# Neural-accelerated numerical model for packed bed latent heat storage system

Dessie Tadele Embiale, Shri Balaji Padmanabhan, Mohamed Tahar Mabrouk,

Stéphane Grieu, Bruno Lacarrière

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Dessie Tadele Embiale, Shri Balaji Padmanabhan, Mohamed Tahar Mabrouk, Stéphane Grieu, Bruno Lacarrière. Neural-accelerated numerical model for packed bed latent heat storage system. Energy and AI, 2025, 22 (1), pp.100602. 10.1016/j.egyai.2025.100602. hal-05285919

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Submitted on 8 Oct 2025

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# Neural-accelerated numerical model for packed bed latent heat storage system

Dessie Tadele Embiale<sup>a, *</sup> <sup>1, D</sup>, Shri Balaji Padmanabhan<sup>a</sup>, Mohamed Tahar Mabrouk<sup>a, *</sup> <sup>1, D</sup>, Stéphane Grieu<sup>b</sup>, Bruno Lacarrière<sup>a</sup>

## HIGHLIGHTS

* A neural accelerated numerical model for latent heat storage systems is proposed.

* It fuses fast, less accurate model with slow, accurate model via a neural network.

* It is 764 times faster than the slow and accurate model with error of below 0.235 %.

* Offers good generalization, interpretability, and strong extrapolation capability.

* It is suitable for model-based optimization and control applications.

## GRAPHICAL ABSTRACT

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

### Keywords:

Thermal storage

Latent heat

Phase change material

Machine learning

Neural networks

Numerical models

## ABSTRACT

Developing accurate and computationally efficient dynamic models for packed-bed latent-heat storages (PBLHS) is crucial for reliably predicting their performance across different operating scenarios and enabling their use in planning and real-time control. In this study, a novel neural-accelerated numerical model for PBLHS is proposed by coupling a neural network (NN) into a coarsely discretized equations of the Continuous-solid Phase (CP) model. The embedded NN predicts the surface temperature of the phase change material (PCM) given the fluid temperature and enthalpy of the PCM as inputs, which the CP model fails to capture. This allows the neural-accelerated model to replicate the accuracy of a high-fidelity and computationally expensive model namely Concentric Dispersion (CD) model. An innovative data generation process to generate training data for NN involving both CD and CP model is proposed. Two versions of neural-accelerated model are proposed, one with conventional NN and another using NN with a custom activation function. Both versions demonstrate an excellent accuracy, achieving MSE as low as 0.117 °C, R² values closer to 0.995 and error percentage below 0.394% compared to the highly accurate CD model. As for computational efficiency, the proposed models achieved 342 times and 764 times acceleration respectively. The gain in more acceleration for the later version of the proposed model is achieved through the use of a compact architecture that benefits from the custom activation function, while also enhancing model explainability. These results highlight the model's suitability for scenarios demanding both high accuracy and computational efficiency.

<footnote><sup>a</sup> IMT Atlantique, Department of Energy Systems and Environment, GEPEA UMR CNRS, 6144, F-44307, Nantes, France</footnote>

<footnote><sup>b</sup> PROMES-CNRS (UPR 8521), Université de Perpignan Via Domitia, rambla de la thermodynamique, Tecnosud 66100 Perpignan, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: dessie-tadele.embiale@imt-atlantique.fr (D.T. Embiale), mohamed-tahar.mabrouk@imt-atlantique.fr (M.T. Mabrouk).</footnote>
## Nomenclature

<table><tr><td>ρ</td><td>Density, kg/m<sup>3</sup></td></tr><tr><td>C<sub>p</sub></td><td>Specific heat capacity, kJ/kgK</td></tr><tr><td>u</td><td>Velocity of the fluid, m/s</td></tr><tr><td>k</td><td>Thermal conductivity, W/mK</td></tr><tr><td>μ</td><td>Dynamic viscosity, kg/m<sup>2</sup>s</td></tr><tr><td>c</td><td>Porosity of the bed, -</td></tr><tr><td>h<sub>v</sub></td><td>Volumetric heat transfer coefficient, W/m<sup>2</sup>K</td></tr><tr><td>L<sub>f</sub></td><td>Latent heat of fusion of the PCM, kJ/kg</td></tr><tr><td>T</td><td>Temperature, °C</td></tr><tr><td>H</td><td>Enthalpy of PCM, kJ/kg</td></tr><tr><td>T<sub>ps</sub></td><td>Temperature where melting starts, °C</td></tr><tr><td>T<sub>pl</sub></td><td>temperature where melting ends, °C</td></tr><tr><td>C<sub>ps</sub></td><td>solid specific heat capacity of PCM, kJ/kgK</td></tr><tr><td>C<sub>pl</sub></td><td>Liquid specific heat capacity of PCM, kJ/kgK</td></tr><tr><td>D<sub>p</sub></td><td>Capsule diameter, m</td></tr><tr><td>D<sub>t</sub></td><td>Diameter of the tank, m</td></tr><tr><td>H<sub>t</sub></td><td>Height of the tank, m</td></tr></table>

## Subscripts

<table><tr><td>f</td><td>Heat transfer fluid</td></tr><tr><td>p</td><td>PCM</td></tr><tr><td>ref</td><td>Reference</td></tr></table>

# 1. Introduction

In the energy sector, thermal energy storage (TES) systems play a crucial role in balancing the mismatch between energy supply and demand. The most widely seen application area of TES is their integration with variable renewable energy sources like solar thermal systems. They are used to store the surplus amount of energy produced during periods of high solar radiation and release the stored energy when solar radiation is low. Other application areas of TES include district heating networks, building energy conservation, refrigeration and air conditioning systems, and industrial waste heat recovery [1,2]. TES is categorized into three types: sensible heat storage (SHS), latent heat storage (LHS) and thermo-chemical storage. SHS uses solid materials, like rock, water and molten salt, to store heat. Whereas, LHS stores heat in materials such as paraffins and salts, which undergoes phase change upon heating. Thermo-chemical energy storages allow heat to be stored through reversible reactions, offering higher energy density than SHS and LHS, but at the cost of increased complexity and expense. SHS is less expensive than the other energy storage systems, but has relatively lower energy density. LHS offers higher energy storage capacity compared to SHS, which significantly reduces the storage tank size, while still being cheaper and less complex than thermochemical storage, making it suitable for various applications [3–6]. A notable limitation of LHS is the inherently low thermal conductivity of phase change materials (PCMs), which can restrict the efficiency of heat transfer. To overcome this, various thermal performance enhancement techniques have been employed, including composite PCMs, adding fins, and encapsulation [1,7,8]. Composite PCMs are formed by embedding the PCM with a high thermal conductivity material like graphite, metal foams or nanoparticles. This enhancement significantly increases the overall thermal conductivity of the composite PCM allowing much faster heat absorption and release, hence, faster charging and discharging [9–14]. Adding fins of different cross-sections is another performance enhancement strategy which increases the heat transfer area between the heat transfer fluid and the PCM [13–20]. However, despite having these advantages, the addition of fins increases the weight of the storage and reduces the

amount of PCM inside the storage, reducing the total energy storage capacity [9]. Macro-encapsulation is another widely adopted approach in LHS where the storage tank is filled with small spherical capsules, commonly referred to as Packed Bed Thermal Energy Storage (PBTES) systems. These storages offer various advantages like significantly larger heat transfer area for rapid heat exchange and higher volumetric energy storage capacity [21]. Each capsule is filled with the storage material, and the Heat Transfer Fluid (HTF) flows through the gap between the capsules. During the charging phase, as the HTF flows over the capsules, the first stage is a sensible heating process in which the temperature of the PCM rises until the melting temperature is reached. Then, the PCM starts to melt and a large amount of energy (latent heat) is stored during this phase transition stage. Once the PCM is completely melted, another sensible heating process continues in the liquid phase. During the discharging phase, the phase change process is essentially repeated, with heat being transferred from the PCM to the HTF [22].

In the literature, several mathematical models have been developed to model the dynamic thermal behavior of PBTES. These mathematical models can be grouped into single-phase models and two-phase models. The single-phase model considers the solid PCM and the HTF as one phase and the temperature of the solid and the fluid phases are considered to be the same. However, due to the significant temperature gradient typically observed between the solid and fluid phases in reality, assuming a uniform temperature across them is not physically accurate. This is mainly because of the difference in the thermal conductivity and specific heat capacity between the HTF and the PCM [1,23]. In two-phase models, the packed bed is considered to be in two distinct phases, solid and liquid, and the temperatures of the solid and liquid phases are determined by solving two separate energy conservation equations. The Continuous-solid Phase (CP) model is one of the two-phase models that considers the PCM capsules as a continuous porous medium instead of individual capsules. Yang and Cai [5], Reddy et al. [24], Wu and Fang [25], Regin et al. [26] used the CP model to study the thermal performance of PBTES. The Concentric Dispersion (CD) model is another two-phase approach similar to the CP model but it accounts for radial temperature gradients within individual solid spheres. Unlike the other models which consider the capsules as a continuous medium, the CD model treats them as individual capsules and the thermal gradient inside the capsules in the radial direction is taken into consideration. The CD model has been used in many studies to investigate the temperature distribution and performance of PBTES [27, 28]. Other studies have also compared this model with experimental results [7,29,30], demonstrating good agreement. Karthikeyan and Velraj [31], Guo et al. [23] and Ismail and Stuginsky [32] performed a comparative study between the CP and CD model, and validated the simulation results experimentally. The authors reported that, the CP model fails to capture the thermal gradients within the capsules, whereas the CD model accurately accounts for these gradients and aligns well with experimental data. However, the CD model is considerably more computationally intensive than the CP model. Overall, the CP model is computationally efficient but lacks accuracy, while the CD model offers higher accuracy at the cost of a significant computational effort. This highlights the need for an intermediate model that balances both speed and accuracy for practical applications.

Machine Learning (ML) has emerged as a promising and efficient alternative to physics-based modeling, with the ability to learn complex patterns directly from data. Linear regression [33], which uses one or more input features to predict a continuous variable, is simple and interpretable but poor in handling nonlinear system dynamics. Support Vector Machines [34] offer improved performance particularly with smaller datasets, but are computationally demanding and sensitive to kernel selection. Gradient boosting algorithms [35] achieve high accuracy in forecasting and modeling nonlinearities but necessitates carful hyperparameter tuning and significant computational resources. Artificial neural networks are well-suited for capturing complex and nonlinear behavior; however, they require extensive training data and
lack transparency [36,37]. Random forest models provide robust and accurate predictions and also handles missing data effectively [38]. K-Nearest Neighbors [39], while easy to implement and effective for low-dimensional problems, is less efficient for large datasets and can be influenced by irrelevant features. Several studies in the literature have applied these ML techniques to predict the performance of TES systems, particularly those using PCM-based latent storage. Anagnostopoulos et al. [40] employed deep learning combined with metaheuristic algorithms to predict the performance of a PBTES system. Similarly, Tavakoli et al. [2] utilized feedforward neural networks to optimize fin geometry in a finned LHS unit. Anand et al. [39] also conducted design analysis and performance prediction of a PBTES using various machine learning techniques, including linear regression, support vector machines, random forests, decision trees, and extreme gradient boosting, reporting the latter as the most effective. Additionally, He et al. [36] provided a comprehensive review of ML applications in TES systems, covering performance prediction, optimal design, and control. These studies point that ML methods are good at accurately predicting the system's performance, facilitating better design and optimization and overcomes the computational expensiveness of physics-based modeling. However, such models strongly depend on both the quality and quantity of data, making them susceptible to overfitting, which limits their ability to generalize beyond the training scenarios. Most of the reported models are not also capable of predicting detail dynamics of the system, rather limited to predicting stored energy and outlet temperature. They can also be computationally intensive to train and often lack interpretability, offering little insight into the underlying physics of PBTES [36].

Effectively integrating ML with physics-based models overcomes the limitations of both purely data-driven approaches and purely physical approaches. When physics-based and ML models are combined, they offer fast and accurate predictions by capturing complex, nonlinear patterns while also providing meaningful insights into the system behavior. This enhances generalization and interpretability, which is critical for understanding the underlying mechanisms. Purely ML models typically lack this capability [41] demonstrated the potential strengths of combining physics-based and ML models in a sensible heat-based PBTES systems. This model achieves high accuracy while being up to 350 times faster than traditional numerical approaches, making it ideal for real-time applications. Physics and ML components can be combined in several ways, with the most natural approaches being either in series or in parallel [42,43], where they may operate independently or interdependently, though this can sometimes cause one component to overshadow the other. Another widely used technique is Physics-Informed Neural Networks (PINNs) which incorporates the domain knowledge directly into the ML architecture to ensure the prediction given by the ML model adhere to the known physical principles. Despite their ability to integrate physics-based constraints with data driven learning, PINNs often suffer from high computational cost for training and need to retrain for different working conditions [44]. A promising approach is to couple them in a way that ensures seamless integration and equal participation. One notable framework for this is the Universal Differential Equations (UDE) framework, where a neural network is embedded within differential equations, which has gained interest in many fields [44,45]. It is a mathematical concept that incorporates universal approximators into differential equations and it needs to be implemented for the targeted application by defining the equation and what part should be encapsulated in the ML part. Furthermore [45], coupled a neural network with simplified differential equations to accelerate complex model for heat pump application.

In the present study, a novel neural-accelerated numerical model is proposed to simulate the dynamic thermal behavior of a latent heat-based PBTES. This model is designed as a hybrid between the CD and CP models, aiming to combine and leverage the advantages of both. To

achieve this goal, the CP model is modified by incorporating a ML component to take into consideration the inhomogeneous heat transfer process inside the capsules. Unlike PINNs which incorporates physical laws as residual loss during training, the proposed neural-accelerated model embeds a pre-trained neural network directly within the equations of CP model. This approach is based on the UDE framework, which combines neural networks with differential equations and to the best of the authors knowledge, this specific method of combining the two modelling techniques of the PBTES using the UDE framework has not been previously done.

## 2. Mathematical models

As depicted in Fig. 1, a cylindrical storage tank with spherical capsules filled with paraffin wax as PCM is considered. During the charging process, the hot HTF enters from the top of the tank and leaves at the bottom after heat is exchanged with the capsules. Whereas during the discharging process the flow direction is reversed, it enters from the bottom and leaves through the top hole, and heat is transferred from the PCM to the HTF. Distributors are provided at both ends of the tank to evenly distribute the HTF. The storage system considered in this study has a height $H_t = 2.0$ m, diameter of $D_t = 0.8$ m with $\epsilon = 0.45$ porosity and capsule of diameter $D_p = 0.02$ m. Thermolab 66, which is well known for its application in a wide range of temperature is selected as a HTF. The thermophysical properties of the HTF and the PCM are shown in Table 1 and Table 2, respectively.

### 2.1. Governing equations

Two models are adopted in this paper: A one-dimensional CP model and a two-dimensional CD model. For both models, energy balance equations for the HTF and PCM are developed subjected to the following general assumptions:

* the HTF is considered incompressible;

* the void fraction is uniform in the bed;

* the thermal resistance of the capsule shell is neglected;

* radiation heat exchange is not considered;

* the HTF is incompressible and the momentum equation is not considered.

### 1. The CP model

For the CP model, the PCM is considered as a homogeneous and isotropic medium. Thermal diffusion is also considered along the height of the tank only, and the one-dimensional energy balance equations for the HTF and the PCM are given in Eq. (1) and (2), respectively [47].

$$ \epsilon \rho_f C_{p-f} \left( \frac{\partial T_f}{\partial t} + u \frac{\partial T_f}{\partial x} \right) = k_f \frac{\partial^2 T_f}{\partial x^2} + h_v (T_p - T_f) + h_w (T_w - T_f) \quad (1) $$

$$ (1 - \epsilon) \rho_p \frac{\partial H}{\partial t} = k_p \frac{\partial^2 T_p}{\partial x^2} + h_v (T_f - T_p) \quad (2) $$

### 2. The CD model

The CD model considers the thermal gradient inside the capsule instead of considering it as a continuous solid medium. For the PCM, in addition to the heat transfer in the axial direction, the heat transfer inside the capsules in the radial direction is also modeled. The energy balance equation for the HTF and the PCM can be written as in Eq. (3) and (4), respectively [1].

$$ \epsilon \rho_f C_{p-f} \left( \frac{\partial T_f}{\partial t} + u \frac{\partial T_f}{\partial x} \right) = k_f \frac{\partial^2 T_f}{\partial x^2} + h_v (T_p - T_f) + h_w (T_w - T_f) \quad (3) $$
<figure><img src="image_4.png" /><figcaption>Fig. 1. Schematic of the PBTES.</figcaption></figure>

<table><caption>**Table 1**

Thermophysical properties of the HTF [46].</caption><thead><tr><th>Property</th><th>Value</th></tr></thead><tbody><tr><td>Density (kg/m<sup>3</sup>)</td><td>- 0.000321T<sup>2</sup> - 0.614254 T + 1020.62</td></tr><tr><td>Thermal conductivity (W/m.K)</td><td>- 1.5<sup>-7</sup> T<sup>2</sup> - 3.3<sup>-5</sup> T + 0.118294</td></tr><tr><td>Specific heat capacity (kJ/(kg.K))</td><td>8.970785<sup>-4</sup> T<sup>2</sup> - 3.313T + 1496.005</td></tr><tr><td>Dynamic viscosity (kg/ms)</td><td>exp (log(T) + 10.773) × 1<sup>-3</sup></td></tr></tbody></table>

<table><caption>**Table 2**

Thermophysical properties of the PCM (Paraffin wax) [22].</caption><thead><tr><th>Property</th><th>Value</th></tr></thead><tbody><tr><td>Melting temperature range (°C)</td><td>58-60</td></tr><tr><td>Latent heat of fusion (kJ/kg)</td><td>189</td></tr><tr><td>Thermal conductivity (W/m.K)</td><td>0.22</td></tr><tr><td>Density (kg/m<sup>3</sup>)</td><td>918</td></tr><tr><td>Solid specific heat (kJ/(kg.K))</td><td>1.984</td></tr><tr><td>Liquid specific heat (kJ/(kg.K))</td><td>2.384</td></tr></tbody></table>

$$ \rho_p \frac{\partial H}{\partial t} = k_p \left( \frac{\partial^2 T_p}{\partial r^2} + \frac{2}{r} \frac{\partial T_p}{\partial r} \right) \quad (4) $$

When phase change occurs over a temperature range rather than at a distinct temperature point, the enthalpy method becomes a suitable approach for accurately modeling the process. The temperature-enthalpy relation of the PCM is determined using Eq (5).

$$ H = \begin{cases} C_{ps}(T_p - T_{ref}) & \text{if } T_p < T_{ps} \\ C_{ps}(T_{ps} - T_{ref}) + L_f \left( \frac{T_p - T_{ps}}{T_{pl} - T_{ps}} \right) & \text{if } T_{ps} \le T_p \le T_{pl} \\ C_{ps}(T_{ps} - T_{ref}) + L_f + C_{pl}(T_p - T_{pl}) & \text{if } T_p > T_{pl} \end{cases} \quad (5) $$

The correlation for the volumetric heat transfer coefficient ($h_v$) between the HTF and the PCM and the velocity of the HTF are given by Eqs. (6) and (7), respectively [7]:

$$ h_v = \frac{6(1 - \epsilon) k_f (2 + 1.1 P_r^{0.33} R_e^{0.36})}{D_p^2} \quad (6) $$

$$ u = \frac{4 \dot{m}_f}{\pi \epsilon \rho_f D_t^2} \quad (7) $$

The correlation for the Prandtl number and Reynolds number is given in Eqs. (8) and (9).

$$ P_r = \frac{C_{pf} \mu_f}{kf} \quad (8) $$

$$ R_e = \frac{\rho_f u D_p}{\mu_f} \quad (9) $$

## 2.2. Initial and boundary conditions

The initial and boundary conditions for both the PCM and HTF in the two models are defined as follows,

**For the HTF:**

Initial conditions: $T_f(t = 0, x) = T_{f0}(x)$

Boundary conditions: $T_f(t, x = H) = T_{inlet}, \left(\frac{\partial T_f}{\partial x}\right)_{x=0} = 0$
**For the PCM:**

Initial conditions for both models: $T_p(t = 0, x) = T_{p0}(x)$
Boundary conditions:

$$ \left(\frac{\partial T_p}{\partial x}\right)_{x=0} = 0, \quad \text{and} \quad \left(\frac{\partial T_p}{\partial x}\right)_{x=H} = 0 $$

In the CD model, two additional boundary conditions are introduced:

$$ k_p \left( \frac{\partial T_p}{\partial r} \right)_{r=r_0} = h_p (T_f - T_{p-s}), \quad \text{and} \quad \left( \frac{\partial T_p}{\partial r} \right)_{r=0} = 0 $$

The initial and boundary conditions are for the charging phase only. For the discharging phase, the flow direction is reversed, the boundary conditions are updated, and the final state of the storage is considered as initial condition, as discussed in Section 2.

## 2.3. Solving methodology

The mathematical models described in Section 2.1 can be spatially discretized using the finite volume method, which is a robust numerical technique for solving conservation equations. The storage is discretized into “N” axial control volumes (CVs) along its height for both models. Additionally, in the CD model, the capsules are further discretized into “R” CVs radially to simulate the heat transfer within the capsule. Differential Equations.jl library, is used to solve the resulting Ordinary Differential Equations (ODE) system.

## 2.4. Grid independency study

A grid independence test is conducted by simulating both models with varying numbers of control volumes under fixed conditions. The results are compared against a reference solution obtained using $N = 2000$ CVs, which is considered to have negligible discretization error. For the CP model, the number of discretizations along its height for both the HTF and PCM equations is varied. The error between the results obtained for each control volume configuration and the reference fine-mesh solution is calculated and presented in Fig. 2. It is observed that, after $N = 1000$ CV, the error becomes negligibly small and further refinement of the grid does not improve the accuracy significantly. In contrast, for the CD model, R is varied while keeping $N = 1000$ CVs. It was found that 40 CVs are sufficient to model the heat transfer inside the capsule with error percentage of below 0.05 as indicated in Fig. 3. Based on the grid-independence analysis, $N = 1000$ CVs are selected for simulating both the PCM and the HTF domains in the CP model. For the CD model, $N = 1000$ CVs are used for the HTF domain, while $R = 40$ CVs are chosen for the PCM domain to capture the radial temperature gradient efficiently.

<figure><img src="image_5.png" /><figcaption>Fig. 2. Variation of error percentage of the CP model for different N values.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Variation of error percentage of the CD model for different R values, $N = 1000$ CVs.</figcaption></figure>

## 2.5. Comparative analysis of the CP and CD models

To highlight the disparity between the two models, simulations using both models are carried under identical boundary and initial conditions, and the resulting numerical solutions are compared to evaluate their performance and accuracy. Time varying inlet temperature and mass flow rate are applied during the simulations, as shown in Fig. 4 and Fig. 5. The inlet temperature varies randomly between 75 °C and 95°C during charging, and between 30 °C and 40°C during discharging. The mass flow rate ranges between 0.2 kg/s and 0.5 kg/s for both charging and discharging phases. Based on the simulation results of both models, the key quantities of interest such as charging and discharging times, PCM temperature distribution along the storage axis during charging, and HTF outlet temperature during discharging are evaluated and compared.

The temperature distribution of the PCM across the height of the storage system at different times during the charging process for the CD and CP models is presented in Fig. 6. The temperature variation of the PCM with time at the middle ($H = 1m$) and the end ($H = 2m$) of the storage for both the charging and discharging processes is shown in Fig. 7. The outlet temperature of the HTF during the discharging process is compared between the two models, as shown in Fig. 8. During the initial sensible heating stage, the temperature of the PCM rapidly increases until it reaches the phase transition temperature for both models as seen in Fig. 7. Then phase transition starts and the PCM melts slowly. After melting is completed, the PCM is in liquid state and its temperature increases until it reaches the charging temperature. From the figures it is also seen that according to the CP model, the storage system takes less time to completely charge and discharge compared to the CD model’s simulation. This is mainly because the heat transfer between the HTF and the PCM is overestimated in the CP model since it doesn’t consider the thermal gradient inside the capsule. Due to the low thermal conductivity of the PCM and its phase change process, there is a slow thermal dynamic inside the capsule and neglecting these dynamics leads to fast charging and discharging. However, the CD model relatively takes more time to charge and discharge the storage system since it considers the thermal gradient inside the capsule. Similar results are also reported by [23,31]. The CD model was also validated against experimental results, demonstrating strong agreement [30,31,48]. However, in terms of computational performance, the CD model is approximately 71 times slower than the CP model, as indicated in Table 3.

## 3. The neural continuous solid phase model

The Neural-Continuous Solid Phase (Neural-CP) model developed in this study combines the accuracy of the CD model and the simplicity of the CP model. The Neural-CP model, primarily based on the CP model strongly coupled with a Neural Network (NN) as a universal approx-
<figure><img src="image_7.png" /><figcaption>Fig. 4. Time varying inlet temperature for the charging and discharging processes.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Time varying inlet mass flowrate.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Temperature distribution of the PCM during the charging process.</figcaption></figure>

imator, as shown in Fig. 9. The coupled NN dynamically predicts with high accuracy the PCM surface temperature given effective states of the system $\tilde{T}_f$ and $\tilde{H}$ instead of using temperature-enthalpy relation to model an average PCM temperature. The NN is trained indirectly using CD model simulations via a specific data generation process. The energy conservation equations of the Neural-CP model are given by Eqs. (10) and (11) and the coupled ODEs are integrated over time with the ODE

solvers. The temperature dynamics at the surface of the PCM is known from the CD model, but the effective states $\tilde{T}_f$ and $\tilde{H}$ are unknown. Snapshots of these states are generated using CD model simulation results via an adaptive physics-guided data synthesis process inspired from [45] and adapted to this specific application, which will be explained in detail in Section 3.1. The Neural-CP model is discretized using the finite volume method, and solved using ODE solvers, during which, for each instance, the NN gives the PCM surface temperature given the effective states and primary inputs ($T_{f,in}$ and $m_{f,in}$). The effective states given to the NN act as hidden states that retain information from past behavior, enabling the system to capture and reflect its dynamics over time.

$$ \rho_f C_{p-f} \left( \frac{\partial \tilde{T}_f}{\partial t} + u \frac{\partial \tilde{T}_f}{\partial x} \right) = k_f \frac{\partial^2 \tilde{T}_f}{\partial x^2} + h_v ( \underline{NN} - \tilde{T}_f ) + h_w (T_w - \tilde{T}_f) \quad (10) $$

$$ (1 - \epsilon) \rho_p \frac{\partial \tilde{H}}{\partial t} = h_v (\tilde{T}_f - \underline{NN}) \quad (11) $$

### 3.1. Adaptive physics-guided data generation for neural networks

The neural network predicts the PCM surface temperature given the effective dynamic states ($\tilde{T}_f$ and $\tilde{H}$). The procedure for generating snapshots of these effective states and generating the training data for
<figure><img src="image_10.png" /><figcaption>Fig. 7. Comparison of PCM temperature at the middle and end of the storage system.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Outlet temperature of the HTF during the discharging process.</figcaption></figure>

<table><caption>**Table 3**

Computational time comparison of the two models.</caption><thead><tr><th colspan="2">Computational time (s)</th></tr></thead><tbody><tr><td>CP model</td><td>CD model</td></tr><tr><td>1.7</td><td>120</td></tr></tbody></table>

the neural network is outlined below. The workflow of the data generation process is also illustrated in Fig. 10. For a given primary input ($T_{f,in}$ and $m_{f,in}$), the CD model with 1000 CVs discretization is solved, and the obtained results including the temperature distribution of the HTF, as well as the enthalpy and temperature distribution of the PCM, are extracted. Then, from these results, the heat transfer term $q = (h_v(T_f - T_p))$ between the HTF and the PCM is calculated. This term accounts for the thermal interaction between the HTF and the PCM and accurately captures the heat exchange dynamics. Since it is derived from the CD model, it also takes into consideration the thermal gradient inside the capsule and accurately represents the physical process. The calculated $q$ is then given as an input to a modified-CP model defined in Eqs. (12) and (13). Solving the modified-CP model generates the effective states $\tilde{T}_f$ and $\tilde{H}$.

$$ \epsilon \rho_f C_{p-f} \left( \frac{\partial \tilde{T}_f}{\partial t} + u \frac{\partial \tilde{T}_f}{\partial x} \right) = k_f \frac{\partial^2 \tilde{T}_f}{\partial x^2} - q + h_w (T_w - \tilde{T}_f) \quad (12) $$

$$ (1 - \epsilon) \rho_p \frac{\partial \tilde{H}}{\partial t} = k_p \frac{\partial^2 \tilde{T}_p}{\partial x^2} + q \quad (13) $$

In this study, a total of ten simulations (five charging and five discharging) are performed to generate snapshot data for the NN training. Each simulation has a time span of 18,000 s, during which the inputs are continuously varied. During charging, the inputs such as the HTF inlet temperature and the mass flow rate are varied randomly between 75°C and 95°C and 0.2 and 0.5 kg/s, respectively. Whereas, during discharging, the mass flow rate is varied randomly in same range used for charging, and the HTF inlet temperature is varied randomly between 30°C and 40°C. The variations of these inputs follow a similar pattern to Fig. 4 and Fig. 5.

$$ \epsilon \rho_f C_{p-f} \left( \frac{\partial \tilde{T}_f}{\partial t} + u (T_{f,out} - T_{f,in}) \right) = k_f \frac{\partial^2 \tilde{T}_f}{\partial x^2} - q_a + h_w (T_w - \tilde{T}_f) \quad (14) $$

$$ (1 - \epsilon) \rho_p \frac{\partial \tilde{H}}{\partial t} = k_p \frac{\partial^2 \tilde{T}_p}{\partial x^2} + q_a \quad (15) $$

To further simplify the model, and to make it computationally more efficient without affecting accuracy, aggregation of control volumes can be performed during the data generation process. In this study, the Neural-CP model is tailored to use only $N_r$ CVs, with $N_r < N$. The details of how the CVs are aggregated for the data generation process is illustrated in Fig. 11 and explained as follows. First, the CD model with $N$ CVs is solved given the primary inputs, the results are extracted and $q$ in each CV is calculated from these results. Then, the CVs are aggregated into $N_r$ by regrouping each $N/N_r$ CVs together. The aggregated heat rate $q_a$ transferred from the fluid to the capsules in the new $N_r$ CVs is calculated by summing $q$ of each $N/N_r$ CVs. This $q_a$ is then given to the CP model and the modified-CP model is solved using only $N_r$ CVs. To accommodate for this aggregation, set of equations are modified, as shown in Eqs. (14) and (15), where the $\frac{\partial T_f}{\partial x}$ term is replaced with $(T_{f,out} - T_{f,in})/dx$, where $T_{f,out}$, $T_{f,in}$, and $dx$ are respectively the input and output fluid temperature of each new CV and its length. This guarantees that the heat accumulation in the new $N_r$ CVs follow the same pattern as in the simulation using $N$ CVs. Forcing these aggregated $T_f$ and $q_a$ and solving Eqs. (14) and (15) yield the modified dynamics states. This aggregation technique helps in reducing the computational time of the Neural-CP model while maintaining its accuracy.
<figure><img src="image_12.png" /><figcaption>Fig. 9. Structure of the Neural-CP model.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. Adaptive physics-guided data generation process.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. Adaptive physics-guided data generation process with aggregation of CVs.</figcaption></figure>

## 3.2. Neural network model

Once the training data are generated, the NN is trained to produce the PCM temperature given the modified states $\tilde{T}_f$ and $\tilde{H}$. The NN used in this study is a simple feedforward Deep Neural Network (DNN) which are a class of artificial neural networks where multiple layers of neurons are connected are used for this study. In this study, the NN has two inputs and one output, and its architecture along with the selected hyperparameters is presented in Table 4. The use of small-scale NN trained on a reduced datasets aligns with the core objective of developing hybrid models that give accurate predictions while maintain computational efficiency and practicality for real-world engineering problems. In these models, the physics-based model captures the majority of the dynamics and the NN model compensates for the missing

<table><caption>**Table 4**
Architecture of the DNN.</caption><tr><td>Number of layers</td><td>4</td></tr><tr><td>Number of neurons per layer</td><td>10</td></tr><tr><td>Activation function</td><td>Leaky Rectified Linear Unit (Leaky-ReLU)</td></tr><tr><td>Optimizer</td><td>Adam</td></tr></table>

Here, the numerical ODE solver tracks of the main dynamics and gives the modified states ($\tilde{T}_f$ and $\tilde{H}$) and the NN only learns the relation between these modified states and $\tilde{T}_p$. The use of a simple feedforward DNN instead of a complex and computationally expensive Recurrent Neural Network (RNN) is possible due to the inputs of the NN which are modified states from the ODEs and when time integrating the ODEs, these modified states act as hidden states and provide the information about the past dynamics of the system to the NN. Simple DNN are easy to train and also offers lower inference time compared to RNN which helps to minimize the overall computational cost of solving the Neural-CP model. The NN is trained on a single core CPU with data collected from a total of 10 simulations with randomly varying inputs (five charging and five discharging), with a batch size of 32 and over a total of 500 training epochs. Grid search is used for hyperparameter tuning process, and a NN with four layers and ten neurons per layer is selected as the optimal architecture. After each layer, Leaky Rectified Linear Unit (Leaky-ReLU) is applied to introduce non-linearity. Adam optimizer with a learning rate scheduler which reduces the learning rate over the time is used to train the NN. Due to the small size of the NN and small amount of data used, the model converged in a total training time of approximately 10 min. The NN is trained only once and used in the
Neural-CP model without any additional training.

## 4. Results and discussion

The performance of the Neural-CP model is regressively validated under various scenarios to showcase its ability to produce accurate results and compatibility for real-time applications. Accuracy, computational performance and extrapolation capabilities are tested and discussed in detail in the following sections. Performance metrics like mean square error, $R^2$ score, and error percentage are used to compare the results of the models.

### 4.1. Model validation and discussion

Based on the aggregation of CVs discussed in Section 3.1, the Neural-CP model is built for different number of CVs and a sensitivity analysis is performed to select the number of aggregation of CVs that provide a good balance between the computational efficiency and accuracy of the model. To achieve this, a reference CD model with 1000 CVs is considered and error percentage between this reference model and the neural-CP model is estimated for different number of CVs. The results are given in Fig. 12 where the error percentage and the computational time is shown for different CVs. From this figure, it is observed that the computational time of the Neural-CP model decreases as the number of CVs are reduced from N to $N_r$. This is explained by the reduction of the number of ODEs to be solved from 2 N to $2N_r$. The computational time reduction comes without any significant loss in accuracy until $N_r = 200$ CVs. Below this threshold, the accuracy of the model decreases significantly. Thus, $N_r = 200$ CVs is selected to further validate the Neural-CP model.

To validate the results of the Neural-CP model, a total of six simulations are performed and the results are compared with the results of the other models. For the charging process, identical initial conditions are considered for all models and the simulation is performed until the temperature of the PCM reaches the HTF inlet temperature. Then, the final state of the storage is used as initial condition for the discharging process. During each simulation the inlet temperature and the mass flow rate are randomly generated and are made time varying, as previously shown in Figs. 4 and 5. For the first validation simulation, multiple plots which compare the different solutions of the Neural-CP, CP and CD models are shown. Fig. 13(a) and (b) compare the PCM temperature distribution along the height of the storage system during charging process at different times $t = 25, 75, 100$ and $200$ min for the three models. Fig. 14 shows the comparison of the temperature of the PCM for all models at the middle ($H = 1m$) and at the end ($H = 2m$) of the storage system during the charging and discharging processes. The outlet temperature of the fluid during the discharging process is

<figure><img src="image_15.png" /><figcaption>Fig. 12. Comparison of computational time and error percentage for different Neural-CP models with number of CVs.</figcaption></figure>

compared in Fig. 15.

From Fig. 13, Fig. 14, and Fig. 15, it is evident that the Neural-CP model accurately captures the dynamics of latent thermal storage and closely matches the results of the CD model. In comparison, the CP model shows significant deviations from the CD model, indicating poor performance. Although the Neural-CP model is built upon the CP model, the integrated NN effectively captures the critical physical behavior that the CP model alone fails to represent, specifically the thermal gradient within the PCM capsule, as captured by the CD model. By effectively learning the missing physics from the data generated using the CD model, the Neural-CP model improves the predictive accuracy of the CP model without explicitly increasing its complexity, owing to the neural network-accelerated framework. Furthermore, for all the validation simulations, the Neural-CP model (200 CVs) is compared against the CD model (1000 CVs) for different performance metrics such as the Mean Squared Error (MSE), $R^2$ score, and error percentage. The same performance metrics are also computed between the CP and CD models for references. The performance metrics comparison for different validation simulations are tabulated in Table 5. The MSE for the CP model and the CD model is between 39.32 °C and 43.07 °C with $R^2$ values below 0.464 and an error percentage of around 4%. This indicates that the CP model has limitations in predicting the performance of the PBTES, as also discussed in Section 2.5. In contrast, the Neural-CP model demonstrates a remarkable accuracy in matching the results of the CD model with MSE values as low as 0.059 °C, $R^2$ values closer to 0.997 and an error percentage below 0.235%. This showcases the ability of the Neural-CP model in capturing the dynamics of the system effectively.

As for computation performance, the average time taken to solve the different models is mentioned in Table 6. On average, The Neural-CP model is 342 times faster than the CD model and almost 5 times faster than the CP model. The good computational performance of the Neural-CP model is primarily due to the base CP model and the aggregation of the control volumes, leading to relatively small number of ODEs (400 ODEs) compared to CD model (2000 ODEs).

In this study the Neural-CP model is tested on a specific tank size and PCM; however, the novel strategy of combining the CP and CD model and building the Neural-CP model can be adapted for any PBTES regardless of the size of the tank, PCM type and thermophysical properties. This is because the strategy relies on creating a relationship between the thermal dynamics of the two modelling approaches, resulting in a modified-CP model that retains the structure of the CP model while achieving the accuracy of the CD model. Overall, the Neural-CP model accurately captures the complex heat transfer behavior between the PCM and the HTF while maintaining low computational demand. It effectively combines the strengths of detailed CP and CD models, resulting in a computationally fast and accurate model, providing a reliable and efficient solution for applications where both accuracy and performance are critical. Additionally, when compared to pure ML method, Neural-CP gains a huge edge by consistently adhering to the governing conservation laws at every instance, demonstrating strong physical consistency and reliability. This reinforces confidence in its predictive performance and establishes the Neural-CP model as a robust and physically grounded model. Furthermore, the Neural-CP model closely aligns with and replicates the behavior of the CD model, which is a widely accepted numerical approach. The CD model has also been experimentally validated in the literatures [30,31,48] and this comparison indirectly supports the validity of the Neural-CP model. This alignment allows the Neural-CP model to reflect the trusted performance and credibility associated with the CD model while offering improved efficiency.

### 4.2. Assessing extrapolation capability of the neural-CP model

Evaluating the extrapolation capability of the Neural-CP model is crucial, as real-world operating conditions often deviate from the
<figure><img src="image_16.png" /><figcaption>Fig. 13. Temperature distribution of the PCM during the charging process at different times during the first validation simulation.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Fig. 14. Comparison of PCM temperature at the middle and the end of the storage system during the first validation simulation.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 15. Outlet temperature of the HTF during the discharging process during the validation first simulation.</figcaption></figure>

<table><caption>**Table 5**

Details of the validation simulations.</caption><thead><tr><th rowspan="2">Simulation</th><th rowspan="2">ṁ<sub>f</sub><br>range<br>(kg/s)</th><th colspan="3">CP model vs CD model<br>(1000CV)</th><th colspan="3">Neural-CP model (200CV)<br>vs CD model (1000CV)</th></tr><tr><th>MSE (°C)</th><th>R<sup>2</sup></th><th>Error %</th><th>MSE (°C)</th><th>R<sup>2</sup></th><th>Error %</th></tr></thead><tbody><tr><td>1</td><td>0.2 to<br>0.5</td><td>41.09</td><td>0.464</td><td>4.015</td><td>0.059</td><td>0.998</td><td>0.181</td></tr><tr><td>2</td><td>0.2 to<br>0.5</td><td>42.86</td><td>0.399</td><td>4.233</td><td>0.092</td><td>0.996</td><td>0.235</td></tr><tr><td>3</td><td>0.2 to<br>0.5</td><td>43.07</td><td>0.476</td><td>3.703</td><td>0.093</td><td>0.998</td><td>0.195</td></tr><tr><td>4</td><td>0.2 to<br>0.5</td><td>42.43</td><td>0.463</td><td>4.119</td><td>0.086</td><td>0.998</td><td>0.172</td></tr><tr><td>5</td><td>0.2 to<br>0.5</td><td>39.32</td><td>0.424</td><td>3.944</td><td>0.079</td><td>0.997</td><td>0.219</td></tr><tr><td>6</td><td>0.2 to<br>0.5</td><td>39.56</td><td>0.317</td><td>3.521</td><td>0.092</td><td>0.997</td><td>0.209</td></tr></tbody></table>

<table><caption>**Table 6**

Comparison of average computational time.</caption><thead><tr><th colspan="3">Computational time (s)</th></tr><tr><th>CP model</th><th>CD model</th><th>Neural-CP model</th></tr></thead><tbody><tr><td>1.7</td><td>120</td><td>0.35</td></tr></tbody></table>

training range. Unlike pure machine learning models, which have very poor extrapolation, a robust physics and ML coupled architecture may retain reliability even when exposed to unseen scenarios. By accessing the extrapolation nature of the Neural-CP model, the generalization of the model can be showcased, to ensure the practical applicability in dynamic and unpredictable environments. To test how well the Neural-CP model performs outside the training range, 4 extrapolation simulations are performed and the details are presented in Table 7. In these simulations, the mass flow rates are varied beyond the training range, and the variation is shown in Fig. 17. The inlet temperature variation is shown in Fig. 16. For the first extrapolation simulations, the results such as temperature distributions at different stages of a process and outlet temperatures are shown in Figs. 18, 19 and 20.

Even when the variation range of the HTF mass flow rate is extended beyond the training data, the Neural-CP model exhibits strong generalization capability and maintains close agreement with the CD model. It continues to accurately capture the essential heat transfer dynamics between the HTF and the PCM capsule. Although the mass flow rate is not a direct input to the neural network within the Neural-CP model, changes in the mass flow rate influence the dynamics of the modified states ($\tilde{T}_f$ and $\tilde{H}$), and the interplay between these modified states and the neural network through the governing physical equations results in strong extrapolation capability without explicitly relying on extrapolative operations within the neural network itself. This extrapolation capability of the Neural-CP model shows that the NN effectively learns the relationship between its inputs and the output. Since the Neural-CP model is built on top of the CP model, it efficiently retains the physical dynamics of the system. This dynamics represents the actual physical

<table><caption>**Table 7**

Details of the validation simulations.</caption><thead><tr><th rowspan="2">Simulation</th><th rowspan="2">ṁ<sub>f</sub><br>range<br>(kg/s)</th><th colspan="3">CP model vs CD model<br>(1000CV)</th><th colspan="3">Neural-CP model (200CV)<br>vs CD model (1000CV)</th></tr><tr><th>MSE (°C)</th><th>R<sup>2</sup></th><th>Error %</th><th>MSE (°C)</th><th>R<sup>2</sup></th><th>Error %</th></tr></thead><tbody><tr><td>1</td><td>0.1 to<br>0.6</td><td>51.41</td><td>0.517</td><td>3.922</td><td>0.129</td><td>0.995</td><td>0.228</td></tr><tr><td>2</td><td>0.1 to<br>0.6</td><td>40.11</td><td>0.246</td><td>3.727</td><td>0.104</td><td>0.982</td><td>0.276</td></tr><tr><td>3</td><td>0.05 to<br>0.6</td><td>53.51</td><td>0.337</td><td>4.853</td><td>0.146</td><td>0.988</td><td>0.396</td></tr><tr><td>4</td><td>0.1 to<br>0.65</td><td>41.69</td><td>0.578</td><td>3.695</td><td>0.146</td><td>0.997</td><td>0.258</td></tr></tbody></table>
<figure><img src="image_19.png" /><figcaption>Fig. 16. Inlet temperature range for the test simulation.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 17. Mass flow rate range for the test simulation.</figcaption></figure>

correlation between enthalpy and temperature of the PCM at the surface of the capsule, capturing both the sensible and latent heat behaviors across the phase transitions and taking into account the thermal resistance inside the capsule. This correlation exists in both the CP and CD models; however, the CP model assumes the PCM to be a continuous medium, which reduces its accuracy. The dynamics that the CP model fails to capture are effectively learned by the neural network, enabling the Neural-CP model to achieve accurate predictions. This performance highlights the Neural-CP model’s ability to generalize to unseen operating scenarios, which is critical for control, optimization, and planning application. Overall, the Neural-CP model provides not only computational efficiency and accurate solutions but also robustness under extrapolative conditions.

### 4.3. Interpretability of the neural-CP model

To assess the interpretability of the ML part of the Neural-CP model, its predictions ($\tilde{T}_p$) are analyzed by varying input values ($\tilde{T}_f$ and $\tilde{H}$). This dependency is shown in Fig. 21 and the relationship between $T_p$ and $H$ used in the CP model is included in the same figure. It is noticed that the prediction of the ML model for fluid temperature values close to the PCM melting temperature is very similar to the theoretical relationship used in the CP model. It is also seen that, for fluid temperature values that are far from the PCM melting temperature, the variation of $\tilde{T}_p$ with respect to $\tilde{H}$ becomes closer to sensible heating process correcting the fact that the melting process in the PCM is not homogeneous inside the capsules. This prediction behavior shows that the ML model is following a pattern

that resembles the ideal behavior of a homogeneous PCM while adding a new behavior remedying the flaw of the assumption of continuous and homogeneous medium used in the CP model.

Post-training analysis revealed that the mapping from the inputs to the output exhibited a smooth and monotonic trend consistent with the underlying physical process as indicated in Fig. 21. Before the phase transition region, it is observed that the relationship is approximately linear. However, this relationship becomes highly nonlinear during the phase transition region. After the phase change region, the temperature again rises approximately linearly with enthalpy. This behavior resembles to an inverted logistic function. This information can be used to increase the interpretability and explainability of the NN block by introducing this behavior inside its architecture as an activation function. As a base for this custom activation function, given in Table 8, which is capable of representing a combination of a linear and logistic behavior similar to the pattern observed in the previous analysis is used. The coefficients of this custom activation function, $a_1, a_2, a_3$ and $k$, are learned during the training of the NN. Grid search is performed to simplify the architecture of the original NN and it leads to a smaller NN with a single hidden layer combined with the custom activation function. The hyperparameters of the new NN model are given in Table 8.

Since the size of the NN is reduced to just one hidden layer, the computational performance of the new Neural-CP is better than the previous model. As indicated in Table 9, the new Neural-CP model is 764 times faster than the CD model and 2.23 times faster than the original Neural-CP model, which has four layers of NN. The architectural simplification from four hidden layers with ten neurons per layer to one
<figure><img src="image_21.png" /><figcaption>**Fig. 18.** PCM temperature distribution during the charging process at different times for the test simulation.</figcaption></figure>

hidden layer and a custom activation function with learnable parameters significantly improves the computational efficiency while maintaining its performance.

Compared to the pure NN block, which operates as a black box with limited interpretability, the new model incorporating the custom activation function provides greater insight into the underlying relationship between the inputs and output. Four different simulations are performed to test the performance of the new Neural-CP model and the comparison with the CD model is presented in Table 10. It is tested with a mass flowrate within the training range in simulation 1 and outside the

training range in simulations 2, 3 and 4. Fig. 22 also shows the comparison of the temperature of the PCM at the middle ($H = 1m$) and at the end ($H = 2m$) of the storage system during the charging and discharging processes. From these comparisons, it can be seen that the new Neural-CP model performs well during each simulation. However, a very slight decrease in performance is noticed in the simulations, but compared to the computational gain, it is an acceptable trade-off.

## 5. Conclusion

A Neural-CP model for PBLHS is proposed that leverages the accuracy of the CD model and the computational efficiency of the CP model. It is a neural-accelerated model, where the NN is embedded into the energy balance equations of the CP model, and the NN dynamically provides the information about the complex PCM temperature gradient which it learnt from accurate CD model. This seamless integration of the NN into the CP model to build the Neural-CP model is achieved by developing a strategy that creates a relationship between the CP and CD model. This strategy incorporates the thermal interaction between the HTF and the PCM from the CD model into the CP model, which results in a modified-CP model. The modified-CP model replicates the behavior of the CD model while having a similar structure as the CP model and is used to generate data for training the NN. To further simplify the model, the number of axial discretization was reduced from 1000CVs to 200CVs by aggregating 5 CVs into one CV. The Neural-CP model is thoroughly

<figure><img src="image_22.png" /><figcaption>**Fig. 20.** Outlet HTF temperature during the discharging process for the test simulation.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Fig. 19.** Comparison of PCM temperature at the middle and the end of the storage system for the test simulation.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Fig. 21. Relationship between the inputs and the output of the ML model.</figcaption></figure>

<table><caption>**Table 8**

Hyperparameters of the NN with custom activation function.</caption><tr><td>Number of layers</td><td>1</td></tr><tr><td>Number of neurons per layer</td><td>10</td></tr><tr><td>Activation function</td><td>a<sub>1</sub> + a<sub>2</sub>x + a<sub>3</sub>(1/(1+e<sup>-kx</sup>))<sup>2</sup></td></tr><tr><td>Optimizer</td><td>Adam</td></tr></table>

<table><caption>**Table 9**

Comparison of average computational time.</caption><thead><tr><th></th><th>CP model</th><th>CD model</th><th>Neural-CP model</th><th>Neural-CP model (With custom activation)</th></tr></thead><tbody><tr><td>Computational time (s)</td><td>1.7</td><td>120</td><td>0.35</td><td>0.157</td></tr><tr><td>Acceleration</td><td>-</td><td>- (ref)</td><td>342x</td><td>764x</td></tr></tbody></table>

validated both within and beyond the training range to assess its accuracy and robustness. Within the training range, Neural-CP model achieves the accuracy of the CD model with 342 times faster computational speed and MSE as low as 0.059 °C, R² values closer to 0.99 and an error percentage below 0.172%. Outside the training range, the model also demonstrates strong extrapolation capabilities with good accuracy, offering good generalization and interpretability. This significant improvement in computational efficiency with good accuracy makes the

<table><caption>**Table 10**

Performance comparison of the new Neural-CP model with the CD model.</caption><thead><tr><th rowspan="2">Simulation</th><th rowspan="2">ṁ<sub>f</sub> range (kg/s)</th><th colspan="3">Neural-CP model (200CV) vs CD model (1000CV)</th></tr><tr><th>MSE (°C)</th><th>R<sup>2</sup></th><th>Error %</th></tr></thead><tbody><tr><td>1</td><td>0.2 to 0.5</td><td>0.117</td><td>0.997</td><td>0.243</td></tr><tr><td>2</td><td>0.1 to 0.6</td><td>0.210</td><td>0.996</td><td>0.289</td></tr><tr><td>3</td><td>0.05 to 0.6</td><td>0.224</td><td>0.993</td><td>0.394</td></tr><tr><td>4</td><td>0.1 to 0.65</td><td>0.166</td><td>0.994</td><td>0.307</td></tr></tbody></table>

model well-suited for applications that require fast computations with accuracy. However, when the mass flowrate deviates significantly from the range used during training, its effect relative to the temperatures of the HTF and the PCM changes hugely. Under these extreme mass flowrate variation scenarios, the performance of the Neural-CP model might degrade. Finally, by analyzing the relationship between the inputs and the output of the NN, the relationship is approximated by a function. This function is used as an activation function and the architecture of the NN is reduced from four hidden layers with ten neurons per layer to one hidden layer and this custom activation function with learnable parameters. This architectural simplification helps to further accelerate the model 2.23 times (764 times compared to CD model) while making the predictions of the NN more interpretable through the custom activation function. This new model offers high fidelity predictions of the dynamics of the PBLHS with very limited computational overhead paving the way to the implementation of optimal real-time control strategies or long-term simulation for planification purposes.

### CRediT authorship contribution statement

**Dessie Tadele Embiale:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Shri Balaji Padmanabhan:** Writing – review & editing, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Mohamed Tahar Mabrouk:** Writing – review & editing, Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Stéphane Grieu:** Writing – review & editing, Supervision.
**Bruno Lacarrière:** Writing – review & editing, Validation, Supervision, Project administration, Methodology, Funding acquisition, Formal analysis.

<figure><img src="image_25.png" /><figcaption>Fig. 22. Comparison of PCM temperature at the middle and the end of the storage.</figcaption></figure>
Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work has benefited from state aid managed by the Agence Nationale de la Recherche under the France 2030 programme, reference 22-PETA-0002.

Data availability

Data will be made available on request.

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