Application of interpretable data-driven methods for the reconstruction of supernova neutrino energy spectra following fast neutrino flavor conversions

Neutrinos can experience fast flavor conversions (FFCs) in highly dense astrophysical environments, such as core-collapse supernovae and neutron star mergers, potentially affecting energy transport and other processes. Simulating fast flavor conversions under realistic astrophysical conditions requires substantial computational resources and poses significant analytical challenges. While machine learning methods such as multilayer perceptrons have been used to accurately predict the asymptotic outcomes of FFCs, their “black-box” nature limits the extraction of direct physical insight. To mitigate this limitation, we employ two distinct interpretable machine learning frameworks, Kolmogorov-Arnold Networks (KANs) and Sparse Identification of Nonlinear Dynamics (SINDy), to learn interpretable surrogates for the asymptotic input-output mapping from an FFC simulation dataset. Our analysis reveals a fundamental trade-off between predictive accuracy and model simplicity. KANs demonstrate high fidelity in reconstructing post-conversion neutrino energy spectra, achieving accuracies of up to 90%. In contrast, SINDy yields a low-rank, compact closed-form approximation of the input-output mapping, at the expense of some predictive accuracy. Critically, using these structured and sparse surrogates as diagnostic tools, we identify that the system’s evolution is most sensitive to the initial number density of heavy-lepton neutrinos when FFCs are triggered, compared with other physical quantities. Ultimately, this work provides a methodological framework for interpretable machine learning that supports genuine data-driven scientific discovery in astronomy and astrophysics, going beyond prediction alone.

💡 Research Summary

The manuscript presents a novel, interpretable machine‑learning framework for reconstructing the post‑fast‑flavor‑conversion (FFC) neutrino energy spectra in core‑collapse supernovae (CCSNe) and neutron‑star mergers (NSMs). Fast flavor conversions, which occur on extremely short spatial scales, are known to influence neutrino transport and thus the dynamics of these astrophysical events. However, fully resolving FFCs within realistic simulations is computationally prohibitive, and existing black‑box neural‑network approaches, while accurate, provide little physical insight.

To address this, the authors employ two distinct interpretable methods: Kolmogorov‑Arnold Networks (KANs) and Sparse Identification of Nonlinear Dynamics (SINDy). Both are trained on a dataset generated from one‑dimensional periodic box simulations of neutrino gases. Each sample is represented by an 11‑dimensional feature vector comprising nine moment‑based quantities (total and bin‑wise flux factors and number densities for electron neutrinos, electron antineutrinos, and heavy‑lepton neutrinos) together with two continuous energy tags for each bin. This design mirrors the information typically available in state‑of‑the‑art CCSN/NSM codes while preserving a continuous energy dependence.

KANs implement the Kolmogorov‑Arnold representation theorem by learning trainable B‑spline activation functions for each input dimension. This yields a network that is both highly expressive and transparent: the learned univariate functions ϕ₍q,p₎(xₚ) and the final aggregators Φ₍q₎ can be inspected, visualized, and related to physical processes. Training uses standard Adam optimization with mean‑squared‑error loss and a validation split to avoid over‑fitting. The resulting model achieves up to 90 % reconstruction accuracy (R²) on held‑out test data, reproducing the detailed shape of the asymptotic neutrino spectra. Notably, the network’s performance is most robust when the initial heavy‑lepton (νₓ) number density is high, suggesting a strong sensitivity of the conversion outcome to this parameter.

SINDy, by contrast, seeks a compact symbolic surrogate. A library of candidate functions (polynomials up to second order, cross‑terms, logarithmic and exponential forms) is constructed, and sparse regression with an L₁ penalty selects a minimal set of non‑zero coefficients that best map the 11‑dimensional inputs to the spectral outputs. The resulting expression is a low‑rank, closed‑form formula that directly reveals which combinations of physical variables dominate the mapping. The dominant term identified is the product of the heavy‑lepton number density and the electron‑neutrino flux factor, confirming that the interaction between νₓ abundance and electron‑neutrino transport controls the onset and saturation of FFCs. While SINDy’s predictive accuracy is modestly lower (≈5–10 % drop relative to KAN), its interpretability is maximal: the surrogate can be inserted into analytic models, used for sensitivity studies, or even guide the formulation of new theoretical approximations.

The authors explicitly discuss the trade‑off between predictive fidelity and interpretability. KANs provide near‑state‑of‑the‑art accuracy while still offering a degree of transparency through learned activation shapes; SINDy sacrifices some accuracy for an analytically tractable model that yields clear physical insight. Both approaches demonstrate that data‑driven surrogates can complement, rather than replace, full quantum‑kinetic calculations, enabling rapid evaluation of FFC outcomes within large‑scale hydrodynamic simulations.

Beyond performance metrics, the paper extracts a key scientific conclusion: the system’s evolution is most sensitive to the initial heavy‑lepton neutrino number density (νₓ) when fast flavor conversions are triggered. This insight emerges consistently from both KAN sensitivity analyses (e.g., perturbation of input features) and the sparsity pattern of the SINDy model. It suggests that future CCSN/NSM modeling efforts should prioritize accurate treatment of νₓ distributions, especially in regions where ELN (electron lepton number) crossings are expected.

The manuscript also outlines future directions: expanding the training set to include non‑periodic geometries, collision terms, and more realistic angular distributions; incorporating physical constraints (e.g., lepton‑number conservation) directly into the KAN activation functions; and enriching the SINDy library with physics‑motivated basis functions such as ELN‑crossing diagnostics. These steps aim to further tighten the link between machine‑learned surrogates and underlying neutrino‑flavor physics.

In summary, this work delivers a methodological blueprint for applying interpretable machine‑learning tools to a challenging problem in neutrino astrophysics. By demonstrating both high‑accuracy reconstruction (KAN) and compact symbolic modeling (SINDy), the authors show that interpretability need not be sacrificed for performance. Moreover, the identification of νₓ density as the dominant driver of FFC outcomes provides a concrete, data‑derived hypothesis that can be tested in future simulations and potentially inform observational strategies for next‑generation neutrino detectors.