Analytical Models

Analytical Models#

Fiesta includes a library of analytical (physics-based) light-curve models implemented in pure JAX. Unlike the surrogate models, which are neural-network approximations trained on simulation data, analytical models evaluate closed-form or ODE-based physics equations directly.

Key properties:

JIT-compilable and differentiable – compatible with flowMC’s MALA gradient sampler via jax.grad.
Drop-in replacements for surrogates – every analytical model exposes the same predict() interface and can be used inside CombinedSurrogate and EMLikelihood without any code changes.
Float32-safe – all internal computations use log10 space to avoid overflow (e.g. explosion energies ~1e49 erg exceed float32 max ~3.4e38).

When to use analytical models vs surrogates:

Use surrogates when a fast, pre-trained approximation of a complex simulation (e.g. radiative-transfer kilonova or jet afterglow) is available and sufficient.
Use analytical models when you need direct physics interpretability, want to combine a surrogate with a simple analytical component (e.g. afterglow surrogate + kilonova analytical), or when no surrogate exists for the transient class of interest.

Usage#

Instantiating a model

Every analytical model takes a list of filter names and an array of source-frame times (days):

import jax.numpy as jnp
from fiesta.inference.analytical_models import OneComponentKilonovaModel

model = OneComponentKilonovaModel(
    filters=["bessellb", "bessellv", "bessellr"],
    times=jnp.geomspace(0.1, 30.0, 100),
)

Calling predict

The predict method accepts a dictionary of parameter values and returns (source_frame_times, {filter_name: apparent_mag_array}):

params = {
    "log10_mej": -1.5,
    "log10_vej": -0.5,
    "log10_kappa": -0.3,
    "luminosity_distance": 100.0,   # Mpc
    "redshift": 0.022,
}

t_days, mag_app = model.predict(params)
# mag_app["bessellb"] is an array of apparent magnitudes

Combining with a surrogate

Analytical models are interchangeable with surrogates inside CombinedSurrogate. For example, to model a kilonova + GRB afterglow where the afterglow is a trained surrogate:

from fiesta.inference.lightcurve_model import AfterglowFlux, CombinedSurrogate
from fiesta.inference.analytical_models import OneComponentKilonovaModel

afterglow = AfterglowFlux(
    name="afgpy_gaussian_CVAE",
    filters=["bessellb", "bessellv", "bessellr"],
)

kilonova = OneComponentKilonovaModel(
    filters=["bessellb", "bessellv", "bessellr"],
    times=jnp.geomspace(0.1, 30.0, 100),
)

combined = CombinedSurrogate(
    models=[afterglow, kilonova],
    sample_times=jnp.geomspace(0.1, 300.0, 200),
)

t, mags = combined.predict(params)

The fluxes of both models are added in each photometric band. If a model falls outside the time range of the sample_times it is set to jnp.inf mag there.

Model Catalogue#

Phenomenological Models#

Empirical shape functions that parameterize light-curve morphology directly in magnitudes, without underlying radiation physics.

BazinModel

Exponential rise and fall (Bazin et al. 2009). Useful as a generic empirical template for any transient.

Parameter	Description	Units
`t0`	Explosion / reference time	days
`log10_tau_rise`	Rise timescale (log10)	log10(days)
`log10_tau_fall`	Fall timescale (log10)	log10(days)
`amp_mag_{f}`	Peak amplitude per filter f	mag
`base_mag_{f}`	Baseline flux per filter f	mag

VillarModel

Piecewise linear rise + power-law decay with a smooth sigmoid transition (Villar et al. 2017).

Parameter	Description	Units
`t0`	Explosion / reference time	days
`log10_tau_rise`	Rise timescale (log10)	log10(days)
`log10_tau_fall`	Fall timescale (log10)	log10(days)
`beta_slope`	Power-law decay slope before peak	–
`log10_gamma`	Transition point (log10)	log10(days)
`amp_mag_{f}`	Amplitude per filter f	mag

AfterglowModel

Smooth broken power-law for GRB/jet afterglow light curves.

Parameter	Description	Units
`t0`	Reference time	days
`log10_t_break`	Break time (log10)	log10(days)
`alpha_1`	Pre-break power-law index	–
`alpha_2`	Post-break power-law index	–
`amp_mag_{f}`	Amplitude per filter f	mag

PhenomenologicalTDEModel

Sigmoid rise + power-law decay for tidal disruption events.

Parameter	Description	Units
`t0`	Explosion / reference time	days
`log10_tau_rise`	Rise timescale (log10)	log10(days)
`log10_tau_fall`	Fall timescale (log10)	log10(days)
`alpha_decay`	Power-law decay index	–
`amp_mag_{f}`	Amplitude per filter f	mag
`base_mag_{f}`	Baseline flux per filter f	mag

EvolvingBlackbodyModel

Empirical piecewise power-law evolution of temperature and radius, converted to broadband magnitudes via blackbody SED. Unlike the other phenomenological models this computes bolometric luminosity from L = 4 pi R^2 sigma T^4.

Parameter	Description	Units
`log10_temperature_0`	Initial temperature (log10)	log10(K)
`log10_radius_0`	Initial radius (log10)	log10(cm)
`temp_rise_index`	Temperature power-law index before peak	–
`temp_decline_index`	Temperature power-law index after peak	–
`temp_peak_time`	Time of temperature maximum	days
`radius_rise_index`	Radius power-law index before peak	–
`radius_decline_index`	Radius power-law index after peak	–
`radius_peak_time`	Time of radius maximum	days

Supernova Models#

Models for core-collapse and thermonuclear supernovae powered by radioactive decay, magnetar spin-down, or circumstellar interaction.

ArnettModel

Nickel-56 / Cobalt-56 radioactive decay without diffusion (Arnett 1982). Uses Gauss-Legendre quadrature for the decay integral.

Parameter	Description	Units
`tau_m`	Diffusion timescale	days
`log10_mni`	Nickel-56 mass (log10)	log10(Msun)
`v_phot`	Photospheric velocity	1e9 cm/s

NickelCobaltModel

Nickel-56 / Cobalt-56 decay with Arnett (1982) diffusion integral.

Parameter	Description	Units
`f_nickel`	Fraction of ejecta in Nickel-56	–
`log10_mej`	Ejecta mass (log10)	log10(Msun)
`log10_vej`	Ejecta velocity (log10)	log10(km/s)
`log10_kappa`	Optical opacity (log10)	log10(cm^2/g)
`log10_kappa_gamma`	Gamma-ray opacity (log10)	log10(cm^2/g)

MagnetarPoweredSNModel

Magnetar spin-down engine with Arnett diffusion.

Parameter	Description	Units
`log10_p0`	Initial spin period (log10)	log10(ms)
`log10_bp`	Polar magnetic field (log10, in units of 1e14 G)	log10(1e14 G)
`mass_ns`	Neutron star mass	Msun
`theta_pb`	Angle between spin and B-field axes	radians
`log10_mej`	Ejecta mass (log10)	log10(Msun)
`log10_vej`	Ejecta velocity (log10)	log10(km/s)
`log10_kappa`	Optical opacity (log10)	log10(cm^2/g)
`log10_kappa_gamma`	Gamma-ray opacity (log10)	log10(cm^2/g)

CSMInteractionModel

Circumstellar medium shock interaction following the Chevalier (1982) self-similar solution for forward and reverse shocks.

Parameter	Description	Units
`log10_mej`	Ejecta mass (log10)	log10(Msun)
`log10_csm_mass`	CSM mass (log10)	log10(Msun)
`log10_vej`	Ejecta velocity (log10)	log10(km/s)
`eta`	CSM density profile exponent	–
`log10_rho`	CSM density amplitude (log10)	log10(g/cm^{eta+3})
`log10_kappa`	Opacity (log10)	log10(cm^2/g)
`log10_r0`	CSM inner radius (log10)	log10(AU)

Kilonova Models#

Models for kilonovae (neutron-star merger ejecta) powered by r-process radioactive heating.

MetzgerModel

Multi-shell ODE kilonova with 300 mass shells (Metzger 2017). Includes early neutron precursor and late r-process heating.

Parameter	Description	Units
`log10_mej`	Ejecta mass (log10)	log10(Msun)
`log10_vej`	Minimum ejecta velocity (log10)	log10(c)
`beta`	Velocity power-law index	–
`log10_kappa_r`	R-process opacity (log10)	log10(cm^2/g)

MetzgerFullModel

Multi-shell ODE kilonova with 200 shells using linear velocity spacing and Barnes+16 thermalisation efficiency.

Parameter	Description	Units
`log10_mej`	Ejecta mass (log10)	log10(Msun)
`log10_vej`	Minimum ejecta velocity (log10)	log10(c)
`beta`	Velocity power-law index	–
`log10_kappa_r`	R-process opacity (log10)	log10(cm^2/g)

Constructor options: neutron_precursor (bool, default True), vmax (float, default 0.7).

OneComponentKilonovaModel

Single-component kilonova using a diffusion integral (no ODE).

Parameter	Description	Units
`log10_mej`	Ejecta mass (log10)	log10(Msun)
`log10_vej`	Ejecta velocity (log10)	log10(c)
`log10_kappa`	Gray opacity (log10)	log10(cm^2/g)

Constructor options: temperature_floor (float, e.g. 4000.0 K).

MagnetarBoostedKilonovaModel

Multi-shell ODE kilonova (200 shells) with additional magnetar spin-down heating injected into the innermost shell.

Parameter	Description	Units
`log10_mej`	Ejecta mass (log10)	log10(Msun)
`log10_vej`	Minimum ejecta velocity (log10)	log10(c)
`beta`	Velocity power-law index	–
`log10_kappa_r`	R-process opacity (log10)	log10(cm^2/g)
`log10_p0`	Initial spin period (log10)	log10(ms)
`log10_bp`	Polar magnetic field (log10, in units of 1e14 G)	log10(1e14 G)
`mass_ns`	Neutron star mass	Msun
`theta_pb`	Angle between spin and B-field axes	radians
`thermalisation_efficiency`	Magnetar thermalisation fraction	–

Constructor options: neutron_precursor (bool), pair_cascade (bool), vmax (float).

Shock-Powered Models#

Models for early emission powered by shock breakout and shock cooling.

ShockCoolingModel

Shock-cooling emission following Piro (2021) with analytic piecewise solution.

Parameter	Description	Units
`log10_Menv`	Envelope mass (log10)	log10(Msun)
`log10_Renv`	Envelope radius (log10)	log10(Rsun)
`log10_Ee`	Explosion energy (log10)	log10(erg)

ShockedCocoonModel

Jet cocoon shock cooling with a fully algebraic (closed-form) solution.

Parameter	Description	Units
`log10_mej`	Ejecta mass (log10)	log10(Msun)
`log10_vej`	Ejecta velocity (log10)	log10(c)
`eta`	Density profile exponent	–
`log10_tshock`	Shock breakout time (log10)	log10(s)
`shocked_fraction`	Fraction of ejecta shocked	–
`cos_theta_cocoon`	Cosine of cocoon opening angle	–
`log10_kappa`	Gray opacity (log10)	log10(cm^2/g)

Other Models#

TDEAnalyticalModel

Tidal disruption event with t^{-5/3} fallback accretion rate and Arnett-style diffusion.

Parameter	Description	Units
`log10_l0`	Luminosity at 1 second (log10)	log10(erg/s)
`t_0_turn`	Turn-on time	days
`log10_mej`	Ejecta mass (log10)	log10(Msun)
`log10_vej`	Ejecta velocity (log10)	log10(km/s)
`log10_kappa`	Opacity (log10)	log10(cm^2/g)
`log10_kappa_gamma`	Gamma-ray opacity (log10)	log10(cm^2/g)

SALT3Model

Type Ia supernova spectral-template model using the SALT3 SED framework. Requires the jax-bandflux package (pip install jax-bandflux).

Parameter	Description	Units
`log10_x0`	SALT3 amplitude (log10)	–
`x1`	SALT3 stretch	–
`c`	SALT3 color	–
`t0`	Time of B-band maximum (observer-frame)	days

Note

Unlike all other analytical models, SALT3Model expects times in the observer frame, not the source frame.

Writing a Custom Model#

To add a new analytical model, subclass AnalyticalModel and implement compute_log10_lbol_rphot:

import jax.numpy as jnp
from fiesta.inference.analytical_models import AnalyticalModel

class MyModel(AnalyticalModel):
    parameter_names = ["log10_luminosity", "log10_radius"]

    def __init__(self, filters, times=None):
        super().__init__(filters=filters, times=times)

    def compute_log10_lbol_rphot(self, x, t_days):
        # Return (log10_Lbol [erg/s], log10_Rphot [cm]) at each t
        log10_L = jnp.full_like(t_days, x["log10_luminosity"])
        log10_R = jnp.full_like(t_days, x["log10_radius"])
        return log10_L, log10_R

The base class handles the blackbody SED, filter integration, redshift correction, and distance modulus automatically. The predict method is JIT-compiled and differentiable out of the box.

For phenomenological models that parameterize magnitudes directly (rather than bolometric luminosity), subclass PhenomenologicalModel instead and implement compute_shape.