ia Supernovae (SNIa), and Cosmic Chronometers (CC)

Cosmic Chronometers The CC data provides measurements of the Hubble parameter within the redshift range 0.09 ≤ z ≤ 1.965, and the loss function is defined as χ 2 CC = Σ (HPINN i - Hobs i)/σ i squared
Baryon Acoustic Oscillations The DESI DR2 catalogue provides measurements of the Hubble distance ratio, comoving angular distance ratio, and volume-averaged distance ratio, and the likelihood is defined as χ 2 BAO = (dPINN BAO - dobs BAO) T C-1 BAO (dPINN BAO - dobs BAO)
Type Ia Supernovae The analysis makes use of three different SNIa compilations, including PantheonPlus (PP), Union3.0 (U3), and DES-Dovekie (DESD), providing measurements of the distance modulus µobs as a function of redshift zi
Data Loss Definition The component related to training given by the cosmological data is defined as LDATA = λCC LCC + λBAO LBAO + λSN LSN, where the loss function for CC is defined as LCC = 1/N CC χ 2 CC

Training and Stability

The training proceeds in two sequential phases using an Adam optimizer for 10000 epochs and an L-BFGS optimizer for up to 200 epochs, with early stopping when the total loss reaches a minimum

Convergence Test The network is trained using different sets of initial conditions to verify whether the PINN converges to a consistent solution, and the solutions provided by the Cosmo-PINN are consistent and insensitive to the choice of initial conditions
Chebyshev Polynomial Order Stability The reconstruction process is studied by training with different degrees of Chebyshev polynomials (N = 2, 3, 5, 6, and 8), and it is found that between N = 3 − 8, the obtained results are consistent
Posterior Uncertainties HMC Sampling Posterior uncertainties on the reconstructed functions are obtained using the Hamiltonian Monte Carlo (HMC) approach with the NUTS sampler, propagating posterior samples through the network to obtain credible intervals

Reconstruction Results

The reconstruction of wDE (z) is performed under two scenarios: wDE(z) can cross the phantom divide line, and wDE(z) has the value −1 as a lower bound such that wDE(z) ≥ -1

Unbounded Scenario The reconstructed wDE (z) is found to be a monotonically decreasing function that crosses the phantom divide line at the redshift z = 0.27 − 0.42, which is in agreement with the CPL model for w0wa < 0
Quintessence Scenario The reconstructed function wDE(z) exhibits a different evolution, and in all data combinations the reconstructed function ρDE (z) is forced to be constant, meaning the equation of state parameter reaches the ΛCDM limit, i.e., wDE(z) → −1
Comparison with Pure ANN The comparison between Cosmo-PINN and a purely data-driven neural network with the same architecture demonstrates that small oscillations appear in the pure ANN reconstruction, which can be attributed to the absence of both the Chebyshev expansion and the physical constraints imposed on the solution in Cosmo-PINN
Physical Consistency Demonstration The comparison shows that without imposing physical constraints, a purely data-driven reconstruction supports a ghost scenario where omegaDE(z) can have values omegaDE(z) < 0 during the matter dominated era, which led to an ill-defined behaviour for the wDE(z) at the transition redshift

Conclusion

The Cosmo-PINN provides a stable and physically consistent framework for the reconstruction of cosmological observables and of the underlying cosmological theory, demonstrating that imposing physical laws within the loss function is necessary for obtaining physically consistent solutions

The paper is 2605.30139. Every sentence must end with a reference token.

Improvements for AI systems

textbfReal-time Physical Consistency Enforcement in Reconstruction Tools: The Cosmo-PINN System allows for ensuring that the reconstructed observables are consistent with the observational data at each point and with the underlying physical law throughout the training domain. This enables AI systems to generate physically viable cosmological models rather than just statistical fits, overcoming the limitation where the recovered solution is driven by the data and it is not necessarily true that it is physically consistent.
textbfModel-Independent Dark Energy Parameter Recovery: The framework can be applied to any cosmological model and reconstructs the dark energy equation of state parameter wDE(z) without restricting the free functional forms a priori. This capability allows AI systems to extract physical theory properties directly from observational data, moving beyond model-dependent Bayesian approaches where the conclusions made are valid only for the specific model assumed.
textbfRobust Parameter Estimation via Constraint Anchoring: The system incorporates soft priors that anchor the learnable cosmological parameters near Planck values, preventing degeneracies. This ensures that when reconstructing wDE(z), it can distinguish between scenarios by forcing a solution, such as in the quintessence case where the equation of state parameter reaches the ΛCDM limit, i.e wDE(z) → −1, or allowing it to cross the phantom divide line.
textbfVisualization of Physical Model Discrimination: The output allows for clear discrimination between physical scenarios; for instance, in Fig. 6, the reconstruction shows that for an unbounded wDE(z), "the wDE(z) at the present wDE (0) > −1," while in the quintessence scenario it is reconstructed to be a monotonically increasing function. This provides a direct visual tool for testing physical hypotheses against observational data.
textbfDetection of Data-Driven Artifacts: By comparing Cosmo-PINN with a purely data-driven neural network with the same architecture, AI systems can identify numerical artifacts, such as small oscillations appear in the NN reconstruction and potential "ghost scenario where the omegaDE (z) can have values omegaDE (z) < 0 during the matter dominated era. This highlights why the imposition of physical laws within the loss function is necessary for obtaining physically consistent solutions."

Abstract

We introduce Cosmo-PINN, a Physics-Informed Neural Network for reconstruction of the cosmological theory. In this work we demonstrate the application of the Cosmo-PINN in the reconstruction of the dark energy equation of state parameter w DE (z) directly from late-time cosmological observations. This framework overcomes the main limitation shared by Gaussian Process and Artificial Neural Network reconstruction approaches, where the recovered solution is driven by the data and it is not necessarily true that it is physically consistent, by embedding the cosmological constraints directly into the loss function as constraints, ensuring that the reconstructed quantities satisfy the physical laws at every point during the training. For the training of the network, we employed background data, and specifically the Baryon Acoustic Oscillation from DESI DR2, the Cosmic Chronometers and three different Supernova compilations, while we simultaneously introduce the cosmological parameters % H 0, Ω m0 and r drag as trained parameters. The reconstruction shows that the trained w DE (z) crosses the phantom divide around z=0.5 in agreement with the value obtained by the Chevallier-Polarski-Linder model. In the quintessence scenario, the dark energy equation of state parameter is forced to the Λ CDM limit in the past, that is w DE (z) to-1. Finally, we demonstrate the significance of imposing the physical constraints within the loss function by comparing the Cosmo-PINN reconstruction against a purely data-driven neural network with the same architecture.