In Zhao et al. (2024), we showed that many factors in a thermophysical model makes the equation-driven deep learning models like physics-informed neural networks (PINNs) (Raissi et al. 2019) unsuitable, despite their successful application in solving ordinary differential equations (ODEs) and partial differential equations (PDEs) (Cai et al. 2020; Zobeiry & Humfeld 2021; He et al. 2021; Martin & Schaub 2022a,b; Mathews & Thompson 2023; Laghi et al. 2023).

Therefore, for asteroid thermophysical modelling, we adopted deep operator neural network (Deep-ONet) with strong generalization abilities (Lu et al. 2021; Garg et al. 2022; He et al. 2023; Branca & Pallottini 2024).

Such a network is well suited for handling problems with variable boundary conditions.

Following the same consideration, we continue to utilize DeepONet for the development of ThermoONet in this work.

The surface of an asteroid is treated as a layer of dust, whose temperature is influenced primarily by the radiation flux and the thermal parameter (a total of two independent parameters).

However, for comets, the distinct physical properties of the dust-ice mixture layer in contrast to the dust mantle, along with the phase transitions of water ice, necessitate a more complex selection of parameters to describe the thermophysical process.

Consequently, we identified a total of eight independent parameters (Eq. 13).

In this work, we modified the structure of the neural network to address the high dimensionality of the parameters and furthermore manually optimized the distribution of the training dataset, based on the relative impact of different parameters on the results.

The specific methods are introduced in detail below.
3.1.

Architecture of neural network The principles and structure of DeepONet are in details explained in Lu et al. (2021), and we just provide a brief summary here.

The core idea is to utilize a neural network to map the different variables from the partial differential equations into a common mathematical space for processing.

The network that maps the parameters or functions of the equations is referred to as the branch network, while the network that maps the independent variables is called the trunk network.

If we denote the parameters or functions, independent variables as u, y, these two mapping relationships can be expressed as u →G(u) and y →G(u)(y), where G is called the operator.

In the thermophysical modelling of cometary systems, the operator mapping process involves the radiation flux function E(t) and eight characteristic parameters from Equation (13), with depth and time serving as independent variables.

A DeepONet architecture could theoretically be constructed with branch networks encoding radiation flux function and parameters, trunk networks handling independent variables.

The large number of parameters renders the training of an accurate neural network particularly challenging.

This necessitates a simplification of the network architecture through problem-specific adaptations.

Theoretically the sublimation flux fundamentally depends on the temperature at the dust mantle/dust-ice mixture interface.

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physical understanding enables significant architectural simplification by reducing the network output to this subsurface temperature.

Consequently, the trunk network becomes redundant and can be eliminated, with the branch network outputs directly providing the temperature predictions.

For comprehensive modeling of non-gravitational effects induced by sublimation processes, additional consideration must be given to the surface temperature of the dust mantle, which should be incorporated as another output.

This dual-temperature approach enables simultaneous prediction of both the subsurface temperature governing sublimation dynamics and the surface temperature essential for calculating the velocity of the gas (detailed analyses of these effects are provided in Attree et al. (2019) and Jewitt et al. (2020)).

In the asteroid network, we model the thermal behavior of the asteroid using two separate branch networks: one for encoding the radiation flux function and the other for encoding the thermal parameter (Zhao et al. 2024).

The treatment of generating random radiation flux functions can be directly adapted to the comet model (see Section 3.2 for more about the dataset).

However, the comet model introduces a larger number of independent parameters exhibiting heterogeneous sensitivity to the solutions of the model, so direct adoption of the parameter encoding scheme for asteroids proves ineffective.

A single branch network collectively encoding all parameters can reduce sensitivity to each individual parameter, especially for those parameters significantly influencing the results.

Conversely, constructing separate branch networks for individual parameters would result in excessive computational memory allocation and time cost.

To resolve this challenge, we propose a parameter grouping method based on sensitivity analysis.

Parameters governing the thermal processes are partitioned into two distinct categories, high-sensitivity and moderate-sensitivity parameters.

Each category is encoded through dedicated branch network, achieving an optimal balance between parameter sensitivity and computational efficiency.