A Physics-Informed Neural ODE for Non-Parametric Dark Energy Reconstruction: Methodology and Application to DESI DR2

Abstract

The 2025 Dark Energy Spectroscopic Instrument (DESI) results, from baryon acoustic oscillation (BAO) measurements analyzed under the two-parameter Chevallier–Polarski–Linder (CPL) ansatz, report a 2.8–4.2σ preference for evolving dark energy over the cosmological constant Λ. Because CPL admits only monotonic $w(z)$ (the dark energy equation of state as a function of redshift $z$), any non-monotonic feature in the true equation of state is projected onto a slope. All prior non-parametric reconstructions (Gaussian processes (GPs), splines, bins) reconstruct the Hubble rate $H(z)$ or luminosity distance $d_L(z)$ first and numerically differentiate to recover $w(z)$, amplifying noise until error bands diverge past $z \approx 1.2$.

We introduce a physics-informed neural ordinary differential equation (neural ODE) that makes the Friedmann equation the forward pass of the computational graph. A small multilayer perceptron (MLP) parameterizes $w(z)$ directly; an ODE solver integrates the dark energy density $\rho_{DE}$ to produce predicted BAO ratios, Type Ia supernova distance moduli, and compressed cosmic microwave background (CMB) shift parameters; gradients flow back via the adjoint method. Consistency is structural (there is no Friedmann loss penalty), and the reconstruction is stable to $z \approx 2.5$ where GP-based methods diverge. We apply the method to DESI Data Release 2 (DR2) BAO, Pantheon+ Type Ia supernovae (with Union3 and the Dark Energy Survey five-year supernova sample, DES-SN5YR, as robustness checks), Planck 2018 compressed CMB parameters $(R, \ell_A)$, and 36 cosmic chronometers plus 3 DESI DR1 $H(z)$ points, using Λ cold dark matter (ΛCDM) baseline initialization, a 20-model deep ensemble for headline numbers, and a 50-seed profile likelihood.

The canonical ensemble recovers $w(1.0) = -1.18$. A Feldman–Cousins calibration against 50 ΛCDM mocks through the same pipeline yields $p = 0.10$–$0.20$ pinned-target (~0.8–1.3σ) and $p_{\rm LEE} = 0.15$–$0.40$ after correcting for the look-elsewhere effect (LEE) (~0.3–1.0σ), converging at the ~1σ level with a mock-calibrated $\sigma_{\rm mock}(w(1)) = 0.130$. A per-bin decomposition and drop-bin refit show that the luminous red galaxy bin at $z = 0.706$ (LRG2) contributes roughly half of the phantom depth; without it, the preference drops to ~0.6σ. A pipeline-correction audit tracks reported significance from 3.8σ (Gaussian CMB prior + $w_0 w_a$ initialization) down to ~1.3σ (canonical with cosmic chronometers), with every correction published as a feature rather than buried.

The headline is reported as a method demonstration with a marginal hint in current data, not a detection. The durable contributions are architectural and methodological: a neural ODE with Friedmann as forward pass, Feldman–Cousins-calibrated profile likelihoods as the honest replacement for Wilks' theorem in regularized non-convex regimes, and an attribution methodology (per-bin chi-squared + drop-bin refit + outlier injection) for decomposing where non-parametric cosmological signals come from. Code and reproduction instructions are at github.com/mruckman1/dark_energy1.