) = a exp (−b
T ), α(T) = c0 + c1 1 + exp (c2 −c3/T), (7)
with constants a, b, c0, c1, c2, c3 taking the values 3.23×1012 Pa, 6134.6 K, 0.146, 0.854, 57.78, 11580 K.

The parameter Ψ in Eq. (5) represents the reduction factor in the sublimation flux from the bare icy surface, accounting for the permeability of the mantle to gas flow.

It is given by the expression 1/(1 + pH) (Gundlach et al. 2011; Hu et al. 2017), where p is a constant and H = X/d, with d denoting the diameter of the spherical dust aggregates in the dust mantle (Skorov & Blum 2012).

We take p = 0.14 and d = 1 mm in this work, same as Hu et al. (2017). f in Eq. (5) is used to approximate the reduction in sublimation flux from the pure ice caused by the presence of mixed dust, as the Hertz-Knudsen formula is derived for pure ice (Crifo 1997).

The last boundary condition describes the invariability of temperature at infinitely large depth, ∂ ∂xTx→∞= 0. (8)
2.2.

Normalization of thermophysical model Direct application of deep learning methods to the model described above presents several challenges: (1) the parameters have a large dynamic range, leading to the fact that the resulting temperature distribution could span over a wide numerical range. (2) The model is influenced by the rotational velocity of comet, which is not explicitly represented in the established equations.

Instead, it is implicitly embedded in the time-dependent function sψ and the time steps used in the subsequent numerical simulations.

Therefore, normalization becomes particularly crucial in this context.

Article number, page 2 of 13

We denote the rotational angular velocity as ω and introduce the following normalized variables (Lagerros 1996),
2.3.

Numerical treatment
xd = x
, t = ωt, T = T
, (9)
lsd , xm = x
lsm
Te where lsd and lsm are the skin depth of the dust mantle and the dust-ice mixture, Te is the theoretical maximum surface temperature at the heliocentric distance r, called characteristic temperature, given by � (1 −AB)F⊙
κd ρdCdω, lsm = �
κm ρmCmω, Te =
lsd = �
ϵσ , (10)
the heat conduction equation Eq. (1) and three boundary conditions Eq. (3), Eq. (4), Eq. (8) can then be normalized as:
∂t = ∂2T
∂T
∂x2 , ∂ ∂xd
4 xd=0 −Φd
T xd=0 = E(t),
T (11)
∂ ∂xm
−Φd ∂ ∂xd
−= −Φm
T xd=X
T xm=X
+ + lZ(X),
∂ ∂xm
T xm→∞= 0,
where Φd and Φm are the thermal parameters (Spencer et al. 1989) of the dust mantle and dust-ice mixture, defined by
, Φm = √ρmCmωκm
. (12)
Φd = √ρdCdωκd ϵσT 3e
ϵσT 3e
E(t) is the radiation flux rescaled in the range [0, 2π] for variables t and [0, 1] for E.

Z(X) = Z(X)/(1−AB)F⊙is the sublimation flux after variation.

It is evident that the radiation flux and the derived temperature are successfully rescaled in the range [0, 1], with all physical parameters explicitly represented in the mathematical expression of the equations.

This transformation can significantly enhance the efficiency and ease of dataset creation for deep learning.

For instance, when studying different comets, the original model for the radiation flux functions leads to varying domains of definition due to differences in rotational velocities.

Additionally, the range of values fluctuates with variable parameters such as Bond albedo AB and heliocentric distance r.

This variability complicates the construction of a comprehensive set of radiation flux functions.

However, the normalized model can effectively address this issue through the parameter extraction.