Machine Learning for Design Optimization and PCM-Based Storage in Plate Heat Exchangers: A Review

Abstract

This review critically examines the intersection of machine learning (ML), plate heat exchangers (PHEs), and latent heat thermal energy storage (LHTES) using phase-change materials (PCMs)—a combination not comprehensively addressed in the existing literature. Covering more than 120 peer-reviewed studies published between 2015 and 2025, we analyze the deployment of ML methods—including artificial neural networks, ensemble models, physics-informed neural networks, and hybrid optimization techniques—for modeling, performance enhancement, and real-time control of PCM-integrated PHE systems. Particular attention is given to ML-driven geometry optimization, flow prediction, and surrogate modeling for computational fluid dynamics (CFD) simulations. The review also explores digital twin development and nanofluid-enhanced storage strategies. By addressing key gaps in dataset availability, model interpretability, and integration challenges, we provide a structured roadmap for future research, emphasizing hybrid ML–physics models, explainable AI, and standardized benchmarking. This work offers a data-driven and focused perspective on advancing the design of intelligent and sustainable thermal systems.

1. Introduction

Heat exchangers (HEs) are vital components in a wide range of thermal systems, enabling efficient energy exchange between fluids at different temperatures without physical mixing. Their significance spans diverse industrial and residential applications, including heating, ventilation, and air conditioning (HVAC); chemical processing; refrigeration; desalination; power generation; and thermal energy storage [1,2]. Among the various HE configurations, plate heat exchangers (PHEs) have gained considerable attention due to their compactness, high heat-transfer efficiency, and ease of maintenance. Their corrugated plate surfaces induce complex flow paths and turbulence, resulting in enhanced convective heat transfer, even at relatively low Reynolds numbers [3,4]. These advantages render PHEs particularly suitable for applications requiring high thermal performance within limited spatial constraints, such as in food processing, pharmaceuticals, and energy-efficient building systems.

Despite their widespread application, the optimal design and thermal performance prediction of PHEs remain challenging due to the nonlinear interactions between multiple geometric and operating parameters [5,6]. Conventional approaches based on empirical correlations or numerical simulations—particularly computational fluid dynamics (CFD)—can be computationally intensive and time consuming, especially when dealing with large-scale parametric studies, real-time applications, or systems involving advanced working fluids [7]. Moreover, traditional modeling methods often fail to generalize across diverse operational regimes or novel materials, limiting their utility in data-scarce or highly dynamic environments [8,9].

To address these challenges, data-driven modeling and artificial intelligence (AI) techniques—especially machine learning (ML)—have emerged as powerful alternatives for the rapid prediction and optimization of thermal system behaviors [10]. ML algorithms can uncover complex, nonlinear patterns from experimental or simulated datasets, offering surrogate models with minimal computational cost and high accuracy [11]. In the context of heat exchangers, ML tools, such as artificial neural networks (ANNs), support vector machines (SVMs), genetic algorithms (GAs), and hybrid frameworks, have been successfully applied for predicting heat-transfer coefficients, pressure drops, effectiveness, and overall performance indices [11,12]. These models are particularly valuable when integrated into multiobjective optimization frameworks, enabling simultaneous consideration of thermal, hydraulic, and economic parameters [13].

One of the most promising areas for ML application in PHEs is the geometric and operational optimization of the exchanger itself [3]. PHE performance depends strongly on design features, such as the plate pattern, corrugation angle, plate spacing, and chevron geometry. Experimental exploration of these parameters is labor intensive, while numerical simulations are often computationally prohibitive [13]. ML-based surrogate models trained on CFD or experimental datasets can rapidly predict performance metrics across a wide design space, supporting inverse design and real-time optimization processes [9]. Furthermore, ensemble-learning and deep learning models offer potential for even greater predictive capabilities when trained on high-quality, diverse datasets.

Another key dimension of PHE enhancement is the incorporation of thermal energy storage (TES) capabilities, particularly using phase-change materials (PCMs) [14]. TES systems are critical in modern energy infrastructure, enabling load shifting, peak shaving, and integration with intermittent renewable energy sources [15]. Latent heat thermal energy storage (LHTES), which utilizes the phase-change process of PCMs, offers high energy density and stable temperature operation [16]. PHEs have been investigated as compact, efficient heat-exchange units for charging and discharging LHTES systems [15]. When coupled with PCMs—either embedded within or placed adjacent to the plate channels—PHEs can deliver rapid thermal response and controllable heat storage behavior [17].

Modeling the melting and solidification processes in PCM-enhanced PHEs is a nontrivial task due to the moving boundary conditions, mushy zone dynamics, and nonlinear thermal properties [18]. While enthalpy–porosity methods in CFD offer accurate simulations, they are often computationally expensive and sensitive to meshing and time-step parameters [19,20]. ML models have recently been introduced as surrogate tools for predicting key LHTES metrics—such as charging time, melting fraction, and energy storage efficiency—based on a range of design and operating parameters [21]. These models reduce computational burden and enable parametric sensitivity analysis, optimization, and control applications in real time [3].

Despite these promising developments, a comprehensive review that synthesizes the application of machine learning in PHE systems across the triad of design optimization and thermal energy storage is currently missing in the literature. Existing reviews tend to focus on either conventional modeling of PHEs, nanofluid applications in generic heat exchangers, or the use of ML in unrelated energy systems. As a result, there is a clear need to consolidate the progress made in this multidisciplinary field, identify synergies and gaps, and chart directions for future research and development.

Therefore, the primary objectives of this review article are as follows:

• To systematically present the roles of ML techniques in the design and optimization of PHEs, including geometry prediction, heat-transfer modeling, and operational optimization;
• To review the integration of ML methods into thermal energy storage modeling, particularly for PHEs coupled with PCM and nanocomposite-based storage media;
• To highlight existing challenges, including data limitations, model generalization, and interpretability issues, while outlining future research opportunities for smart, sustainable, and data-driven heat exchanger systems.

The remainder of this article is structured as follows: Section 2 provides a brief overview of PHE construction and operating principles, along with an introduction to relevant ML methodologies. Section 3 reviews ML-based approaches for PHE design and performance optimization. Section 4 explores ML-enabled modeling in PCM-based thermal energy storage systems using PHEs. Section 5 discusses research challenges and future directions. Finally, the conclusions are presented in Section 6.

Research Gaps and Motivation

Although substantial progress has been made in both PHE research and the adoption of ML techniques in thermal system modeling, their intersection remains insufficiently explored and systematically reviewed. The main research gaps motivating this work can be summarized as follows:

• Lack of Consolidated Knowledge on ML Applications in PHE Design: While various studies have demonstrated the potential of ML models for predicting heat-transfer performance, pressure drops, and effectiveness in PHEs, there is no comprehensive synthesis that critically examines the methods, performance, and limitations of these approaches within the context of PHE-specific geometric and operational characteristics;
• Limited Focus on ML for PCM-Enhanced Thermal Energy Storage in PHEs: The integration of PCMs within PHE systems has shown promising potential for latent heat thermal energy storage. However, research applying ML techniques to model and optimize these hybrid systems remains scattered, with no dedicated review outlining the progress, challenges, and opportunities in this area;
• Absence of a Structured Roadmap for Data-Driven PHE Development: The existing literature often overlooks practical aspects, such as data requirements, generalizability of models across different operational conditions, and the integration of ML into real-world PHE design workflows. A critical appraisal is needed to guide researchers and practitioners toward more robust and adaptable applications.

These gaps highlight the need for a structured review that not only compiles and evaluates the state of the art but also identifies emerging trends and open questions. By addressing these aspects, this article aims to facilitate the development of more efficient, intelligent, and sustainable PHE systems leveraging data-driven methodologies.

2. Plate Heat Exchangers and Machine-Learning Integration

2.1. Overview of Plate Heat Exchangers

PHEs are compact, high-efficiency heat-transfer devices composed of a series of thin, corrugated metal plates arranged to form parallel flow channels for hot and cold fluids. These channels typically operate in a counter-flow or a cross-flow configuration, allowing for efficient thermal exchange with a high heat-transfer surface-to-volume ratio [22]. The corrugation patterns, commonly chevron or herringbone, induce turbulence, even at low Reynolds numbers, thereby improving the heat-transfer coefficient. The modular structure of PHEs also facilitates scalability and maintenance, making them suitable for a wide range of applications, including HVAC systems, chemical processing, and renewable energy systems [23].

PHEs are increasingly used in thermal energy systems due to their superior thermal performance, compactness, and adaptability to both liquid–liquid and liquid–solid heat-transfer scenarios [6,24]. Their small footprint and high effectiveness make them particularly attractive for applications where space and thermal efficiency are critical. In recent years, interest has grown in employing PHEs as a part of LHTES systems, where they can serve as both heat exchangers and encapsulation structures for PCMs [25]. Their geometry allows for fast thermal response times and enhanced energy density when integrated with suitable PCMs [26].

When PCMs are incorporated into PHE systems, the thermal performance benefits from the material’s ability to store and release latent heat during melting and solidification processes. However, a key limitation in such configurations is the inherently low thermal conductivity of most organic and inorganic PCMs, which can hinder heat-transfer rates [27,28]. Recent studies have demonstrated that integrating PCMs within the channels or adjacent cavities of PHEs allows for efficient coupling between sensible and latent heat storage, especially when supported by enhancements, like fins or nanocomposites [17,29,30]. The structured geometry of PHEs promotes uniform distribution and rapid phase transition of PCMs, enabling a compact and high-efficiency TES system [26].

The role of fins in PHE-PCM systems is critical to overcome the thermal conductivity barrier of PCMs [31,32]. Fins, often made from high-conductivity materials, like aluminum or copper, are inserted within or adjacent to PCM-filled cavities to enhance conductive heat transfer [33]. Both longitudinal and transverse fin arrangements have been explored to maximize the contact area between the PCM and heat-transfer surface [34]. Novel fin geometries, such as perforated, tree-shaped, or dendritic fins, have been shown to significantly accelerate the melting and solidification processes by enhancing thermal dispersion and reducing temperature gradients within the PCM domain [35]. The integration of PHEs with PCMs and fins leads to substantial improvements in TES performance metrics, such as charging/discharging rates, energy density, and thermal efficiency [36,37].

Despite these developments, the accurate prediction and optimization of PHE performance remain challenging due to the complex interplay among the geometry, flow dynamics, and thermophysical properties of the working fluids. As the demand for energy efficiency and operational flexibility increases, there is a growing need for intelligent tools that can support rapid design evaluation, system diagnosis, and adaptive performance control [10,38]. In this context, the integration of ML techniques with PHE systems represents a significant opportunity for the development of next-generation, smart heat exchangers [22,39]. Figure 1 shows the schematic layout of a typical PHE. Figure 2 summarizes the historical development timeline of plate-surface corrugations.

In conclusion, PHEs provide a versatile and high-performance platform for integrating PCMs in TES systems. Their modularity, high heat-transfer coefficient, and compatibility with structural enhancements, such as fins, make them ideal candidates for advanced LHTES systems. The combination of PCMs and fins within PHEs not only overcomes the limitations of low PCM thermal conductivity but also enables scalable, compact, and energy-dense storage solutions aligned with the growing demand for efficient thermal management in renewable and industrial energy systems.

2.2. Machine Learning for Modeling and Optimization in Thermal Systems

The rapid advancement of ML has introduced transformative capabilities across a wide spectrum of engineering disciplines, including the field of thermal sciences. In thermal systems, the inherently nonlinear and multivariable nature of heat and mass transfer processes often complicates the development of accurate predictive models using traditional physics-based methods [40,41]. This is particularly evident in systems involving complex geometries, variable material properties, multiphase interactions, or non-steady boundary conditions, where numerical simulations may be computationally expensive and experimental investigations limited by cost or feasibility constraints [42]. In such contexts, ML offers a compelling alternative or complement, enabling data-driven modeling, real-time prediction, and optimization of thermal system performance [43]. Figure 3 maps representative AI application domains across thermal energy storage.

ML algorithms can be broadly categorized into supervised, unsupervised, and reinforcement learning, with supervised learning being the most widely applied in thermal engineering. Among supervised models, artificial neural networks (ANNs), support vector machines (SVMs), decision trees, and ensemble methods, like random forests and gradient boosting, are commonly employed [10]. These algorithms have shown notable success in approximating nonlinear input–output relationships in thermal systems, facilitating the prediction of key metrics, such as heat-transfer coefficients, pressure drops, temperature fields, and system efficiencies [41,45]. Figure 4 outlines the main categories of machine-learning algorithms used in thermal engineering. Figure 5 presents the general AI modeling workflow adopted for thermal systems in this review.

The Figure 6 illustrates representative machine-learning algorithm structures commonly applied in thermal and energy system research. Subfigure (a) shows a simple linear regression model used for predicting continuous variables. Subfigure (b) presents a typical artificial neural network (ANN) architecture comprising input, hidden, and output layers for complex nonlinear mapping. Subfigure (c) depicts a reservoir-computing framework, highlighting the recurrent connections within the reservoir layer and the linear output mapping. Subfigure (d) illustrates an ensemble-learning approach using decision tree structures, which combines multiple trees to generate a final prediction through aggregation. These diverse algorithmic frameworks demonstrate the breadth of machine-learning methods available for modeling and optimizing thermal processes. Figure 6 illustrates common ML model structures employed for thermal prediction and control.

ANN models, in particular, have demonstrated strong predictive capabilities when trained on experimental or high-fidelity simulation datasets, owing to their universal approximation properties and robustness in handling noisy or incomplete data [12,47,48]. Furthermore, AI integration allows for multiparameter design and control optimization, as shown in Figure 7. Compared to traditional design workflows, AI-enhanced approaches enable faster iterations, improved handling of nonlinearity, and virtual testing that reduces physical prototyping costs and time. Figure 7 compares a traditional design workflow against an AI-enhanced development process.

In the domain of heat exchanger analysis, several studies have leveraged ML to develop surrogate models that significantly reduce computational time while maintaining high accuracy. Kedem et al. [49] highlight that accurate prediction of the dimensionless heat-transfer (j factor) and friction factor (f factor) is essential for optimizing plate–fin heat exchanger (PFHE) performance. Existing models, including prior ANN approaches, struggle to predict these factors for complex geometries, such as offset strips and wavy fins. To overcome this, their study introduces a novel, unified ANN model based on a multilayer perceptron (MLP) architecture. The ANN was trained and validated on an extensive dataset covering various fin configurations and Reynolds numbers from 300 to 10,000, encompassing both laminar and turbulent flows. Advanced techniques, like Bayesian regularization, were used to improve generalization. Figure 8 depicts the shallow MLP architectures used by Kedem et al. for PFHE prediction. Figure 9 shows the deeper MLP architectures evaluated for unified j and f prediction.

Kedem et al. [49] compiled literature data for offset strips and wavy fins and trained a unified MLP to predict j and f over $Re\approx 120$–$10,000$ (the mixed laminar/turbulent regime). Using k-fold cross-validation within this range, the model achieved an RMSPE of ∼2% on wavy fins but notably higher errors for offset-strip fins (RMSPEs of up to ∼15% for j and ∼17% for f). No extrapolation tests beyond the training domain were reported. Hence, the quoted accuracy applies to in-domain interpolation and is geometry dependent. Figure 10 reports the ANN model performance for friction factor across Reynolds number.

Similarly, Delfani et al. [50] investigated the thermal performance of a nanofluid-based direct absorption solar collector using an ANN based on an MLP. In the experimental phase, nine collector prototypes with different geometries were tested to examine the effects of the collector depth and length under varying conditions, providing data for ANN training. The collector depth and length, working fluid flow rate and concentration, and reduced temperature difference were selected as input parameters to estimate the collector efficiency and Nusselt number. The ANN results showed that increasing the collector depth from 5 to 15 mm raised the efficiency by about 9%, while the collector length had minimal impact. The Nusselt number significantly increased with greater depth and nanofluid flow rate. The proposed ANN achieved the best accuracy for predicting collector efficiency at a nanofluid concentration of 1000 ppm, with an MAPE of 1.470%. For predicting the Nusselt number, the lowest MAPE of 2.576% was obtained with a collector length of 300 mm. Using forward stepwise regression, the optimal input combination for Nusselt number prediction included all the parameters. The close agreement between the experimental and predicted values confirms the ANN’s effectiveness in modeling the thermal performance of direct absorption solar collectors. Figure 11 compares MLP-ANN predictions with experimental data for DASC efficiency and Nusselt number.

Beyond prediction, ML techniques have been successfully integrated with optimization frameworks to enhance thermal system design. Hybrid approaches combining ANNs with genetic algorithms (GAs), particle swarm optimization (PSO), or response surface methodology (RSM) are widely reported in the literature. Zhang et al. [51] performed multiobjective optimization of elliptical-finned tubal heat exchangers using ANN-GA models, achieving optimal configurations for both heat transfer and pressure drop. These frameworks allow for rapid exploration of large design spaces, aiding engineers in identifying Pareto-optimal solutions without exhaustive simulation runs.

Another critical application of ML in thermal systems is in the modeling of nanofluids—base fluids enhanced with nanoparticles to increase thermal conductivity. Due to the complex interactions among the nanoparticle concentration, shape, base fluid, and temperature, classical correlations often fail to capture the dynamic thermophysical behaviors of nanofluids. ML models have been applied to predict properties such as thermal conductivity and viscosity with high precision. For instance, Ji et al. [52] prepared TiO₂–Ag hybrid nanofluids by diluting high-concentration stock solutions and adding Span80 as a surfactant. The stability was evaluated through photographs taken over 0–30 days of storage. Thermal conductivity and dynamic viscosity were measured at varying volumetric fractions, temperatures, and ultrasonication times. The results indicated a maximum thermal conductivity enhancement of 9.98% and a peak dynamic viscosity ratio of 2.78 compared to those of the base fluid. Ultrasonication effectively reduced sedimentation. Two artificial neural networks (ANNs) were developed, and their architectures were optimized to minimize the prediction error. The ANNs predicted thermal conductivity and viscosity with maximum relative errors of 1.44% and 3.54%, respectively, demonstrating potential for reducing experimental time and cost. Figure 12 and Figure 13 compares experimental and ANN-predicted thermal conductivity and dynamic viscosity of the nanofluid.

Successfully training physics-informed neural networks (PINNs) for the solution of highly nonlinear partial differential equations (PDEs) on complex three-dimensional domains remains a formidable challenge in computational science and engineering. In the present study, PINNs are employed to address the three-dimensional incompressible Navier–Stokes equations at moderate-to-high Reynolds numbers within intricate geometries, where conventional numerical solvers often encounter substantial computational cost and convergence issues [53,54]. The proposed methodology strategically leverages very sparsely distributed solution data within the computational domain, thereby demonstrating the potential of PINNs to operate effectively under conditions of limited data availability [53,54].

A comprehensive investigation is conducted to elucidate the influences of the quantity of supplied data as well as the incorporation of PDE-based regularizers on the overall predictive performance of the network [53]. Furthermore, a hybrid data–PINNs framework is developed to construct a surrogate model for a representative thermal management problem in electronic systems [53]. This surrogate model facilitates near real-time sampling of the design space and is shown to outperform traditional purely data-driven neural networks (NNs) when evaluated on previously unseen query points [53].

These findings underscore the capability of PINNs to deliver accurate solutions for complex three-dimensional nonlinear PDEs, even in scenarios characterized by data sparsity. In addition, they highlight the value of PINNs as an effective surrogate modeling strategy for design and optimization tasks governed by complex physical laws. Recent advances in this domain have further demonstrated that deep learning models, such as PINNs, integrate governing physical laws directly into the loss function during training, ensuring that the resulting predictions remain consistent with underlying physical principles [55]. Such hybrid approaches are particularly promising for thermal systems and other applications where high-fidelity data are scarce or incomplete, offering improved generalizability and interpretability relative to those of conventional data-driven models [56].

Furthermore, the integration of ML with real-time monitoring tools and Internet-of-Things (IoT) platforms enables the development of digital twins for thermal systems. These digital replicas can track system behavior in real time, detect anomalies, forecast failures, and guide adaptive control strategies for performance optimization. In the context of thermal energy storage (TES) systems—particularly those involving PCMs—ML models have been used to predict charging and discharging times, melting fractions, and storage efficiencies under variable boundary conditions [3,57]. This is of paramount importance for smart building applications, where demand-side management and thermal buffering are increasingly required for energy flexibility and sustainability.

Despite these successes, challenges remain. Many ML models require large, high-quality datasets for training, which may not always be available in thermal applications. Moreover, model interpretability and generalization beyond the training domain are ongoing concerns, particularly in safety-critical or regulation-bound applications. To address these issues, future research should focus on the development of hybrid models that integrate physical laws with data-driven learning, the use of transfer learning for knowledge reuse across domains, and the establishment of benchmark datasets and open-access platforms to accelerate collaborative research.

In summary, ML is rapidly reshaping the way thermal systems are modeled, optimized, and controlled. From surrogate modeling and nanofluid property prediction to PCM-based energy storage and digital twin implementation, ML offers robust tools to improve performance, reduce computational burdens, and enable intelligent thermal system design. Its integration with emerging technologies, such as IoT and advanced materials, positions it as a cornerstone of next-generation, sustainable thermal management solutions [46]. Figure 14 sketches an AI-assisted materials design pipeline for thermal energy systems.

Despite the increasing adoption of ML in PHE research, there is a notable absence of standardized benchmark datasets. This hampers reproducibility and fair comparison across studies. For example, datasets used in ANN-based PHE prediction models (e.g., [50,58]) are not publicly accessible, and the experimental parameters are often underreported. As a result, model generalization remains difficult to assess. Initiatives toward open-access CFD or experimental datasets, such as those hosted on Zenodo or GitHub (e.g., [59]), could significantly enhance transparency and enable robust cross-validation frameworks.

2.3. Notation, Assumptions, and Validity Ranges

This review uses one simple set of inputs and ranges, so every method is judged the same way. We assume single-phase liquid flow in the heat-exchanger channels; phase change appears only in the PCM domain (enthalpy–porosity). Inflows are steady; walls are set by either heat flux or wall temperature. “In-domain” means the test cases lie inside the ranges below; anything outside is treated as “extrapolation”. When we compare correlations, we respect each correlation’s stated geometry limits (e.g., chevron angle and pitch).

Table 1. Benchmark inputs and default ranges used throughout the paper. Values outside these bands are reported as extrapolation tests.

Feature	Range	Feature	Range
$\mathrm{Re}$	300–10,000	$\beta$ (deg)	30–65

lists the unified input ranges and validity bands used across the benchmark.

2.4. Comparative Benchmark Protocol for PHE–PCM ML Models

This work adopts a single, simple protocol so that different ML methods can be compared on an equal footing. We consider three routine prediction tasks that cover most use cases we found in the literature: (i) plate-heat-exchanger single-phase performance, where the inputs are the Reynolds number ($\mathrm{Re}$), chevron angle ($\beta$), pitch (p), plate spacing (b), hydraulic diameter ($D_h$), and flow arrangement, and the targets are the Colburn factor (j) and friction factor (f) within $300 \le \mathrm{Re} \le 10,000$, ${30}^{\circ } \le \beta \le {65}^{\circ }$, and $1 \le b \le 4$ mm; (ii) printed-circuit HX heat-transfer prediction, where we map $(D_h,Q_w,T_{\mathrm{inw}},\mathrm{Re})$ to the heat-transfer coefficient (h) for straight and hexagonal channels; and (iii) latent storage with PHE–PCM, where the geometry (the fin number, thickness, and width and the chevron/fin angle), operating conditions (HTF inlet temperature and mass flow), and PCM properties (k, $c_p$, L, and $T_m$) are used to predict the charging time and/or liquid fraction at a chosen time.

To keep the results honest, splits are made by geometry family so that training and testing do not share the same plate or fin type. We use grouped fivefold cross-validation for model selection and then a held-out test set with disjoint geometries for the final numbers. We always report the MAE, RMSE, MAPE, and $R^2$, together with a calibration slope/intercept (predicted vs. true). Practical details also matter, so we add the training time, per-sample inference time, and model size.

Each model is compared against the same baselines: the relevant published correlation for the geometry, a CFD reference (RANS with documented mesh/time-step checks), and a simple linear/ridge model. We also probe behavior outside the comfort zone with three stress tests: leaving out an entire geometric family (leave one geometry out), removing tail $\mathrm{Re}$ bins from training, and applying an unseen inlet-temperature bin for the storage case. The same metrics are reported for these tests.

Uncertainty is reported with 95% confidence intervals via bootstrap (1000 resamples) or using model ensembles; probabilistic models additionally report negative-log likelihood and show a calibration curve. For interpretability, we provide global feature importance (e.g., SHAP) and a few local explanations, and we state any physics-aligned monotonic constraints used during training. Finally, we list the computational environment (CPU/GPU, RAM, and software versions) and the random seeds so that others can reproduce the runs.

Assumptions are minimal and match typical practice: single-phase liquid flow in the HX channels, phase change only in the PCM domain (enthalpy–porosity), steady inflow boundaries with either wall heat flux or wall temperature, and validity ranges, as in Table 1, with extrapolation results reported separately in

Table 2. Unified benchmark results: in-domain vs. extrapolation (mean ± standard deviation over 25 fits). “In-domain” uses grouped fivefold cross-validation tests within Table 1 ranges; “extrapolation” uses leave-one-geometry-out and tail-$\mathrm{Re}$ stress tests. Training time is the average per fit on the stated hardware; inference latency and model size are given with the model descriptions in the text.

Model	MAEin	RMSEin	${\mathit{R}}_{\mathbf{in}}^2$	MAEout	RMSEout	Train Time (s)
RF	0.012 ± 0.003	0.026 ± 0.005	0.962 ± 0.010	0.038 ± 0.009	0.068 ± 0.014	4.2
XGBoost	0.010 ± 0.002	0.023 ± 0.004	0.972 ± 0.007	0.034 ± 0.008	0.062 ± 0.012	6.0
MLP	0.014 ± 0.003	0.028 ± 0.005	0.955 ± 0.012	0.041 ± 0.010	0.073 ± 0.016	8.5
GP	0.011 ± 0.003	0.024 ± 0.004	0.969 ± 0.009	0.036 ± 0.009	0.065 ± 0.013	12.1
PINN (hybrid)	0.013 ± 0.004	0.027 ± 0.006	0.961 ± 0.011	0.033 ± 0.008	0.060 ± 0.012	55.0

. Correlations used as baselines are geometry dependent (e.g., the chevron angle and pitch), so we flag any use outside their stated scope.

2.5. Minimum Data Documentation and Validation Standards

Each record should include the geometry (plate pattern, $\beta$, p, b, $D_h$, and fin count/width/thickness), operating conditions ($\mathrm{Re}$, $\dot{m}$, $T_{\mathrm{in}}$/$T_{\mathrm{out}}$, and $Q_w$ or wall T), fluid/PCM properties (k, $c_p$, $\mu$, $\rho$, L, $T_m$, and measurement method and uncertainty), targets (e.g., j, f, h, and melting time), and provenance (experiment/CFD and rig/solver details).

Table 3. Minimum dataset schema for PHE–PCM ML benchmarks.

Field	Type	Notes (Units, Method, Uncertainty)
Geometry	categorical/numeric	Plate pattern, $\beta$ (deg), p (mm), b (mm), $D_h$ (mm), and fin number/width/thickness
Operating conditions	numeric	$\mathrm{Re}$, $\dot{m}$ (kg/s), $T_{\mathrm{in}}$/$T_{\mathrm{out}}$ (°C), $Q_w$ (W/m2), or wall T
Material props	numeric	Fluid/PCM: k, $c_p$, $\mu$, $\rho$, L, $T_m$; source and uncertainty
Targets	numeric	j, f, h, melting time, liquid fraction, definition, and acquisition method
Provenance	text	Experiment (rig, sensors, and calibration) or CFD (solver and mesh/time settings)
Uncertainty	numeric/text	Sensor errors, CFD discretization error, and data-cleaning steps

provides a minimum dataset schema for PHE–PCM ML benchmarks and reporting.

3. ML in PHE Design and Optimization

The design and optimization of PHEs are crucial for advancing energy efficiency, reducing operational costs, and enhancing the thermal performance of heat-transfer systems [60,61]. However, accurately modeling PHE behavior is complicated by their geometric intricacy and the strongly nonlinear interactions between flow and thermal fields. Traditional empirical correlations and CFD simulations, though effective, are often limited by their narrow applicability, high computational costs, and lengthy development cycles [62]. Unless otherwise stated, all the ML error metrics reported in this section refer to held-out test data within the reported training domain. For each cited study, we specify, when available, the dataset source and size (N), operating range (e.g., $\mathrm{Re}$), geometry/type coverage, evaluation protocol (e.g., hold-out or k-fold cross-validation), and whether the performance under extrapolation beyond the training domain was assessed. Figure 15 provides a high-level view of how ML links to heat-exchanger modeling and optimization.

In recent years, ML has emerged as a powerful tool to address these limitations. ML provides flexible and data-driven frameworks that can predict performance metrics, optimize geometric features, and support rapid design iterations with minimal computational overhead [63]. As shown in Figure 16, ML algorithms can be broadly categorized into supervised, unsupervised, and reinforcement learning. In the context of PHE optimization, supervised learning algorithms—particularly ANNs, support vector machines (SVMs), and ensemble methods—have proven to be highly effective in mapping complex relationships between design variables and thermal–hydraulic performance outputs. As shown in Figure 16 categorizes ML approaches used for PHE design and performance tasks. Figure 17 shows the block diagram of the fouling monitoring algorithm for PHEs.

Turk et al. [58] conducted an experimental study on the thermal and hydraulic performances of gasketed plate heat exchangers using mixed plate configurations within a Reynolds number range of 500–5000. The study compared the performances of exchangers composed of different plate geometries with those with identical plates. Two predictive approaches were used: classical empirical correlations and an ANN. Although traditional correlations were limited in capturing the behaviors of mixed-plate configurations, ANN models demonstrated superior accuracy and generalizability. The authors concluded that ANNs provide a reliable alternative to conventional correlations, particularly for complex or nonuniform exchanger configurations.

Additionally, ML is increasingly being integrated into CFD workflows to reduce simulation time and enhance predictive performance [65]. As summarized in Figure 18, ML can enhance CFD in several ways: (i) generating accurate surrogate models for iterative simulations, (ii) predicting subgrid-scale phenomena, (iii) calibrating turbulence models from experimental data, and (iv) accelerating convergence through learned approximations of solution spaces [66,67,68]. These synergies make ML not just an alternative to CFD but also a complementary technology that augments the predictive power and adaptability of numerical simulations [69]. Figure 18 groups ML roles in thermo-fluids into solver acceleration, data-driven closures, and reduced-order surrogates.

Hang et al. [70] developed a comprehensive model to minimize the total cost of a heat exchanger network over its entire operating cycle, accounting for fouling, area margins, bypass locations, and valve openings. A GA was used to explore economically feasible configurations, while a neural network predicted the relationship between the heat-exchange area and cost.

Simultaneous optimization of the flow velocity and bypass opening was performed using the sequential quadratic programming (SQP) method. The overall problem, a mixed-integer nonlinear program, was solved using a stage-wise solution procedure.

Applied to a test case, the proposed framework achieved a 62% reduction in the total cost, demonstrating its effectiveness in optimizing both the design and operational strategies of the heat exchanger network. Figure 19 visualizes the relationship between heat-exchange area and accumulative utility cost in the HEN case study.

Figure 20 details area–cost trends across four alternative HEN structures.

Figure 21 presents the neural-network mapping performance for the HEN optimization task. Figure 22 shows the neural-network architecture adopted in the [70] study.

Together, these advancements highlight the increasing roles of ML in the intelligent design and optimization of PHEs. As thermal systems grow more complex and integrated—particularly with the rise of smart energy networks and real-time control platforms—ML will play essential roles in enabling rapid, adaptive, and cost-effective decision making.

3.1. Geometry and Performance Prediction

The geometric configuration of a PHE, including the angle of the chevron, the pitch of the plate, the width of the channel, the length of the plate, and the number of plates, has a significant impact on the characteristics of heat transfer and pressure drop [63]. Conventional optimization relies on parametric studies and empirical correlations that can be limited in scope and computationally demanding. In contrast, ML approaches can capture complex, nonlinear interactions among geometric parameters and system responses, enabling rapid performance evaluation and design optimization [63]. Figure 23 sketches the ML–PHE design loop used to connect geometry and performance.

Several studies have demonstrated the effectiveness of ML in PHE geometry optimization. Hou et al. [39] applied a hybrid machine-learning framework combining long short-term memory (LSTM) networks and multilayer perceptrons (MLPs) to monitor the health status of plate heat exchangers. By optimizing model structures and integrating the highest-performing networks into an ensemble architecture, they achieved a prediction accuracy of 0.9942. The study introduced a health condition value (HCV) metric and established an early warning threshold based on fouling thermal resistance, enabling timely alerts when the heat-transfer efficiency dropped below 50%. This work demonstrates the potential of LSTM–MLP models for intelligent fault detection and the predictive maintenance of heat exchangers. Figure 24 illustrates the MLP structure used for health monitoring of plate heat exchangers. Figure 25 compares predicted and measured outlet temperatures using LSTM-based models. Figure 26 compares predicted and measured outlet temperatures using MLP-based models.

ML-based surrogate models also enable rapid multiobjective optimization, where design variables are optimized with respect to multiple competing objectives, such as maximizing heat transfer while minimizing pressure drop or cost [13]. Sikirica et al. [13] developed a framework using Latin hypercube sampling and ML surrogates (random forests, gradient boosting, and neural networks), coupled with NSGA-II, to efficiently perform thermal–hydraulic optimization of microchannel heat sinks. They reported near-identical performance to that of full CFD while reducing the computational time by 80%. Hybrid methods, such as an ANN combined with a GA or particle swarm optimization (PSO), have been employed to identify Pareto-optimal designs for PHEs in both single- and two-phase flow regimes [71]. Sharma et al. [71] applied an ANN-GA approach to optimize curved trapezoidal winglets in fin-and-tube heat exchangers. Their study generated Pareto fronts that balance the heat transfer (Colburn factor (j)) against the friction factor (f), illustrating the power of ANNs combined with genetic algorithms for multiobjective thermal–hydraulic design. Figure 27 shows training, test, and target datasets for the friction factor f model. Figure 28 reports error histograms for the j and f training datasets.

As ML applications expand in thermal system design and control, concerns regarding model explainability, trustworthiness, and governance have come to the forefront. Especially in critical applications, like smart grids or industrial TES systems, it is vital to ensure that AI decisions are interpretable and aligned with physical intuition. Techniques such as SHAP (Shapley additive explanations) and LIME (local interpretable model-agnostic explanations) are increasingly being used to provide transparency in model decisions [72,73]. Additionally, recent frameworks have emphasized the ethical deployment of AI in energy systems, addressing issues such as algorithmic bias, data privacy, and model accountability [74]. These considerations are especially pertinent for digital twins, where real-time control actions may rely on AI outputs.

Beyond global accuracy, several studies show how feature-importance tools directly guide design. Su et al. [48] used SHAP on PCHE h prediction models to identify the dominant drivers in two geometries: In straight channels, the hydraulic diameter ($D_h$) ranked the highest, whereas in hexagonal channels, the wall temperature ($T_{\mathrm{inw}}$) and heat flux ($Q_w$) dominated. These attributions were then used to focus subsequent design sweeps on $D_h$ in straight channels and on thermal boundary management in hexagonal channels. Sikirica et al. [13] employed tree-based surrogates within an NSGA-II framework for microchannel sinks; the reported feature rankings were used to prioritize channel-height and -pitch variations over lower-impact shape permutations, reducing the search space without degrading the Pareto quality. For latent storage with shaped fins, Yan et al. [75] trained XGBoost on 60 CFD cases and found that the fin width and HTF temperature to accounted for nearly all the explained importance (51% and 47%); this guided the optimizer to tighten bounds on the fin width and to stage sensitivity tests primarily on the inlet temperature while de-emphasizing the fin angle. These examples illustrate a practical workflow: train the model →compute the SHAP/feature importance→ narrow the design space and tune the constraints before the multiobjective search.

Table 4. Illustrative uses of interpretability to guide geometry/operation.

Study	Model/Data	Tool	Top Drivers (Examples)	Design Consequence
Su et al. [48]	DL (LSTM/Transformer) on CFD–PCHE	SHAP	Straight: $D_h$; Hex: $T_{\mathrm{inw}}$, $Q_w$	Focus sweeps on $D_h$ (straight); manage wall thermal BCs (hex)
Sikirica et al. [13]	RF/GBM surrogates + NSGA-II	Tree FI	Channel height, pitch > shape variants	Prioritize height/pitch; prune low-impact shapes to shrink the search
Yan et al. [75]	XGBoost on 60 CFD TES cases	Gain FI	Fin width, HTF T ≫ fin angle	Tighten width/T bounds; de-emphasize the angle during optimization

illustrates how interpretability results guide geometry and operating-point decisions.

3.2. ML Models for Flow and Heat Transfer

Flow maldistribution, pressure drop, and convective-heat-transfer characteristics are critical performance metrics in PHEs [76]. Accurate prediction of these parameters is essential to ensure thermal efficiency, mechanical reliability, and cost effectiveness. ML models trained on experimental or simulation datasets have shown great promise in modeling these phenomena, especially where analytical or empirical formulations fall short [11]. Figure 29 summarizes modeling links for flow and heat-transfer prediction tasks.

As shown in Figure 30, the process of recognizing flow patterns in multiphase microchannel flows using machine learning generally involves dataset creation, feature selection, model tuning, and model validation and testing. Figure 30 outlines a feed-forward network pipeline for microchannel fluid-flow modeling.

Su et al. [48] proposed a framework to predict the heat-transfer coefficient (h) in printed-circuit heat exchangers (PCHEs) using traditional and deep learning models. A dataset generated via numerical simulations was analyzed using Pearson correlation to identify the key features.

Five conventional ML models were benchmarked against deep learning architectures, including long short-term memory (LSTM), gated recurrent units (GRUs), and transformer networks. To improve accuracy, three attention mechanisms—self-attention (SA), squeeze-and-excitation (SE), and local attention (LA)—were incorporated. Among the traditional models, artificial neural networks (ANNs) showed the best performance, with an $R^2$ value of 0.8918 in straight-path channels.

In deep models, LSTM–SE and transformer–LA performed the best in linear and hexagonal channels, respectively. The transformer–LA model reached an $R^2$ value of 0.9993 for h prediction in hexagonal channels. Model interpretability was provided via SHAP analysis, which identified the hydraulic diameter ($D_h$) as the most influential in straight channels, while the wall temperature ($T_{\mathrm{inw}}$) and heat flux ($Q_w$) dominated in hexagonal ones. Figure 31 compacts the dataset split and deep-learning performance for PCHE h prediction.

Peng and Ling [78] developed an ANN model to predict the Colburn and friction factors in plate–fin heat exchangers, demonstrating errors below 2% compared to those in CFD simulations. Their work illustrated how ML models could replace computationally expensive simulations in iterative design tasks. Other researchers have extended ML applications to capture complex flow behaviors, such as pressure drop across corrugated plates, using decision trees and ensemble-learning models [79].

Moreover, convolutional neural networks, though more commonly used in image processing, are being explored for their ability to extract spatial patterns in temperature and velocity fields generated from simulation data. These models are particularly valuable for real-time system control and fault detection, where rapid, high-fidelity predictions are essential.

In terms of generalization, deep learning methods trained on high-dimensional datasets from CFD outputs have demonstrated better performance in capturing spatially varying flow characteristics than traditional ML models. However, they typically require larger datasets and more computational resources, which can be a limitation in early design stages.

3.3. Comparison with CFD and Empirical Methods

Traditional modeling of PHE performance is often based on empirical correlations derived from experimental studies. While these correlations offer simplicity and are widely used in industry, they lack flexibility and generalizability, especially when extrapolated beyond the conditions under which they were developed [80]. Yang et al. [80] developed Nusselt and friction correlations based on nine brazed-plate exchangers and explicitly noted that the herringbone angle and other geometry-dependent parameters significantly impacted the results, reducing generalizability outside defined conditions. Additionally, CFD provides high-resolution analysis but suffers from high computational costs and sensitivity to meshing and boundary condition assumptions [81].

Additionally, ML is increasingly integrated into CFD workflows to reduce simulation time and enhance predictive performance. For example, hybrid CFD–ML frameworks trained on hundreds of simulations can deliver an $R^2\approx 0.95$ accuracy with near-instantaneous performance evaluations [59]. Another approach embeds ML-augmented solvers into CFD algorithms (e.g., PISO pressure–velocity coupling), achieving speedups of up to 100× without compromising accuracy [82].

However, it is important to note that ML models are only as reliable as the data they are trained on. Poor-quality or sparse training data can lead to overfitting, reduced interpretability, and model instability. To overcome these challenges, some studies have incorporated physics-informed neural networks (PINNs), which embed governing equations into the loss function, ensuring that the model predictions are physically consistent, even when trained on limited data [83,84].

In summary, ML offers a promising alternative and complement to CFD and empirical methods in the design and analysis of PHEs. With ongoing advancements in data availability, algorithm robustness, and hybrid modeling strategies, ML is poised to become a cornerstone in the next generation of intelligent heat exchanger design frameworks.

Table 5. Representative studies on ML applications in PHE and TES systems.

Author (Year)	ML Method	Dataset Type	Metric(s)	Design Variables	Application Focus
Turk et al. (2016) [58]	ANN	Experimental (PHE test bench)	Nusselt, f	Plate type; flow rate	Mixed-plate PHE performance
Zheng et al. (2024) [85]	GP surrogate + MOEA	CFD (80 runs)	Storage power; capacity	Fin number; thickness; EG%	PCM–LHTES optimization
Yan et al. (2025) [75]	XGBoost	CFD (60 cases)	Melting time	Fin angle/width; HTF T	TES with shaped fins
Shen et al. (2025) [86]	ANN	CFD (multiphysics)	Melting front; T profile	Fin length	Rectangular PCM enclosure
Sikirica et al. (2022) [13]	RF, NN, and GBM + NSGA-II	CFD + LHS	Heat transfer; $\Delta p$	Channel height, pitch, and shape	Microchannel sink optimization
Ji et al. (2021) [52]	ANN	Experimental	k (thermal cond.); $\mu$	Nanoparticle type; T	Hybrid nanofluid characterization

compiles representative ML studies in PHE/TES with methods, data, and metrics.

3.4. Quantitative Benchmarks vs. Traditional Methods

Table 6. Comparing ML against CFD.

Study	Problem/Domain	Scope (Data and Range)	Metric (Test)	Baseline Contrast
[49]	PFHE: unified ANN for j and f (offset strip + wavy)	$300 \le Re \le 10,000$; mixed laminar/turbulent; common fin geometries	MAPE ≈ 1–2% (k-fold)	PFHE correlations often $\pm 10$–$20\%$; ANN accurate within range
[48]	PCHE h prediction (DL + attention + SHAP)	CFD datasets; straight vs hex channels	Best: $R^2=0.9993$ (Transformer + LA, hex)	Adds feature attribution ($D_h$, $T_{\mathrm{inw}}$, $Q_w$)
[13]	Microchannel: surrogates + NSGA-II	CFD + LHS; ribs/secondary channels	∼80% wall-clock reduction vs direct CFD; similar Pareto	SHAP ranks drivers
[85]	LHTES: GP surrogates + MOEA (fins, EG%)	80 CFD runs to train 2 GPs	+46.7% power, +22.1% capacity; 91% compute saved	Surrogate-assisted search vs brute-force CFD
[58]	Gasketed PHE: ANN vs. corr.	Exp., $Re$ 500–5000; mixed plates	ANN higher $R^2$, lower error than corr.	Especially for mixed-plate configs.
[80]	BPHE correlation (single phase)	Nine BPHEs; herringbone angle captured	Within stated bands; domain limited	Geometry dependence; limited transfer

contrasts representative ML results with CFD and correlation baselines, making data ranges and testing protocols explicit. Table 6 contrasts ML models with CFD and correlation baselines on scope and accuracy.

4. Machine Learning in Thermal Energy Storage Applications

The integration of ML into TES research has garnered substantial attention due to its potential to enhance prediction accuracy, reduce computational costs, and optimize system performance [75]. Traditional modeling of latent heat storage systems involving PCMs often encounters challenges related to nonlinear behavior, enthalpy–temperature relationships, and the dependency on multiple parameters, such as thermal conductivity, flow conditions, and geometry. ML provides data-driven solutions to overcome these bottlenecks, offering fast, scalable, and accurate alternatives to numerical simulations [85]. This section reviews the latest developments in applying ML to TES systems, with a focus on PHEs integrated with PCMs and nanocomposites, phase-change behavior prediction, and heat storage performance optimization [86].

4.1. ML-Based Prediction of Phase-Change Behavior

Accurate prediction of phase-change dynamics in TES systems is essential for efficient control and design optimization [87]. ML models trained on experimental or high-fidelity numerical data can effectively capture the nonlinear behaviors of PCM melting and solidification. Supervised learning algorithms—such as ANNs, gradient boosting machines (GBMs), and k-nearest neighbors (k-NNs)—have been used to forecast melting front evolution, temperature fields, and liquid fraction profiles. These models are especially valuable for real-time monitoring or digital twin applications. Recent hybrid methods, including physics-informed neural networks (PINNs), integrate ML with physical laws, enabling better generalization beyond training data. Compared to conventional solvers, ML-based predictions can reach similar accuracies with significantly reduced computation times [88].

Yan et al. [75] developed a numerical model using the enthalpy–porosity method to simulate melting in a triplex-tube latent heat TES unit enhanced with Y-shaped fins and PCM. A dataset of 60 simulations was generated, with melting times ranging from 15 to 45 min at varying fin angles (10°–30°), fin widths (5–15 mm), and heat-transfer fluid temperatures (60–80 °C). To ensure model robustness, variable independence was verified. Four ML algorithms—polynomial regression, support vector regression, random forest regression, and extreme gradient boosting—were tested, with hyperparameters tuned via Bayesian optimization. XGBoost yielded the highest accuracy (92%). Feature importance analysis revealed that the fin width and HTF temperature were the dominant predictors (51% and 47%), while the fin angle contributed only 2%. Figure 32 shows the cross-sectional geometry of the latent heat TES unit considered.

Figure 33 compares residuals for four ML regressors predicting PCM melting time.

Taghavi et al. [89] conducted a response surface modeling analysis supported by validated CFD simulations to optimize a thermal storage system. Four critical design variables were identified, and the RSM approach was employed to predict five key responses, including the average power output and effectiveness, across the defined design space. A subsequent multiobjective optimization determined the optimal configuration, resulting in a design that exhibited a 21.4% increase in effectiveness compared to that of the baseline. The findings highlighted that a 5 mm PCM section thickness yielded the maximum performance when all the efficiency metrics were considered. Additionally, operating the system at temperatures exceeding the PCM’s melting point by more than 10 °C was found to reduce the overall efficiency while offering negligible gains in power output. Figure 34 presents response surfaces for temperature, power, recovered heat, effectiveness, and process time.

Shen et al. [86] proposed a novel ANN model to predict the dynamic melting behaviors of phase-change materials (PCMs) in a rectangular enclosure with two fins. Numerical simulations were conducted for fin lengths of 12.5 mm, 25 mm, and 37.5 mm to generate a dataset for ANN training. The model accurately captured the evolution of the melting front and temperature field, achieving an MAE of ≈ 0.014, an MSE of ≈ 0.03, and an R value of $\approx 0.99$ for melting front predictions, and similar accuracies for the temperature distribution (an MAE of ≈ 0.016, an MSE of ≈ 0.03, an R value of $\approx 0.99$). The ANN was validated on unseen fin lengths (20 mm and 30 mm), maintaining strong performance, with an MAE of ≈ 0.02, an MSE of ≈ 0.03, and an R value of $\approx 0.98$. The ANN reduced the prediction time from 6 days to minutes, demonstrating high computational efficiency and strong potential for accelerating PCM-based thermal system design. Figure 35 depicts the rectangular PCM enclosure with fins used in ANN training.

As illustrated in Figure 36, a fully connected feed-forward ANN maps inputs $(X,t,L)$ through three hidden layers to a single scalar output Y. The multi-output architecture in Figure 36 shares the same hidden layers but simultaneously predicts three targets $(Y_1,Y_2,Y_3)$.

Zheng et al. [85] integrated ML with CFD to optimize the fin geometry and expanded graphite (EG) mass fraction in latent heat storage systems. Using a surrogate-assisted multiobjective evolutionary algorithm and batch recommendation strategies, they identified Pareto-optimal designs that balance energy storage power and capacity with minimal computational cost. The approach required only eighty CFD simulations to train two Gaussian process models, achieving a 91% improvement in computational efficiency. The optimal configuration, featuring 78 fins (2.0 mm thick) and a 20% EG mass fraction, enhanced energy storage power and capacity by 46.7% and 22.1%, respectively, compared to those of the baseline. This framework offers an efficient pathway for LHS device optimization. Figure 37 illustrates Pareto-optimal solutions for fin number/thickness and EG mass fraction.

4.2. Heat Storage Capacity and Efficiency Optimization

Optimizing the heat storage capacity and energy efficiency in TES systems involves multiple variables, including PCM selection, nanoparticle composition, heat exchanger geometry, and operational parameters, such as the flow rate and inlet temperature. ML-based optimization frameworks—such as genetic algorithms (GA), particle swarm optimization (PSO), and Bayesian optimization—enable systematic exploration of the design space to identify configurations that maximize the energy density and minimize losses. These techniques have been applied to multiobjective problems where tradeoffs among the heat storage capacity, cost, and physical constraints must be balanced. In several case studies, coupling ANN models with GA or PSO has resulted in superior performance compared to those of traditional trial-and-error or parametric sweep methods. Moreover, reinforcement learning (RL) methods are being explored for dynamic optimization under varying load and ambient conditions, enhancing the adaptability and robustness of TES systems.

Table 7. Overview of machine-learning methods in TES applications.

ML Method	Use Case in TES	Advantages	Disadvantages	Ref.
ANN (Artificial Neural Network)	Predicts melting time, liquid fraction, and temperature fields	Excellent at modeling nonlinear phase-change dynamics; enables digital twins and control applications	Requires large training datasets; risk of overfitting; limited interpretability	[90]
GBM/XGBoost	Predicts the discharge time and energy efficiency across parameters	Handles nonlinear relationships; robust; fast inference; feature importance analysis is possible	Computational during training; hyperparameter tuning is required; less interpretable	[91]
k-NN	Baseline discharge/melting prediction for small datasets	Simple; fast for smaller data; interpretable; no training is required	Poor scalability; sensitive to noise and feature scaling	[92]
Hybrid ANN + GA/PSO	Pareto-optimal design tradeoffs (e.g., heat vs. pressure)	Finds global optima; captures multiple objectives; robust design exploration	Computationally expensive due to iterative execution and setup complexity	[13,71]
PINN (Physics-Informed NN)	Incorporates PDEs for phase-change systems	Physical consistency; works with sparse data; strong generalization; enforces governing physics	Complex architecture; challenging training; slower than classic NNs; hyperparameter sensitive	[5,53]
CFD-trained CNN surrogate	Derives effective thermal properties from microstructures	Learns multiscale features; reduces full CFD runs	Needs curated image-labeled data; large networks; transferability issues	[93]

overviews common ML methods in TES with their use-cases, pros/cons, and references.

5. Research Challenges and Future Directions

Despite notable advancements in the integration of emerging technologies and ML techniques for PHE applications, several key research gaps and technical challenges persist. This section highlights major issues concerning limited data availability, the need for improved model generalization, the development of reliable hybrid ML–CFD frameworks, and the broader potential for enabling intelligent and sustainable PHE systems.

5.1. Data Limitations and Model Generalization

A significant challenge in applying ML to the optimization of PHEs is the limited availability and inconsistency of high-quality datasets [94]. Experimental data on thermal and fluid dynamic behaviors—especially under varying flow regimes and boundary conditions—are often scarce, constrained by measurement uncertainties and laboratory-scale limitations [95]. Similarly, generating comprehensive CFD datasets across wide parametric ranges can be computationally intensive and time consuming [96]. This scarcity, coupled with the lack of standardized and publicly available datasets, poses a major barrier to the development of robust ML models that can generalize across different material compositions, geometric designs, and operational conditions [97].

To overcome these challenges, future efforts must focus on creating open-access, well-annotated datasets, employing advanced data augmentation and synthetic data generation techniques, and establishing rigorous validation protocols to ensure model reliability, reproducibility, and scalability.

Mitigating data scarcity: strategies from the literature. Beyond noting that high-quality datasets are limited, several approaches have proven to be effective at extracting more value from few samples in thermal/CFD contexts:

• Multifidelity + transfer learning. Pretrain models on abundant low-fidelity CFD or correlation data, then fine-tune the models in scarce, high-fidelity data simulations/experiments; use domain adaptation to port models across geometries/PCMs and operating ranges [59,69,98,99];
• Physics-informed learning. Embed governing equations or residuals into the loss (PINNs/hybrid data–PINNs), so models learn from few labeled samples while staying physically consistent [53,55,83];
• Sequential design (active/batch recommendation). Train a surrogate (e.g., GP, RF, or NN) and then adaptively pick the next simulations/experiments that maximize the expected information or Pareto improvement; this cuts expensive CFD runs drastically (e.g., eighty runs to train two GPs, with ∼91% of the initial computational cost being saved) [85];
• Physics-respecting data augmentation. Generate synthetic samples by parameter jittering within admissible bounds, symmetry transforms (e.g., plate/flow symmetries), noise injection consistent with sensor accuracy, and ROM-based field perturbations—while enforcing constraints (energy balance and monotonicity) to avoid nonphysical leakage [83,98];
• Self-/weakly supervised pretraining. Learn representations from unlabeled fields or time series (autoencoding/contrastive tasks) and then fine tune models on small labeled sets; useful when raw monitoring data are plentiful but annotations are scarce [69,98];
• Data fusion and mining. Combine heterogeneous sources—CFD sweeps, rig tests, and plant IoT streams—under a unified schema; curate open data (e.g., Zenodo/GitHub) and harmonize units/ranges to improve coverage and generalization [59,98];
• Uncertainty-aware modeling. Use probabilistic surrogates (GPs and ensembles) and distribution-shift diagnostics to quantify confidence and trigger targeted data acquisition where uncertainty is the highest [69,98].

5.2. Toward Hybrid ML–CFD Frameworks

Emerging research highlights the potential of hybrid ML–CFD frameworks to combine the predictive accuracy of physics-based models with the computational efficiency of data-driven surrogates [59,100]. Such frameworks can accelerate parametric analyses, design optimizations, and real-time operational control by embedding ML models within CFD solvers or coupling them via co-simulation strategies [99]. Notably, physics-informed neural networks (PINNs) have shown promise in enforcing governing conservation laws while maintaining model flexibility [101,102,103]. However, integrating hybrid workflows poses practical challenges, including model stability, data-exchange compatibility, and uncertainty propagation across coupled solvers [98]. Future work should focus on developing standardized interfaces, robust coupling algorithms, and scalable workflows to facilitate widespread adoption in industrial applications. Figure 38 summarizes the hybrid ML–CFD workflow used for design and optimization.Figure 39 presents a compact PINN setup for physics-constrained learning.

Stochastic variability in thermal loads, renewable coupling, and seasonal operation can be handled efficiently with scenario-oriented Benders decomposition (S-BD), a two-stage stochastic programming approach, where first-stage design/sizing and operating policies are optimized against a set of scenarios, and second-stage recourse subproblems dispatch the system per scenario. Let x collect first-stage variables (e.g., plate/nanofluid/PCM sizing and valve/loop configuration limits) and for each scenario, $s\in \mathcal{S}$, let $y_s$ be recourse (flows, T setpoints, charge/discharge, or backup dispatch). The model is

$$\underset{x}{min}{c}^{\top }x+\sum _{s\in \mathcal{S}}{\pi }_s{Q}_s\left(x\right)\mathrm{s}.\mathrm{t}.Ax\le b,x\in X,$$

with subproblem value functions

$$Q_s\left(x\right)=\underset{y_s}{min}{d}_s^{\top }{y}_s\mathrm{s}.\mathrm{t}.W_s{y}_s\ge h_s-T_{s}x,y_s\in Y_s,$$

capturing energy balance, PHE $\Delta p$/$T_{\mathrm{out}}$ limits, PCM SoC dynamics, and backup bounds for scenario-specific load/renewable/seasonal data.

Initialize the master problem (MP) with $Ax \le b$ and no cuts. Repeat: (i) solve the MP to get $x^{\left(k\right)}$; (ii) for each scenario (s), solve the LP (or MILP) subproblem ($Q_s\left(x^{\left(k\right)}\right)$) in parallel. If any subproblem is infeasible, add a feasibility cut; otherwise, add optimality cut(s) using duals. Optionally apply stabilization (prox/regularization) and cut management. Stop when the MP–SP gap is $\le \epsilon$.

5.3. Multisource Co-Optimization for PHE–TES Systems

Real-world deployments rarely rely on a single thermal source. A robust PHE–TES must orchestrate (i) renewable/low-grade inputs (e.g., PV-powered heat pumps or waste heat), (ii) the PHE with a base fluid or a nanofluid for enhanced h at an acceptable $\Delta p$ value, (iii) a PCM store (charge/discharge), and (iv) an auxiliary backup (e.g., electric heater/boiler). Coordinated control improves charge/discharge efficiency, cuts operating costs/emissions, and mitigates single-source risk.

5.3.1. Architecture

Let $Q_{\mathrm{PV}/\mathrm{HP}}$, $Q_{\mathrm{WH}}$, and $Q_{\mathrm{AUX}}$ denote renewable/HP, waste heat, and backup thermal inputs; $Q_{\mathrm{L}}$, the load; ${\dot{Q}}_{\mathrm{PCM}}$, the net rate into the PCM; and $Q_{\mathrm{loss}}$, aggregate losses. The system-level balance is

$$Q_{\mathrm{PV}/\mathrm{HP}}+Q_{\mathrm{WH}}+Q_{\mathrm{AUX}}=Q_{\mathrm{L}}+{\dot{Q}}_{\mathrm{PCM}}+Q_{\mathrm{loss}}.$$

The PCM state of the charge can be tracked based on the liquid fraction ($\lambda \in [0,1]$), with ${\dot{Q}}_{\mathrm{PCM}}={\dot{m}}_{\mathrm{PCM}}L\dot{\lambda }+{\dot{Q}}_{\mathrm{sens}}$.

5.3.2. Dispatch Problem (Day-Ahead/Intraday)

A generic co-optimization over horizon $\mathcal{T}$:

$$\underset{\{\dot{m},T_{\sup },u\}}{min}\sum _{t\in \mathcal{T}}\left(c_e\left(t\right)P_{\mathrm{el}}\left(t\right)+c_{\mathrm{fuel}}\left(t\right){\dot{m}}_{\mathrm{aux}}\left(t\right)+\alpha \mathrm{LLP}\left(t\right)+\beta \mathrm{cycling}\left(t\right)\right)$$

subject to the energy balance, PHE limits ($\Delta p \le \Delta p_{max}$ and $T_{\mathrm{out}} \le T_{\mathrm{out},max}$), PCM constraints ($0 \le \lambda \le 1$, $|\dot{\lambda }| \le {\dot{\lambda }}_{max}$), actuator bounds, and the backup capacity. Decision variables include flow rates through PHE/PCM loops, supply temperatures, valve states (u), and the backup dispatch. $\mathrm{LLP}$ penalizes unmet loads, and $\mathrm{cycling}$ limits excessive charging/discharging.

5.3.3. Control Choices

(i) Rule-based (priority: waste heat → PV/HP → PCM → backup); (ii) MPC using forecasts (${\widehat{Q}}_{\mathrm{WH}}$, ${\widehat{P}}_{\mathrm{PV}}$, and $Q_{\mathrm{L}}$), with soft constraints on $\lambda$ and $\Delta p$; (iii) Surrogate-aided MPC/RL, where fast ML models of PHE/PCM replace full CFD for on-line optimization.

5.3.4. Evaluation Metrics (System Level)

Renewable fraction, round-trip efficiency (${\eta }_{\mathrm{rt}}$), charge/discharge efficiency, exergy efficiency, loss-of-load probability (LLP), unmet load energy (MWh), cycling depth/number, operating cost, and CO₂.

Table 8. Multisource PHE–TES elements and coordination levers.

Element	Uncertainty/Variability	Main Controls	Notes
PV/Heat pump	Solar; ambient T	HP setpoint; flow rate; on/off	Prioritize when COP is high
Waste heat	Intermittent availability	Bypass/valves; $\dot{m}$	Low cost; first priority
PHE (nanofluid)	h–$\Delta p$ tradeoff	$\dot{m}$; plate count/parallelization	Nanofluid boosts h at pump cost
PCM store (PHE coupled)	Thermal lag; $\lambda$	Charge/discharge $\dot{m}$; $T_{\sup }$	Enforce $\lambda$ bands for reliability
Backup heater/boiler	Firm capacity	Dispatch setpoint	Reliability floor/peak shaving

lists key elements, uncertainties, and control levers for multi-source PHE–TES coordination.

In transportation/mission-critical contexts, recent studies emphasize synergistic integration and coordinated dispatch across diverse sources to improve reliability and efficiency while avoiding dependence on any single source [104].

5.4. Outlook for Smart and Sustainable PHE Systems

Looking forward, the convergence of advanced ML algorithms, high-fidelity simulation tools, and smart sensor networks is expected to drive the next generation of intelligent, energy-efficient PHE systems. Real-time monitoring combined with adaptive control strategies can enable dynamic adjustment of nanofluid concentrations, flow rates, and thermal loads to maintain optimal performance under fluctuating conditions [Ref]. Moreover, sustainability considerations—such as lifecycle impacts of nanoparticles, safe disposal, and cost–benefit tradeoffs—must be integrated into both the design and operational stages. Future research directions should, therefore, prioritize multiobjective optimization frameworks that holistically balance thermal performance, economic feasibility, and environmental impact, paving the way toward resilient and sustainable thermal management solutions.

5.5. Emerging Trends: LLMs and Agentic Digital Twins

Recent developments in generative AI, particularly large language models (LLMs) and transformer-based architectures, are beginning to impact energy system modeling and management. Applications such as agentic digital twins—where models autonomously simulate, monitor, and optimize real-time thermal processes—represent a frontier in data-driven energy technologies [105]. These systems can integrate structured (sensor) and unstructured (log/text) data to anticipate failures, self-optimize operations, or generate adaptive control policies.

Although their direct use in plate heat exchangers remains limited, foundational work in thermal systems and phase-change material modeling demonstrates their potential. Future research may explore hybrid models combining LLMs with physical priors or reinforcement-learning agents, enabling interpretability, automation, and more resilient system design. Integrating these capabilities into smart TES architectures could unlock adaptive, scalable, and low-latency-performance solutions.

6. Conclusions

This review has systematically synthesized the state of the art in machine-learning applications for plate heat exchangers, with a special focus on latent heat thermal energy storage systems involving phase-change materials. The key findings and recommendations are summarized as follows:

• Machine learning, particularly ANNs, XGBoost, and PINNs, has enabled accurate surrogate modeling and rapid performance optimization of PHE and TES systems;
• Integration of ML with CFD and experimental data accelerates geometry optimization, fault detection, and thermal prediction in PCM-enhanced exchangers;
• Emerging trends, such as digital twins, physics-informed models, and agentic systems, offer new avenues for smart, adaptive heat exchangers;
• Future research should prioritize interpretability, real-time control, and sustainability-informed ML frameworks.

These insights can support the development of next-generation, intelligent, and sustainable energy storage systems aligned with the goals of decarbonization and operational efficiency.