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Adaptive Uncertainty-Guided Surrogates for Efficient phase field Modeling of Dendritic Solidification
Eider Garate-Perez, Kerman López de Calle-Etxabe, Oihana Garcia, Borja Calvo, Meritxell Gómez-Omella, Jon Lambarri
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Summary
This paper proposes an adaptive uncertainty-guided surrogate modeling framework for efficient phase field simulations of dendritic solidification in metals, particularly for additive manufacturing. The approach utilizes XGBoost and Convolutional Neural Networks (CNNs) with Monte Carlo dropout and bagging for uncertainty estimation. It employs adaptive sampling within hyperspheres to refine the design space, significantly reducing the number of expensive phase field simulations required compared to classical Optimal Latin Hypercube Sampling (OLHS-PSO). The study evaluates computational cost, surrogate accuracy, and CO2 emissions, highlighting the benefits of domain-informed feature extraction versus data-driven CNNs.
Entities (10)
Relation Signals (7)
Phase Field Model → iscomputationallyexpensivefor → Dendritic Solidification
confidence 95% · The high computational cost of phase field simulations remains a major limitation for predicting dendritic solidification
Surrogate Model → replaces → Phase Field Model
confidence 94% · This work presents a surrogate model for dendritic solidification... to efficiently approximate the spatio-temporal evolution while reducing costly phase field simulations
CNN → uses → Monte Carlo Dropout
confidence 93% · approximated via Monte Carlo dropout for CNNs
XGBoost → uses → Bagging
confidence 93% · approximated via Monte Carlo dropout for CNNs and bagging for XGBoost
Adaptive Sampling → outperforms → OLHS-PSO
confidence 92% · achieving accurate predictions with significantly fewer phase field simulations than an Optimal Latin Hypercube Sampling optimized via discrete Particle Swarm Optimization
Phase Field Model → governs → Allen-Cahn Equation
confidence 90% · Systems evolution is described by the Allen-Cahn equation
Surrogate Model → reduces → CO2 Emissions
confidence 88% · Evaluation considers... associated CO 2 emissions, providing a comprehensive assessment of model performance as well as their related environmental impact
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Abstract
Abstract:The high computational cost of phase field simulations remains a major limitation for predicting dendritic solidification in metals, particularly in additive manufacturing, where microstructural control is critical. This work presents a surrogate model for dendritic solidification that employs uncertainty-driven adaptive sampling with XGBoost and CNNs, including a self-supervised strategy, to efficiently approximate the spatio-temporal evolution while reducing costly phase field simulations. The proposed adaptive strategy leverages model uncertainty, approximated via Monte Carlo dropout for CNNs and bagging for XGBoost, to identify high-uncertainty regions where new samples are generated locally within hyperspheres, progressively refining the spatio-temporal design space and achieving accurate predictions with significantly fewer phase field simulations than an Optimal Latin Hypercube Sampling optimized via discrete Particle Swarm Optimization (OLHS-PSO). The framework systematically investigates how temporal instance selection, adaptive sampling, and the choice between domain-informed and data-driven surrogates affect spatio-temporal model performance. Evaluation considers not only computational cost but also the number of expensive phase field simulations, surrogate accuracy, and associated $CO_2$ emissions, providing a comprehensive assessment of model performance as well as their related environmental impact.
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- Source: https://arxiv.org/abs/2603.00093v1
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Adaptive Uncertainty-Guided Surrogates for Efficient 1 phase field Modeling of Dendritic Solidification 2 Eider Garate-Perez 1,2,* , Kerman López de Calle-Etxabe 1 , Oihana Garcia 1 , 3 Borja Calvo 3 , Meritxell Gómez-Omella 1 , and Jon Lambarri 4 4 1 Intelligent Information Systems, Tekniker, Iñaki Goenaga 5, Eibar (20600) Spain 5 2 Faculty of Informatics UPV-EHU, Paseo Manuel de Lardizabal 1, Donostia (20700) Spain 6 3 Department of Computer Sciences and Artificial Intelligence, UPV-EHU, Donostia (20700) 7 Spain 8 4 Advanced Manufacturing Technologies, Tekniker, Iñaki Goenaga 5, Eibar (20600) Spain 9 * Corresponding author. eider.garate@tekniker.es 10 Abstract 11 The high computational cost of phase field simulations remains a major limit- 12 ation for predicting dendritic solidification in metals, particularly in additive man- 13 ufacturing, where microstructural control is critical. This work presents a sur- 14 rogate model for dendritic solidification that employs uncertainty-driven adaptive 15 sampling with XGBoost and CNNs, including a self-supervised strategy, to effi- 16 ciently approximate the spatio-temporal evolution while reducing costly phase field 17 simulations. The proposed adaptive strategy leverages model uncertainty, approx- 18 imated via Monte Carlo dropout for CNNs and bagging for XGBoost, to identify 19 high-uncertainty regions where new samples are generated locally within hyper- 20 spheres, progressively refining the spatio-temporal design space and achieving ac- 21 curate predictions with significantly fewer phase field simulations than an Optimal 22 Latin Hypercube Sampling optimized via discrete Particle Swarm Optimization 23 (OLHS-PSO). The framework systematically investigates how temporal instance 24 selection, adaptive sampling, and the choice between domain-informed and data- 25 driven surrogates affect spatio-temporal model performance. Evaluation considers 26 not only computational cost but also the number of expensive phase field simula- 27 tions, surrogate accuracy, and associated CO 2 emissions, providing a comprehensive 28 assessment of model performance as well as their related environmental impact. 29 Keywords: adaptive sampling, surrogate model, phase field, additive manufacturing, 30 dendritic solidification, spatio-temporal modeling. 31 1 arXiv:2603.00093v1 [physics.comp-ph] 17 Feb 2026 1. Introduction32 The ecological awareness that has emerged in recent decades has reshaped manufacturing, 33 prioritizing durability, sustainable materials, and production processes that reduce energy 34 consumption and waste, while emphasizing recyclability and reusability (Jarfors et al., 35 2024; Panagiotopoulou et al., 2022). To this transformation is added the other design 36 conflicts that until now were found in manufacturing: improving the physical properties 37 of the materials and maximizing the economic performance of the products (Fassi & 38 Aniyou, 2023). This convergence highlights the importance of designing products that 39 simultaneously meet ecological, functional, and economic requirements. 40 In this context, computational models offer a significant advantage by capturing the 41 underlying physical relationships of the phenomena involved (J. Wang et al., 2022), avoid- 42 ing unnecessary material waste and reducing the associated economic costs (Nyamekye et 43 al., 2024). This approach not only accelerates innovation but also aligns with sustainable 44 practices.45 However, traditional and accurate physical models capable of simulating the properties 46 associated with a design are often computationally expensive, particularly when searching 47 for optimal configurations across large design spaces (Duquesnoy et al., 2023; Würth et 48 al., 2023; Xue et al., 2024). Artificial Intelligence (AI) and Machine Learning (ML) have 49 emerged as powerful alternatives to complex simulations, offering the ability to rapidly 50 and reliably predict material properties (Usman et al., 2024; J. Wang et al., 2022). This 51 more efficient prediction enables the optimization of design problems that would otherwise 52 be prohibitive when relying solely on conventional computational models. 53 This approach, commonly termed a surrogate model (Azarhoosh & Ilchi Ghazaan, 54 2025; Kaminsky et al., 2021; Ravutla et al., 2025), aims to replace complex computa- 55 tional models or experiments with ML approximations that minimize accuracy loss while 56 enabling optimization, design exploration, and performance improvement at manageable 57 computational cost (Khatouri et al., 2022). Surrogate models are generally trained on 58 reduced datasets that capture the essential characteristics of the phenomenon, generated 59 either by the original model or derived from experimental data (Forrester & Keane, 2009). 60 The construction of such datasets, known as sampling (Di Fiore et al., 2024), is critical 61 to ensuring representativeness while avoiding excessive computational expense. 62 In the literature, two main sampling strategies are distinguished: classical sampling, 63 which is performed at the initial stage when limited information about the phenomenon is 64 available, and therefore focuses on ensuring adequate exploration of the design space; and 65 adaptive sampling, which iteratively integrates with surrogate model training to exploit 66 model information and thereby enhance both exploration and exploitation of the space 67 (Herten et al., 2017; H. Liu et al., 2018; Samadian et al., 2025). Many studies have 68 shown that adaptive sampling is able to achieve more accurate surrogate models with less 69 data than classical sampling, and therefore, the training of these is more efficient with 70 2 particularly expensive models or experiments (Viana et al., 2021). 71 Building on these advances, it becomes possible to address domain-specific challenges 72 that traditionally rely on expensive simulations. One such challenge is the prediction of 73 dendritic solidification geometries in metals, a complex phenomenon whose accurate mod- 74 eling is computationally demanding (Bhagat & Rudraraju, 2023). These microstructural 75 patterns are directly linked to the physical and mechanical properties of the solidified 76 material (Kurz et al., 2021), and their accurate characterization is essential for ensuring 77 the performance of the final component. The importance of this problem is intensified in 78 additive manufacturing (AM), where metals undergo rapid solidification processes that 79 critically determine the reliability of the produced parts (Yu et al., 2022). AM technolo- 80 gies, recognized for their precision, reduced material waste, and environmental advantages 81 (Javaid et al., 2021; Jiang et al., 2022), require models capable of capturing solidification 82 phenomena to align with both ecological and functional demands. 83 Among the computational approaches employed to address this challenge, the phase 84 field model stands out as one of the most widely used (Dobravec et al., 2023; Tang & Du, 85 2022; Zeng et al., 2024). This method enables the simulation of dendritic solidification 86 geometries in 1D, 2D, or 3D, providing a detailed description of microstructural evolution 87 during solidification (Ji et al., 2025; S. Mao et al., 2024; Seiz et al., 2023). Although higher- 88 dimensional phase field models yield greater accuracy in simulating dendritic solidification, 89 they also entail a steep increase in computational cost. In additive manufacturing, this 90 makes surrogate models particularly valuable, as they approximate phase field behavior 91 with far lower computational demand (Choi et al., 2024; Lee & Kim, 2023; Viardin et 92 al., 2025). In this context, adaptive sampling is crucial for efficient surrogate modeling. 93 By guiding data acquisition based on model predictions, it reduces the computational 94 cost of high-fidelity simulations while ensuring that key features of dendritic solidification 95 are accurately captured. Nevertheless, these approaches still rely on an initial dataset 96 generated by the phase field model. Together, this strategy enables the construction of 97 precise surrogate models with fewer simulations, balancing accuracy and efficiency in the 98 exploration of solidification processes.99 One of the biggest challenges when replacing phase field models is dealing with their 100 inherent spatio-temporal nature (Lavinas et al., 2024; Migdady et al., 2025). Most of 101 the literature focuses on using several temporal instances to predict the final stage of 102 solidification, as well as other properties related to the evolution of the microstructure. 103 For that reason, the challenge lies in accurately capturing both the temporal progression 104 and the spatial complexity of the system while minimizing reliance on the original phase 105 field model during training and prediction. 106 In the literature, different strategies have been proposed to address the spatio-temporal 107 complexity of phase field simulations. For instance, Montes De Oca Zapiain et al., 2021 108 employs the first 80 temporal instances to predict the last 5, applying a two-point statist- 109 ics transformation followed by PCA and an LSTM to estimate the final microstructure. 110 3 Peivaste et al., 2022 predicts subsequent instances using an encoder–decoder (U-net) ar- 111 chitecture, while Yan et al., 2024 combines PCA, autoencoders, and LSTM networks to 112 infer the final state from 50 instances. Similarly, Ahmad et al., 2023 investigates the num- 113 ber of temporal instances required to predict the final microstructure using a convolutional 114 neural network coupled with LSTM layers. Although these approaches successfully reduce 115 the temporal dimensionality of the problem, they still rely on phase field simulations to 116 generate the predictor instances. Consequently, the choice and number of temporal snap- 117 shots become critical, as they directly determine the computational cost and efficiency of 118 the overall framework. This highlights the need for strategies that not only learn from 119 reduced temporal information but also minimize dependence on high-fidelity simulations 120 when constructing predictive models.121 In this framework, surrogate models play a central role in capturing the spatio- 122 temporal correlations inherent to phase field simulations while alleviating their compu- 123 tational burden. Consequently, the selection of predictive instances becomes a critical 124 design decision, as it directly affects model accuracy, complexity, and convergence, while 125 also determining the degree to which dependence on the high-fidelity simulator can be 126 reduced. When instances must be generated by the real physical model up to a time t f , 127 computational savings are restricted to the interval between t f and the final solidification 128 stage.129 Beyond these key aspects, it is also essential to evaluate whether surrogate models 130 truly deliver efficiency in a broader sense. When proposing such models to design more 131 effective manufacturing processes—whether to enhance functionality, reduce economic 132 costs, or mitigate environmental impact—it becomes crucial to measure and compare 133 their performance against sustainability constraints. 134 The present work introduces an adaptive sampling methodology to train a surrog- 135 ate model for dendritic solidification in metals, with a focus on additive manufacturing. 136 Three main objectives guide this study: (i) to analyze the effect of temporal training 137 instance selection on model performance, training cost, and potential computational sav- 138 ings relative to the original phase field simulations; (i) to propose and compare two 139 sampling strategies, one classical based on Optimal Latin Hypercube Sampling optimized 140 via Particle Swarm Optimization, and one adaptive guided by model uncertainty, to as- 141 sess whether adaptive sampling reduces the number of required samples while improving 142 performance and overall training efficiency; and (i) to evaluate two feature extraction 143 approaches, a domain-knowledge-informed transformation coupled with XGBoost and a 144 convolutional neural network (CNN), to compare generalization capabilities in the absence 145 of domain knowledge against convergence and training cost. Furthermore, all analyses 146 are conducted from a sustainability perspective, quantifying the CO 2 emissions associated 147 with each approach.148 The remainder of this paper is organized as follows. Section 2, methods section, de- 149 scribes the phase-field modeling framework, data generation process, surrogate modeling 150 4 approaches, sampling strategies, and experimental setup. Section 3, results and discus- 151 sion section, presents the evaluation of computational cost, CO 2 emissions, and predictive 152 performance, together with a discussion of the findings. Finally, Section 4, conclusions 153 section, summarizes the main outcomes and implications of the study. 154 2. Methods155 2.1 Phase field model for dendritic solidification in metals 156 The geometry of the solidification microstructure in metals plays a critical role in man- 157 ufacturing processes due to its strong influence on the resulting mechanical properties. 158 Among the manufacturing processes in which microstructure is studied are casting, weld- 159 ing, and additive manufacturing. These are typically the initial steps in complex process 160 chains, and the quality of the resulting materials has a direct impact on the final products. 161 In materials science, the complexity of solidification processes is studied across multiple 162 scales, from the atomic to the macroscopic. In general, solidification involves complex 163 interactions between thermal, mechanical, and chemical phenomena. 164 Phase field model is one of the most widely used and effective methodologies for 165 studying, approximating, and simulating materials microstructural evolution. This model 166 is capable of predicting complex, random geometries within continuous field (Tourret et 167 al., 2022). Phase Field methods employ an order parameter, hereafter denoted as OP or 168 φ, which can take values in the range φ ∈ [−1, 1] or φ ∈ [0, 1]. The intermediate values 169 between the interval extremes represent the interface between two distinct physical states 170 of the system: liquid and solid.171 Phase field model is a numerical method capable of simulating microstructural evolu- 172 tion. As with many other numerical methods, solving the physical differential equations 173 governing solidification processes requires significant computational cost. When large 174 parameter spaces are to be explored, the use of the model can become computationally 175 prohibitive (Alizadeh et al., 2020).176 The model analyzed in this work simulates two-dimensional solidification with dend- 177 ritic geometries characteristic of metal solidification (Biner, 2017). This simplified model 178 does not account for phenomena resulting from fluid dynamics, phase transition-induced 179 expansion or shrinkage, nor thermal noise. The microstructural evolution is described 180 through the temporal changes of a non-conserved order parameter, as well as spatial 181 variations in temperature across the domain. 182 Systems evolution is described by the Allen-Cahn equation, 183 τ ∂φ ∂t =− δF δφ ,(1) where τ is the characteristic time scale. The term on the right-hand side of the equation 184 denotes the functional derivative of Ginzburg-Landau free energy, which is defined by, 185 5 F (φ,m) = Z V 1 2 ε 2 |∇φ| 2 + f (φ,m)dv,(2) where f (φ),m is the density of the local free energy. In this work, f (φ,m) is defined 186 by a double-well potential, in such a way that it has two stable states or local minima, 187 located at φ = 0 and φ = 1, Thus, f (φ,m) is defined in the following way, 188 f (φ,m) = 1 4 φ 4 − 1 2 − 1 3 m φ 3 + 1 4 − 1 2 m φ 2 .(3) .189 The ε parameter describes the thickness of the interface. Anisotropy is related to ε 190 and depends on the the outward-pointing normal vector of the interface. In particular, 191 ε = ̄εσ(θ),(4) where ̄ε defines the mean value of ε and the anisotropy σ(θ) is expressed as 192 σ(θ) = 1 + δ cos (j (θ− θ o )).(5) Anisotropy is related to the degree of symmetry observed in the resulting dendritic 193 crystals. For example, when j = 4, cubic symmetry is obtained, and when j = 6, 194 hexagonal symmetry arises. The parameter δ in Equation 5 defines the strength of the 195 anisotropy. The variable θ represents the angle relative to a reference direction, as de- 196 scribed in Equation 6, while θ 0 denotes the initial orientation angle (Dantzig & Rappaz, 197 2009).198 θ = tan −1 ∂φ/∂y ∂φ/∂x ! (6) In Equation 3, the parameter m defines the thermodynamic driving force. As the melt 199 becomes undercooled, the well tilts toward the solid phase, causing the order parameter to 200 shift in that direction. The value of m is defined as a function of temperature as follows: 201 m(T ) = α π tan −1 [γ (T eq − T )](7) where α is a positive constant satisfying α < 1 (Kobayashi, 1993). The T eq value 202 of Equation 7 is the equilibrium temperature. Considering the explicit form of the local 203 energy density, it is possible to obtain the evolution of the order parameter in the following 204 way:205 τ ∂φ ∂t = ∂ ∂y ε ∂ε ∂θ ∂φ ∂x ! − ∂ ∂x ε ∂ε ∂θ ∂φ ∂y ! +∇· ε 2 ∇φ + φ(1− φ) φ− 1 2 + m . (8) Finally, the evolution of the normalized temperature is defined as follows: 206 6 ∂T ∂t =∇ 2 T + κ ∂φ ∂t .(9) From Equation 9, it can be inferred that the equilibrium temperature is 1. The 207 parameter κ defines the diffusion constant, which is assumed to have the same value in 208 both the liquid and solid phases.209 2.2 Phase field data210 As previously described, the physical model is governed by seven parameters that define 211 the thermophysical conditions of the dendritic solidification. Each specific combination 212 of these parameters leads to a distinct dendritic solidification pattern associated with a 213 given metal under well-defined processing conditions. Table 1 summarizes the parameters 214 and their corresponding admissible ranges considered in this study. 215 ParameterDefinitionMin valueMax value τCharacteristic time scale1.72· 10 −4 6· 10 −4 εInterface thickness0.0051.95· 10 −2 κDiffusion constant13.6 δRelated to anisotropy0.0010.04 jDendritic symmetry46 αConstant related to thermodynamic driving force0.40.9 θ 0 Initial orientation0.10.4 Table 1: Minimum and maximum feasible values for each physical model in the phase-field model. For each admissible combination of the seven parameters, a full spatio-temporal simu- 216 lation of the dendritic solidification process is performed. The spatial domain is discretized 217 into a uniform grid of 100× 100 cells, while the temporal evolution is resolved over 4000 218 discrete time steps. Consequently, at each time step, the phase-field model produces two 219 matrices of size 100× 100: one corresponding to the temperature field and the other to the 220 phase (solidification) field. These outputs can equivalently be interpreted as two-channel 221 images of size 100× 100 pixels.222 The objective of the surrogate modeling strategy is to predict the final solidification 223 geometry at t = 4000 from partial spatio-temporal information generated by the physical 224 phase-field model. By replacing the full phase-field simulation with a trained artificial 225 intelligence model, a significant reduction in computational cost can be achieved. 226 However, this task is non-trivial. Let t f denote the last temporal instance provided 227 as input to the surrogate model. When t f is closer to t = 4000, the prediction accuracy 228 is expected to improve, since the solidification geometry at t f more closely resembles the 229 final state. Nevertheless, this choice implies that only the computational cost associated 230 with the simulation interval [t f , 4000] is saved. Conversely, selecting a smaller t f increases 231 the potential computational savings but makes the prediction task more challenging. 232 7 Therefore, the selection of t f must be carefully balanced. An appropriate choice should 233 ensure sufficient predictive accuracy while maximizing computational savings with respect 234 to the full phase-field simulation. In addition, minimizing the number of training samples 235 required to reach a satisfactory level of accuracy remains a primary objective, as it directly 236 impacts the overall efficiency of the surrogate modeling approach. 237 2.3 Surrogate Models238 As mentioned, when the Phase field model is surrogated, the greatest difficulty lies in 239 approximating the spatio-temporal nature of the problem. 240 Several proposals have been made in the literature to address such problems. For 241 example, the work by (Montes De Oca Zapiain et al., 2021) extracts the spatial charac- 242 teristics of the problem using statistical methods and dimensionality reduction techniques. 243 To approximate the temporal relationship between images and predict the final state, it 244 employs an LSTM (Long Short-Term Memory), which is a type of recurrent neural net- 245 work. The study by (Herman et al., 2020) follows a similar methodology for describing 246 spatial features but then uses polynomial chaos for regression and final state prediction. 247 In contrast, the work by (Peivaste et al., 2022) uses a U-Net neural network to learn 248 spatial features and predict the final state. 249 The studies by (Li et al., 2024) and (Goswami et al., 2020) propose a neural network 250 that incorporates physical information (Physics-Informed Neural Network, or PiNN). Al- 251 though the former presents the approach for a 1D Phase field model, it also explains the 252 necessary modifications to apply it to 2D and 3D problems and how to use spatial data 253 in the neural network. The later, on the other hand, is used to predict fracture geometry 254 and specifically solves a 2D problem.255 The work by (Yan et al., 2024) extracts spatial information using an autoencoder, and 256 then approximates the temporal relationship between images using an LSTM. Similarly, 257 (Vasylenko et al., 2021) also proposes autoencoders to automatically learn the character- 258 istics of the images.259 As mentioned in the introduction, other works have proposed neural networks cap- 260 able of learning both spatial and temporal relationships simultaneously Ahmad et al., 261 2023. Finally, there are also proposals that incorporate information from the differential 262 equations of the Phase field model into the neural networks themselves Goswami et al., 263 2020.264 One of the main hypotheses of this work is that, although CNN models are capable of 265 learning complex spatial problems without prior information, training such models may 266 require a large amount of data in order to adjust all the model parameters effectively. 267 However, if prior knowledge about the problem is available, through proper extraction 268 of spatial features, the problem can be approximated using a simpler surrogate model, 269 which requires fewer data to converge.270 In a context where data generation is expensive, defining surrogate models that can be 271 8 Figure 1: Extracted spatial features for the XGBoost model and for (i,j) pixel. efficiently trained is essential. Therefore, the following modeling approaches are proposed: 272 1. Spatial Features + XGBoost: The spatial features of the images are extracted 273 using statistical methods. These features are then used to train an XGBoost model, 274 which leverages the spatial characteristics of the input images to predict the final 275 solidification state.276 2. Convolutional Neural Network (CNN): The convolutional neural network uses 277 convolutional layers to automatically learn the spatial features of the input images. 278 These learned features are then used to predict the final solidification state. 279 3. Self-supervised Convolutional Neural Network: In this method, the neural 280 network is pre-trained using a variational autoencoder. The autoencoder learns to 281 reconstruct images based on their features by first distorting the input images used 282 for training. The convolutional layers of the autoencoder are the same as those used 283 in the CNN model. During CNN training, the convolutional layer parameters are 284 initialized with the pre-trained weights from the autoencoder. The goal in this case 285 is to leverage the image representations learned by the autoencoder. 286 2.3.1 XGBoost surrogate287 Spatial features extraction288 For each image used to predict the final solidification state, spatial features are extracted. 289 For each cell (i,j) in the image, the spatial features consist of the values of the order para- 290 meter and temperature within a radius of r = 2 around that cell. Figure 1 schematically 291 illustrates the extraction of pixel features used for each (i,j) cell. 292 For each pixel (i,j) and a radius of r = 2, as shown in Figure 1, the values of tem- 293 perature and order parameter from the 24 neighboring pixels are used, in addition to 294 the values at (i,j). Therefore, if n t time steps or images are used to predict the final 295 solidification state, there will be 2· 25 spatial features per pixel, resulting in a total of 296 2· 25· n t spatial features for the regression problem. 297 9 In addition to the spatial features, the values of the physical simulation parameters 298 from the physical model are also included. 299 300 Symmetry simplification301 To reduce the complexity of the regression problem solved using XGBoost, the symmet- 302 ries of the dendritic geometries are exploited. As explained earlier, this work analyzes 303 dendrites with different anisotropy levels, exhibiting 4-fold and 6-fold symmetry patterns. 304 Following symmetries can be identified: 305 • Anisotropy with 4-fold symmetry: 306 In this case, the dendritic geometry exhibits symmetry with respect to the axes 307 x = 0 and y = 0. Therefore, considering the quadrant defined by the cell range 308 0 ≤ i ≤ 50 and 0 ≤ j ≤ 50 as the reference, the full image can be reconstructed 309 by applying rotations of 90 ◦ , 180 ◦ , and 270 ◦ to this quadrant. Alternatively, if only 310 one axis of symmetry is used, the half-image defined by 0≤ i≤ 50 and 0≤ j ≤ 100 311 can serve as the reference, and the full image can be recovered by applying a 180 ◦ 312 rotation.313 • Anisotropy with 6-fold symmetry: 314 In this case, the dendritic pattern exhibits rotational symmetry that repeats 6, 3, and 315 2 times. Using the 6-fold rotational symmetry, if the region defined by 0 ≤ i ≤ 50 316 and 0≤ j ≤ 1 3 i + 100 3 is taken as the reference, the entire image can be reconstructed 317 by applying rotations of 60 ◦ , 120 ◦ , 180 ◦ , 240 ◦ , and 300 ◦ . Alternatively, using the 318 3-fold symmetry and the region 0 ≤ i ≤ 50 and 0 ≤ j ≤ − 1 3 i + 200 3 as reference, 319 the full image can be reconstructed through 120 ◦ and 240 ◦ rotations. The 2-fold 320 rotational symmetry case is handled in the same way as the 4-fold anisotropy case. 321 In any case, thanks to the symmetry-based simplification, the model can be trained 322 using only parts of the image. In this way, the final solidification state of a region can be 323 predicted using less information. Then, by applying the corresponding rotations, the full 324 image can be reconstructed.325 In the work described in this chapter, the 4-fold rotational symmetry simplification 326 is used for dendrites with anisotropy 4. In contrast, for dendrites with anisotropy 6, the 327 6-fold rotational symmetry simplification is applied. 328 XGBoost329 The XGBoost algorithm combines multiple tree-based estimators using bagging, and re- 330 fines the model iteratively through boosting by focusing on samples that are more difficult 331 to predict.332 In this work, the XGBoost Python library is used. Regarding the algorithm’s hyper- 333 parameters, default values are used in all cases except for the learning rate, maximum 334 depth, and number of estimators, which are set to 0.01, 5, and 200, respectively. The 335 mean squared error is used as the loss function. 336 10 Convolution Convolution Convolution Convolution Convolution 32, 3x3, ReLU 32, 3x3, ReLU Convolution 32, 3x3, ReLU 32, 3x3, ReLU 32, 3x3, ReLU 32, 3x3, ReLU t 0 t 1 . . . t p MaxPooling2D MaxPooling2D MaxPooling2D MaxPooling2D MaxPooling2D MaxPooling2D Combined convolution 64, 3x3, ReLU Physical parameters Dense 32, ReLU Dense 16, ReLU UpSampling2D Combined convolution 32, 1x1, ReLU Transposed convolution 1, 3x3, sigmoid Figure 2: Architecture of the convolutional neuronal network. t 0 , t 1 , ..., t n are the training temporal instances. Feature Selection337 For variable selection, the feature importances provided by the XGBoost model are used. 338 To do this, all variables are ranked from most to least important, the cumulative import- 339 ance is computed, and the top variables that account for up to 0.98 of the total importance 340 are selected.341 2.3.2 Convolutional Neural Network 342 The architecture of the convolutional neural network used in this work is summarized in 343 Figure 2. The following paragraphs provide a detailed description of each component of 344 the network.345 For each training instance and for each sample, one image is generated for temperature 346 values and another for the order parameter. The convolutional neural network applies 347 one convolutional layer to each of these input images. Each of these layers uses 32 filters 348 with kernel sizes of 3× 3, which are applied with a stride of one along both spatial axes. 349 The activation function used is ReLU (Rectified Linear Unit). 350 Following the convolution, the spatial dimensions of the resulting feature maps are re- 351 duced through a downsampling operation that simplifies spatial information. Specifically, 352 max pooling is applied over non-overlapping 2× 2 regions, retaining the maximum value 353 in each case. This reduces the image dimensions by half. 354 Once all the input images have passed through their respective convolutional layers, 355 their outputs are merged through an additional convolutional layer that combines the ex- 356 tracted features. In this layer, 64 filters are used, with the same kernel size and parameters 357 as in the previous layers. After this operation, the resulting feature map is upsampled to 358 11 restore the original image dimensions.359 The physical parameters used in the simulation, the seven parameters described earlier, 360 are integrated into the neural network. To do this, two fully connected (dense) layers are 361 applied: the first with 32 neurons and the second with 16, both using the ReLU (Rectified 362 Linear Unit) activation function.363 The output from the tabular (parameter) layers is then reshaped to match the spatial 364 scale of the image data and combined with the image features through a convolutional 365 layer. This layer uses 32 filters with 1× 1 kernels and again applies the ReLU activation 366 function.367 Finally, the prediction of the output image is performed using a transposed convolu- 368 tional layer followed by a sigmoid activation function. This final layer uses a single filter 369 with a 3× 3 kernel and preserves the spatial dimensions of the input, enabling the network 370 to generate a full-resolution output image. 371 To train the model, the mean absolute error (MAE) is used as the loss function, and 372 the ADAM optimizer is employed for optimization. 373 2.3.3 Self-supervised Convolutional Neural Network 374 The self-supervised neural network uses the same convolutional architecture described in 375 the previous section to predict the final solidification state. However, in this case, the 376 convolutional layers of the neural network are pre-trained using a variational autoencoder. 377 The variational autoencoder adds noise to the input images before processing. Using 378 the same convolutional layers as those in the main network, that is, the layers used initially 379 to process the images, the autoencoder learns to reconstruct the original images from 380 the noisy versions. Through this process, the autoencoder effectively learns to extract 381 meaningful features from the images.382 The convolutional layer parameters learned by the variational autoencoder are used 383 to initialize the convolutional layers of the main convolutional neural network. Self- 384 supervised learning is a methodology that has been applied to improve the training of 385 complex neural networks when only few data is available (W. Mao et al., 2023). In the case 386 of convolutional neural networks, which typically involve a large number of parameters, 387 the more parameters a model has, the more data it requires to achieve accurate results. 388 This makes properly training convolutional networks particularly challenging (Chen et al., 389 2024).390 When the data originates from a complex physical model, it becomes highly desirable 391 to obtain accurate surrogate models using as little data as possible. In this context, self- 392 supervised learning may provide an effective means to develop more efficient convolutional 393 neural networks, particularly when used as surrogate models. 394 The architecture of the variational autoencoder used in this work is shown in Figure 3. 395 As illustrated, temperature and order parameter images are processed separately through 396 dedicated convolutional layers. These layers are the same ones later used in the main con- 397 12 Convolution 32, 3x3, ReLU Convolution 32, 3x3, ReLU Tenperature Noisy tenperature OP Noisy OP MaxPooling2D MaxPooling2D UpSampling2D UpSampling2D Convolution 1, 3x3, sigmoid Convolution 1, 3x3, sigmoid Figure 3: Architecture of the Variational Auto Encoder used in the self-supervised CNN volutional network, each corresponding to its specific image type. After passing through 398 an encoding–decoding (compression and reconstruction) process, the outputs are further 399 processed by an additional convolutional layer. These final layers use a single filter and 400 apply the sigmoid activation function. The kernel size remains the same as in the initial 401 layers, i.e., 3× 3.402 In this way, the autoencoder learns to reconstruct the original images from their noisy 403 versions, effectively capturing the key spatial features of the input images. As mentioned 404 earlier, the parameters obtained after training the first two convolutional layers are then 405 used to initialize the corresponding convolutional layers in the main neural network, as 406 shown in Figure 2. Therefore, the convolutional network begins training from a repres- 407 entation that already captures the predicted spatial features. 408 2.4 Sampling Methods409 One of the main objectives of this work is to analyze how the sampling strategy influ- 410 ences the number of samples required to accurately surrogate a computationally expensive 411 problem such as the Phase field model, with the ultimate goal of minimizing the number 412 of simulations needed to achieve a good fit. 413 To this end, two sampling strategies are considered: classical sampling and adaptive 414 sampling. Classical sampling relies on an optimal Latin hypercube design, whereas the 415 adaptive approach incorporates new samples based on model uncertainty. The following 416 sections describe both methodologies in detail. 417 2.4.1 Classical sampling: Optimal Latin Hypercube 418 Latin Hypercube Sampling (LHS), is one of the most widely used sampling methods 419 in the literature (Luo et al., 2023). The technique divides the space into m n identical 420 hypercubes to place m points, where n is the dimension of the space. The method 421 consists in distributing the m sample points so that each partition of every coordinate is 422 used exactly once. Specifically, if one of the m points is placed at (a 1 ,...,a n ), and a 1 lies 423 13 in the first cell of the m-partition along the first coordinate, then no other point is allowed 424 to share that same cell in the first coordinate. This restriction applies to all remaining 425 coordinates: for each a i , no other point may occupy the same partition cell along that 426 coordinate (McKay et al., 1979).427 As is evident, for fixed values of n and m, multiple Latin Hypercube Samplings (LHS) 428 can exist. Techniques have been proposed to select the most suitable LHS (S. Wang 429 et al., 2019). While standard LHS ensures a certain level of point dispersion, it does 430 not necessarily guarantee the optimal distribution. For this purpose, various methods 431 have been developed to compute an Optimal Latin Hypercube Sampling (OLHS), which 432 maximizes the distance among all points. 433 To obtain an OLHS, an optimization problem that maximizes the distance between 434 sample points must be solved. A metric space (most commonly Euclidean) is selected, 435 and an appropriate optimization algorithm is applied. Several methods have been pro- 436 posed in the literature to generate optimal Latin hypercubes. For instance, (Borisut & 437 Nuchitprasittichai, 2023) reviews different techniques for optimizing LHS, most of which 438 aim to maximize inter-point distances, although some approaches also maximize entropy 439 or incorporate other objective functions. 440 Classical, metaheuristic, and genetic algorithms have all been employed to solve OLHS 441 problems. For example, in distance-maximization scenarios, Park et al., 2025 uses an evol- 442 utionary algorithm, while Aziz & Tayarani- N., 2014 employs a genetic algorithm based 443 on Particle Swarm Optimization (PSO). Other approaches reported in the literature in- 444 clude simulated annealing and various adaptive evolutionary algorithms (Escobar-Cuevas 445 et al., 2024), as well as propagation–translation LHS methods (Park et al., 2025). 446 In this work, a discrete Particle Swarm Optimization (PSO) approach is proposed to 447 obtain an OLHS. In this optimization problem, each LHS represents a feasible solution. 448 An LHS with m points, in any n-dimensional space, can be represented as a permutation 449 of m values along each of the n coordinates. Indeed, in the space X 1 ×X 2 ×·×X n , if a 450 point (a 1 ,...,a n ) is placed, each a i corresponds to a cell in the LHS along coordinate i. 451 Consequently, no other point in the sample may occupy the same LHS cell, i.e., no other 452 point satisfies X i = a i for i = 1,...,n.453 Thus, if m uniform cells are defined for each coordinate, each cell contains exactly one 454 point. For instance, consider a two-dimensional case: for the first coordinate, each dis- 455 cretized cell contains a value, which simultaneously corresponds to a coordinate between 456 1 and m in the second dimension. The permutation positions indicate the cell in the first 457 coordinate, while the values represent the corresponding positions in the second coordin- 458 ate. More generally, in an n-dimensional sampling, if coordinates are considered in pairs 459 and the sample is represented relative to one another across all coordinates, the LHS can 460 be fully defined by n permutations of m values. 461 As an example, if we want to place 7 points in the [0, 1] × [0, 1] space, the LHS 462 illustrated in Figure 4 can be represented as (4, 3, 1, 2, 6, 5, 7), (3, 4, 2, 1, 6, 5, 7). 463 14 X 1 X 2 Figure 4: 7 points LHS sampling for the [0, 1]× [0, 1] space. The fitness value of each population is based on the Morris-Mitchell dispersion (Xiao 464 et al., 2022). A small dispersion value represents a large distance between points. It is 465 calculated as:466 fitness = m X i=1 m X j=i+1 d(p i ,p j ) −δ 1/δ ,(10) where d(p i ,p j ) is the distance between the i-th and j-th points in the LHS, and δ is a 467 pre-defined constant. In this work, the Euclidean distance is used and δ = 50. 468 Each population is transformed in every iteration using the following two techniques: 469 • Move toward optimum. The goal is to move the population toward local and 470 global optima. For each parameter X i , movements are performed toward both the 471 local and global optima. The procedure follows these steps, also illustrated in Figure 472 5:473 1. Positions whose values will be randomly modified are selected. 474 2. For each value of the first point, the variable is assigned the value of the local 475 or global optimum at that position. 476 3. The position where the particle originally held the newly assigned value is 477 identified, and the value at that position is replaced by the value that was just 478 moved.479 The transformation is first applied toward the local optimum and then toward the 480 global optimum. The number of positions to be changed and the specific positions 481 are randomly selected in each case. 482 • Random swap. Two positions are randomly selected, and their values are ex- 483 changed.484 Movement toward the optima is applied in every iteration, but the movement “speed,” 485 i.e., the number of positions changed, varies for each transformation. The random swap is 486 15 32154 12345 32154 12345 12155 32154 12345 1 particle local optimum indexes to change =1,5 23 Figure 5: Transformation toward the local and global optimum. The example shows the local optimum case; the global case is symmetric. applied probabilistically: for each variable, a value is sampled from a uniform distribution. 487 If this value is smaller than a pre-defined threshold R, a random swap is performed. In 488 this work, R = 0.7 is used.489 Finally, the new population is evaluated, and the global and local optima are updated 490 along with the population. The whole algorithm is defined in the pseudocode defined in 491 Algorithm 1.492 2.4.2 Uncertainty based adaptive sampling 493 Adaptive sampling aims to iteratively improve surrogate models or optimization processes 494 by selecting new samples based on model performance and uncertainty. Classical sampling 495 strategies typically emphasize exploration only, which can be inefficient when the relation- 496 ship between input variables and the output is highly nonlinear and unknown. Balancing 497 exploration and exploitation is therefore essential for constructing accurate and efficient 498 adaptive sampling (Viana et al., 2021). In surrogate-based modeling, adaptive sampling 499 allows extracting information effectively from computationally expensive simulations as 500 the studied in this work (Viana et al., 2021). As the relationship between the input space 501 and the response is initially unknown, and because the number of samples strongly in- 502 fluences performance, strategies that adaptively refine the design are crucial (Lookman 503 et al., 2019).504 Adaptive sampling methods address these challenges by iteratively enriching the train- 505 ing dataset with points that improve the model, striking a balance between exploration 506 and exploitation (Fuhg et al., 2021; Lookman et al., 2019). Among the most effective 507 strategies are those based on epistemic uncertainty estimation. These methods quantify 508 the predictive uncertainty of the surrogate model over the entire design space and add 509 samples in regions where uncertainty is highest. High-uncertainty regions can arise either 510 from sparse sampling (lack of exploration) or from areas with steep response gradients 511 (need for exploitation), even if nearby points exist (Hüllermeier & Waegeman, 2021). 512 Numerous approaches for uncertainty estimation have been proposed in the literat- 513 ure (Fakour et al., 2024; Heid et al., 2023), and their effectiveness is closely tied to the 514 16 Algorithm 1 PSO for Optimal Latin Hypercube Sampling Require: n_points,n_features,n_pop,max_iter,tol,distance_method,p,R Ensure: Best LHS design and fitness history 1: Initialize population: 2: for particle i = 1 to n_pop do 3: Generate random LHS of size (n_points× n_features) 4: end for 5: Compute fitness of each particle using Morris-Mitchell dispersion 6: Identify initial global best (gbest) and local best (lbest) 7: Record gbest fitness history 8: for iteration = 1 to max_iter do 9: if tol ≥ tolerance_threshold then 10:break 11: end if 12: for particle i = 1 to n_pop do 13:Extract current LHS i for particle 14:for feature j = 1 to n_features do 15:Randomly select number of positions to move toward lbest and gbest 16:Select indices of positions: same_num_p, same_num_g 17:Change particle 18:new_LHS i ← move_design(feature j , LHS i , lbest, same_num_p) 19:new_LHS i ← move_design(feature j , LHS i , gbest, same_num_g) 20:if rand() < R then 21:new_LHS i ← rand_swap(LHS i , feature j ) 22:end if 23:end for 24: end for 25: new_fitness ← compute_fitness(new_population, distance_method, p, R) 26: for particle i = 1 to n_pop do 27:if new_fitness > previous_fitness then 28:LHS i ← new_LHS i 29:fitness i ← new_fitness i 30:if new_fitness > gbest fitness then 31:Update gbest 32:Record gbest fitness 33:Update tolerance 34:end if 35:end if 36: end for 37: Update lbest to particle with maximum fitness in current population 38: end for 39: return gbest LHS and fitness history 17 surrogate model employed. Probabilistic models offer a more direct way to quantify un- 515 certainty compared to deterministic ones. Nevertheless, reliable uncertainty measures 516 for deterministic surrogates also exist. For neural networks, for example, Monte Carlo 517 dropout provides a practical Bayesian approximation by randomly deactivating neurons 518 during inference (Gal & Ghahramani, 2015). Other model-agnostic approaches include 519 quantile-based loss functions, numerical methods to estimate joint conditional distribu- 520 tions (Tyralis & Papacharalampous, 2024), or model ensemble methods (Hoffmann et al., 521 2021; Rahaman & thiery alexandre, 2021). 522 Building on these ideas, the adaptive method proposed in this work combines explora- 523 tion and exploitation through uncertainty-driven sampling. By iteratively measuring and 524 leveraging the model’s uncertainty, new points are added in the most informative regions 525 of the design space, enhancing both accuracy and efficiency. 526 Monte Carlo Dropout for uncertainty estimation 527 528 In the case of neural networks, the Monte Carlo dropout method was employed to 529 quantify uncertainty. This approach has been widely adopted to estimate epistemic un- 530 certainty in neural networks (Z. Liu et al., 2025; Padarian et al., 2022; Zeevi et al., 2024). 531 Moreover, as explained in (Seoh, 2020), neural networks trained and evaluated with Monte 532 Carlo dropout are analogous to Bayesian neural networks and provide reliable uncertainty 533 estimates.534 Dropout is a regularization technique originally designed to prevent overfitting in 535 neural networks. The method randomly deactivates a proportion of neurons and can 536 be applied during both training and inference. In this work, dropout is applied at two 537 stages during training: first, after each layer that processes the image data (following the 538 structure shown on the left side of Figure 2); and second, in the convolutional layer where 539 physical parameter features and image features are combined (the second layer from the 540 right in Figure 2).541 After training, dropout is also applied during inference, where random network con- 542 nections are deactivated and ignored for prediction. For each input image, 100 stochastic 543 forward passes are performed using dropout, resulting in 100 predictions. The mean of 544 these predictions is taken as the final network output, while the standard deviation across 545 the 100 predictions serves as the measure of predictive uncertainty. 546 To obtain a spatial distribution of the model uncertainty and an overall uncertainty 547 estimate for each model, cross-validation is employed. All available samples are divided 548 into 10 groups of equal size, and during 10 iterations, one group is used for validation 549 while the remaining 9 groups are used for training. In each case, uncertainty is computed 550 on the validation samples.551 Bagging XGBoost Trees for uncertainty estimation 552 18 553 For XGBoost, the methodology used combines bootstrap and bagging techniques to 554 estimate the model’s epistemic uncertainty. Similar approaches have been described in 555 the literature for quantifying uncertainty in tree-based machine learning models (Brophy 556 & Lowd, 2022; Malinin et al., 2021). Although the variance induced by bootstrap res- 557 ampling is not a purely epistemic measure—because subsampling of instances and features 558 introduces additional sampling noise—it has been widely adopted as a practical approx- 559 imation to epistemic uncertainty in complex regressors, as demonstrated for example in 560 Lookman et al., 2019, Fakour et al., 2024, and Heid et al., 2023. Moreover, Palmer et al., 561 2022 shows that while bootstrap-based uncertainty is not an exact estimator of epistemic 562 error, it remains strongly and monotonically correlated with the true prediction error, 563 supporting its use as a reliable and informative approximation. 564 As mentioned earlier, the XGBoost implementation employed in this work uses 200 565 estimators. Each estimator is trained on a random subset of both features and samples. 566 During inference, a random %75 of the trained estimators is selected for each prediction, 567 and the prediction process is repeated 100 times per input sample. The mean value of 568 these repetitions is taken as the final prediction, while the standard deviation across the 569 100 repetitions serves as the uncertainty measure. 570 Similar to the methodology adopted for neural networks, cross-validation is also em- 571 ployed here to approximate the overall uncertainty across the entire input space. In this 572 case, %20 of the samples are randomly selected for validation, and 10 repetitions are 573 performed. In each repetition, uncertainty is computed on the validation samples. 574 Definition of new samples575 576 Using the uncertainty estimates obtained for each model, regions of high uncertainty 577 are identified in order to introduce new samples within those regions. 578 As previously mentioned, uncertainty is computed on the validation samples using 579 the two methodologies described above. In any case, this approach only provides uncer- 580 tainty values for samples that are already part of the design set. However, the use of 581 cross-validation allows representing uncertainty across the entire input space, particularly 582 considering that the initial sampling is performed via OLHS. 583 In this work, if the number of samples at iteration k is denoted as n k , then for iteration 584 k + 1, an additional ⌊0.15n k ⌋ samples are added to the design set, resulting in n k+1 = 585 ⌊1.15n k ⌋ total samples. At each iteration, these ⌊0.15n k ⌋ new samples are selected based 586 on the model uncertainty as follows:587 1. At iteration k + 1, the ⌊0.15n k ⌋ candidate points with the highest uncertainty are 588 identified.589 2. For each of these points, the distance to all other points in the design set is computed, 590 and the nearest neighbor is identified. 591 19 3. Hyperspheres are defined with centers at the high-uncertainty points and radius 592 equal to half the distance computed in step 2. 593 4. For each hypersphere, one new sample is randomly generated within it. This set of 594 points constitutes the ⌊0.15n k ⌋ new samples added to the design set. 595 As an illustrative example, Figure 6 depicts the adaptive sampling procedure for two 596 physical parameters at iteration k.597 2r 1 2r 2 2r 3 2r 1 2r 2 2r 3 123 Figure 6: Definition of new samples based on uncertainty estimation. Red points denote the samples with the highest uncertainty, while green points represent those added for iteration k + 1. Adaptive Sampling Based on Mixed Uncertainty 598 599 The predictive uncertainty of each sample is computed from the pixel-wise uncertainty 600 of its predicted solidification image, using the Monte Carlo Dropout and bagging strategies 601 explained above.602 Each sample or image is partitioned into three zones according to the solidification 603 phase predicted for each pixel: pixels with values below δ are assigned to the liquid zone, 604 those above 1− δ to the solid zone, and the remaining pixels to the interface zone (the 605 transition between liquid and solid), where δ is a small positive value close to 0. 606 For each sample i, the mean uncertainty is calculated separately over each zone, de- 607 noted by ̄ U (i) liq , ̄ U (i) int , and ̄ U (i) sol .608 The interface region, being the most difficult to predict and determining the final 609 geometry of the solidified metal, and consequently its mechanical properties, is assigned 610 the largest weight when aggregating the uncertainty. The mixed uncertainty is therefore 611 defined as:612 U (i) mixed = w liq ̄ U (i) liq + w int ̄ U (i) int + w sol ̄ U (i) sol , with w liq = 0.1, w int = 0.6, and w sol = 0.3. 613 These mixed uncertainty values are used within the adaptive sampling algorithm (Al- 614 gorithm 2) to rank the samples, select those with the highest uncertainty, and generate 615 new candidate points to enrich the design space. 616 20 Algorithm 2 Adaptive Sampling Based on Mixed Uncertainty Require: p,uncertainty_df,doe,w liq ,w int ,w sol ,δ Ensure: New sample points and updated DOE 1: Compute mixed uncertainty for all samples: 2: for each sample i in uncertainty_df do 3: Partition predicted solidification image into zones: liquid: pixels ≤ δ, solid: pixels ≥ 1− δ, interface: remaining pixels 4: Compute mean uncertainty per zone: ̄ U (i) liq , ̄ U (i) int , ̄ U (i) sol 5: Compute mixed uncertainty: U (i) mixed = w liq ̄ U (i) liq + w int ̄ U (i) int + w sol ̄ U (i) sol 6: end for 7: Sort samples by U (i) mixed in descending order 8: Select top p most uncertain samples: p_most_uncertain 9: for each sample in p_most_uncertain do 10: Compute scaled coordinates within [0,1] range for all features 11: Find nearest existing sample and compute half-distance 12: Define hypersphere centered at sample with radius = half-distance 13: Generate new candidate point randomly inside hypersphere 14: Rescale new point back to original feature space 15: end for 16: Assign new experimental IDs to generated points 17: Concatenate new points with existing DOE 18: return new sample points, updated DOE 2.5 Experimental setup617 This section describes the experimental framework developed to evaluate the hypotheses 618 presented in this work.619 2.5.1 Training Temporal Instances 620 For spatio-temporal problems such as phase field simulations, it is essential to capture both 621 spatial and temporal features of the system. Spatial features are handled as described 622 in Section 2.3.1. Temporal features require careful selection of discrete time instances 623 used to train the surrogate model. The final state prediction is strongly influenced by the 624 temporal instances closest to the target time. 625 The computational cost of surrogate model predictions depends on the number and 626 choice of temporal instances simulated using the original high-fidelity model. Let t f denote 627 the last time instance used for training; the potential computational savings are inversely 628 proportional to the cost of simulating the instances between the initial time and t f . Based 629 on this principle, three temporal instance sets are defined for evaluation: 630 • ID 1 =1000, 1500, 2000, 2500631 21 • ID 2 =1000, 1500, 2000, 2500, 3000 632 • ID 3 =1000, 1500, 2000, 2500, 3000, 3500 633 These configurations correspond to executing approximately half, three-quarters, and 634 seven-eighths of the full simulations required to predict t = 4000 using the surrogate. 635 2.5.2 Sampling Size636 For adaptive sampling, the initial design follows the recommendations of Fuhg, 2019; 637 Tenne, 2015, and Afzal et al., 2017, setting the initial sample size to 10n, where n = 638 7 is the number of physical parameters, yielding 70 initial points. At each iteration 639 k + 1, additional samples are added according to Section 2.4.2 as ⌊0.15n k ⌋, resulting in a 640 progressive growth of the sample set: 70, 80, 92, 105, 120, 138,.... 641 Classical fixed designs do not allow incremental adjustment based on model uncer- 642 tainty. For comparison, fixed designs of 70, 150, 300, 500, and 700 points are evaluated, 643 with 700 representing the maximum sample size for both adaptive and classical strategies. 644 2.5.3 Incremental Neural Network Training 645 Adaptive sampling is coupled with incremental surrogate training. Starting from the 646 initial design, the model is retrained as new samples are added, reducing overall model 647 uncertainty progressively. Incremental training is implemented using the Keras library 648 (Chollet et al., 2015), which allows updating model weights with new samples while 649 preserving previously learned information. 650 Cross-validation is applied to estimate uncertainty across the entire sample space. To 651 avoid biasing uncertainty estimation toward newly added points, incremental retraining 652 is reserved for the final model. This approach maintains both computational efficiency 653 and accurate representation of global uncertainty. 654 Overall, this experimental design systematically investigates the influence of temporal 655 instance selection, sample size, and adaptive incremental training on surrogate model 656 performance, directly addressing this work’s hypotheses. 657 2.5.4 Carbon Footprint Analysis658 The accelerating impacts of climate change and global environmental degradation have 659 led to international commitments such as the Paris Climate Change Agreement (PCCA), 660 through which almost all countries have agreed to reduce their carbon emissions (Q. Wang 661 et al., 2024). In this broader context, the rapid expansion of AI has garnered attention due 662 to its substantial energy demands. Despite the potential of AI technologies to optimise en- 663 ergy use and promote environmental sustainability, their development and deployment are 664 themselves energy-intensive processes that generate substantial greenhouse gas emissions. 665 The amount of computational power used for AI increased by 300,000 times between 666 2012 and 2018 (Freitag et al., 2021). Consequently, researchers have emphasised the 667 22 significance of adopting a "Green AI" approach, which prioritises computational efficiency 668 and sustainability, as a contrasting approach to the current prevailing "Red AI" approach, 669 which prioritises model accuracy without considering environmental cost (Freitag et al., 670 2021). This has led to the integration of monitoring tools to quantify and mitigate the 671 energy consumption and carbon footprint of AI workloads. 672 This study employs CodeCarbon (Courty et al., 2024), an open-source software, to es- 673 timate CO 2 emissions and energy consumption generated by training tasks. CodeCarbon 674 records energy consumption from CPUs, GPUs, and memory, and converts these data 675 into carbon-equivalent values based on the carbon intensity of the geographical region. 676 This tool serves to complement traditional performance indicators, such as the R 2 , with 677 quantitative metrics that reflect environmental costs. 678 3. Results and discussion 679 To evaluate and validate the hypotheses and proposed methodology, the results are ana- 680 lyzed along three main dimensions: (1) the environmental impact of each strategy, quan- 681 tified through its carbon footprint at the same steps; (2) computational costs associated 682 with simulation, sampling, and training; and (3) model performance metrics achieved 683 with each approach. Finally, these dimensions are jointly assessed to highlight the overall 684 efficiency and sustainability attainable under each scenario. 685 In the following sections, the results are presented using the following notation for 686 each model: XGB refers to the XGBoost model trained with classical sampling, and XG- 687 Boost_stochastic denotes the same model trained with adaptive sampling; CNN refers to 688 the convolutional neural network trained with classical sampling, and CNN_MC_dropout 689 indicates its adaptive sampling counterpart; finally, CNN_s denotes the self-supervised 690 convolutional neural network trained with classical sampling, and CNN_MC_dropout_s 691 refers to its adaptive sampling version.692 3.1 CO 2 emissions693 The metrics recorded with CodeCarbon during the training sessions provided a detailed 694 account of both CO 2 emissions and energy consumption (kWh). The server was deployed 695 in Sweden, and CodeCarbon automatically applied the Swedish energy data to calculate 696 emissions and energy usage. Sweden was selected because it has one of the lowest emission 697 intensities for electricity generation in Europe, as published in the report presented by the 698 European Environment Agency (EEA) ‘Greenhouse gas emission intensity of electricity 699 generation in Europe’, 2025.700 Considering all the experiments carried out during this work, Figure 7 shows the rela- 701 tions between the CO 2 emissions and energy consumption vs experiments computational 702 temporal costs, subfigure 7a and 7b respectively. 703 23 The visible linear correlation is supported by a linear regression model, where the coef- 704 ficient of determination (R 2 ) approaches 1 (0.99999997 for both CO 2 emissions and energy 705 consumption). This indicates that time, according to CodeCarbon, is the primary factor 706 influencing both emissions and energy use during the training processes. Consequently, 707 the following sections will directly evaluate the environmental impact of the experiments 708 in respect of time, as this is a reliable indicator of the overall impact. 709 020000400006000080000100000 Time (s) 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 CO Emissions (kg) Lineal Regression Data (a) CO 2 emissions (kg) over time (s) ˆy = 0.00000172x +−0.00000013 020000400006000080000100000 Time (s) 0 1 2 3 4 5 Energy Consumed (kWh) Lineal Regression Data (b) Energy consumed (kWh) over time (s) ˆy = 0.00004215x +−0.00000309 Figure 7: CodeCarbon metrics with respect to duration 3.2 Computational cost710 When assessing the computational cost required to fit a surrogate model, it is necessary 711 to measure the execution time of each step in the process. Specifically, the fitting of a 712 surrogate model involves three main stages: (1) data generation through simulations, (2) 713 sample selection using a sampling method, and (3) model training. 714 The methodology proposed in this work focuses on the sampling strategies and the 715 models themselves. Therefore, in order to enable a fair comparison across all experiments, 716 the computational costs analyzed correspond to stages (1), (2) and (3). 717 Regarding data generation, in this case all simulations are performed with the same 718 physical model. Consequently, the computational cost of generating simulated samples is 719 constant, disregarding minor fluctuations due to background processes of the computing 720 system.721 However, the scope of the surrogate model strongly depends on both the quantity and 722 the quality of the available data. For this reason, it is still essential to consider the number 723 of samples, although this aspect will be analyzed in a later section, as the objective is 724 to maximize model performance with the minimum amount of data. Importantly, in 725 the context of more complex physical models, such as full Phase field simulations, the 726 computational cost of data generation becomes a critical factor, where both the number 727 of samples and the total simulation time play a decisive role. 728 The following subsections analyze the computational costs associated with each of the 729 steps outlined above.730 24 101214161820 Simulation time (s) 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 Density Figure 8: Density distribution of the time required to simulate a single sample by the phase field model. 3.2.1 Simulations temporal cost731 To approximate the simulation time of the original physical model, the computational cost 732 of all simulations was recorded. Figure 8 shows the density distribution of the simulation 733 times, where the vertical lines indicate the mean (red) and the median (green), which are 734 14.5s and 13.7s, respectively.735 3.2.2 Sampling temporal cost736 As mentioned above, two sampling methods were considered: classical and adaptive. For 737 the classical method, the computational cost corresponds to the time required to generate 738 the samples for a given sample size, namely 70, 150, 300, 500, 700. 739 In the case of adaptive sampling, the computational cost for a sample size n k must also 740 include the accumulated cost of all previous iterations, i.e., n 0 ,...,n k−1 . This is because 741 reaching n k samples requires k iterations of the method. Thus, in addition to the initial 742 classical sampling, the adaptive method is executed k times. 743 Figure 9 presents the results for the three experimental instances. Each plot shows the 744 computational cost (time) for each model and sampling size. The same colors are used 745 for each type of model, while lighter shades indicate the cost of the classical sampling 746 methodology, in contrast with the adaptive method. 747 The number of training instances and the images used do not influence the sampling 748 method, since the latter only selects which parameters of the physical model should be 749 used to generate new samples in order to improve the surrogate model uncertainty. As 750 shown in Figure 9, the cost of each method depends solely on the sample size, with no 751 impact from the model itself. Moreover, the results clearly indicate that adaptive sampling 752 entails significantly lower computational costs compared to classical sampling. 753 The difference between the two strategies is noteworthy. Adaptive sampling is iterative 754 and accumulates cost over successive iterations. Nevertheless, this does not prevent it from 755 25 100200300400500600700 Sampling size 0 2000 4000 6000 8000 10000 12000 14000 Time (s) Model CNN_MC_dropout XGB_stochastic CNN_MC_dropout_s CNN CNN_s XGB (a) Sampling cost for training instance ID1. 100200300400500600700 Sampling size 0 2000 4000 6000 8000 10000 12000 14000 Time (s) Model CNN_MC_dropout XGB_stochastic CNN_MC_dropout_s CNN CNN_s XGB (b) Sampling cost for training instance ID2. 100200300400500600700 Sampling size 0 2000 4000 6000 8000 10000 12000 14000 Time (s) Model CNN_MC_dropout XGB_stochastic CNN_MC_dropout_s CNN CNN_s XGB (c) Sampling cost for training instance ID3. Figure 9: Computational cost of sampling for each group of training instances, across all models and both sampling methods. 26 being more efficient, since compared to classical sampling it achieves substantially lower 756 computational times. This may be due to the fact that in each iteration only a small 757 number of samples (15% of the previous iteration) are added, which can lead to greater 758 efficiency compared to distance-based sampling strategies. 759 On the other hand, classical methods rely on a single initial exploration of the design 760 space, which results in much higher computational costs. This problem becomes especially 761 challenging in high-dimensional spaces, as the method aims to maximize distances between 762 samples, a task that is computationally expensive. In other words, when using distance- 763 based strategies with large sample sizes, the computational cost of classical sampling 764 grows rapidly, and the method loses efficiency. 765 When comparing the emission values of CO 2 and energy consumption for each sampling 766 method and their corresponding sample sizes and models, we can extrapolate all the results 767 and conclusions obtained for the computational cost in time, as shown in subsection 3.1 768 the corresponding R 2 between the temporal cost and emissions and the temporal cost 769 and energy consumption is practically 1 in both cases. Therefore, carbon emissions are 770 very similar for the 3 models and time instance experiments for the case of the adaptive 771 sampling, with the values showing negligible variation as the number of samples increases, 772 getting all values within 3.469· 10 −4 kg and 3.972· 10 −4 kg, and similar happens for the en- 773 ergy consumption, getting values within 8.501·10 −3 kWh and 9.733· 10 −3 kWh. In the case 774 of the classical method, the values are also similar across all models and time-instance ex- 775 periments; however, both carbon emissions and energy consumption increase considerably 776 as the number of samples grows, reaching values between 3.493· 10 −4 kg and 2.387· 10 −2 kg 777 for the CO 2 emissions and values between 8.561· 10 −3 kWh and 5.849· 10 −1 kWh for the 778 energy consumption.779 The results clearly show this effect: classical sampling struggles to identify spatially 780 scattered sets of distant samples, which has negative consequences for computational 781 efficiency. By contrast, adaptive sampling, through its iterative construction of the design 782 space and strategic selection of new samples, keeps computational costs under control. 783 3.2.3 Training temporal cost784 When training surrogate models with classical and adaptive sampling, it should be noted 785 that in the former case, the model is trained only once, whereas in the latter, training is 786 repeated for each newly added batch of samples. 787 Therefore, for adaptive sampling, the actual time required to train the k-th surrogate 788 model must consider the cumulative computational cost of all models trained from the 789 initial sample set up to the k-th iteration. 790 The computational costs for all models, across different sample sizes and both sampling 791 methods, are shown in Figure 10.792 Comparing the results across the three training instance experiments, it is evident 793 that the trends among models are consistent. It is also important to highlight that 794 27 100200300400500600700 Sampling size 0 10000 20000 30000 40000 50000 60000 Time (s) Model CNN_MC_dropout XGB_stochastic CNN_MC_dropout_s CNN CNN_s XGB (a) Training cost of surrogate models for training instance ID1. 100200300400500600700 Sampling size 0 10000 20000 30000 40000 50000 60000 Time (s) Model CNN_MC_dropout XGB_stochastic CNN_MC_dropout_s CNN CNN_s XGB (b) Training cost of surrogate models for training instance ID2. 100200300400500600700 Sampling size 0 10000 20000 30000 40000 50000 60000 Time (s) Model CNN_MC_dropout XGB_stochastic CNN_MC_dropout_s CNN CNN_s XGB (c) Training cost of surrogate models for training instance ID3. Figure 10: Computational cost of training surrogate models for each group of training instances, for both sampling methods. 28 the computational cost depends on the amount of data and the number of parameters to 795 optimize. Although the trends remain similar across the three instance sets, an increase in 796 the number of training instances directly translates into higher computational cost. This 797 effect is particularly pronounced in neural networks. For example, training a convolutional 798 neural network with classical sampling for ID1 requires approximately 7000s, whereas for 799 the same model, the cost rises to between 10000 and 11000s for ID2 and ID3. 800 The variation of computational cost across the three training instance sets is illustrated 801 in Figure 11 for convolutional neural networks and XGBoost. Similar trends are observed 802 for the remaining models.803 100200300400500600700 Sampling size 2000 4000 6000 8000 10000 Time (s) Convolutional N experiment ID1 ID2 ID3 (a) Computational cost of convolutional neural network training. 100200300400500600700 Sampling size 200 400 600 800 1000 1200 Time (s) XGBoost experiment ID1 ID2 ID3 (b) Computational cost of XGBoost training. Figure 11: Training computational cost of surrogate models across different training instance sets. 29 Again, referring to Figure 10, it is clear that, in general, training with adaptive 804 sampling incurs higher costs compared to classical sampling. As expected, cumulative 805 training costs increase iteratively with each added batch of samples, whereas classical 806 sampling trains only once.807 However, in convolutional neural networks, the relationship between sample size and 808 computational time is approximately linear for classical sampling, but not for adaptive 809 sampling. Specifically, adaptive sampling exhibits a concave curve, with the derivative 810 decreasing as more samples are added. This effect is attributed to incremental training: 811 although cross-validation is required to estimate uncertainty, the model is ultimately 812 retrained with all data incrementally, which reduces the effective training time for each 813 batch of new samples.814 In the case of the environmental cost associated with the training phase, as with the 815 sampling cost, the results and conclusions are also directly extrapolable, because of the R 2 816 values obtained in subsection 3.1. More precisely, in these cases, both CO 2 emissions and 817 energy consumption rise as the number of instances in the temporal experiments increases. 818 In addition, it can be observed that the results of the adaptive method are higher and 819 similar across all models, whereas in the classical method the increase is barely noticeable 820 in comparison. All this leads to carbon dioxide emissions values between 1.888· 10 −3 kg 821 and 8.901· 10 −2 in the case of adaptive sampling, and an energy consumption between 822 4.626· 10 −2 kWh and 2.181kWh for the same sampling method. In the case of the classical 823 sampling, the CO 2 emissions are between 1.202· 10 −3 kg and 1.866· 10 −2 kg, and energy 824 consumptions between 2.945· 10 −2 kWh and 4.573· 10 −1 kWh. 825 Finally, it is noteworthy that XGBoost exhibits particularly low computational costs 826 compared to neural networks. Although adaptive sampling increases the training cost, 827 it still improves performance relative to classical sampling in neural networks, achieving 828 better accuracy for the same or lower computational expense. 829 3.3 Models performance830 In this section, the performance metrics obtained for each model are compared. Specific- 831 ally, the coefficient of determination (R 2 ) was used to evaluate the fitting accuracy of the 832 surrogate models.833 The same plots used for comparing computational costs across models, training in- 834 stance sets, and sampling methods are employed here. These results are shown in Figure 835 12.836 As observed in Figure 12, self-attentive convolutional neural networks do not achieve 837 high performance, with R 2 values remaining below 0.80 regardless of the sampling method 838 or training instance set. However, adaptive sampling clearly improves performance com- 839 pared to classical sampling. Specifically, no classical sampling experiment achieves R 2 > 840 0.50, while adaptive sampling yields values between 0.70 and 0.80 across the three training 841 instance sets.842 30 100200300400500600700 Sampling size 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R squared Model CNN CNN_MC_dropout CNN_MC_dropout_s CNN_s XGB XGB_stochastic (a) Model performance metrics for training instance set ID1. 100200300400500600700 Sampling size 0.2 0.4 0.6 0.8 R squared Model CNN CNN_MC_dropout CNN_MC_dropout_s CNN_s XGB XGB_stochastic (b) Model performance metrics for training instance set ID2. 100200300400500600700 Sampling size 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R squared Model CNN CNN_MC_dropout CNN_MC_dropout_s CNN_s XGB XGB_stochastic (c) Model performance metrics for training instance set ID3. Figure 12: Fitting metrics obtained for each model across training instance sets and both sampling methods. 31 For convolutional neural networks, adaptive sampling consistently outperforms the 843 self-attentive variant across all cases. Moreover, for all three training instance sets, ad- 844 aptive sampling enhances the results obtained with classical sampling. The difference in 845 R 2 between the two methods decreases as the last training instance increases. Beyond 846 reducing the difference, both models achieve higher performance metrics as the number 847 of training instances grows. Notably, the best performance is obtained for ID3, while 848 results in other experiments remain very close to the highest values. 849 In the case of XGBoost, excellent results are obtained across all three training in- 850 stance sets. Specifically, the best outcomes occur for ID1 and ID2, and for ID3, XG- 851 Boost performance closely approaches that of convolutional neural networks. Regarding 852 the sampling methods, adaptive sampling improves results for ID1 relative to classical 853 sampling. For ID2, both methods yield very similar performance, and for ID3, classical 854 sampling slightly outperforms adaptive sampling. 855 Finally, it is important to note that both convolutional neural networks and XGBoost 856 achieve satisfactory initial performance, with R 2 > 0.7 in all cases. As the sample size 857 increases, fitting metrics improve gradually, approaching nearly perfect regression in some 858 instances. However, the rate of improvement is relatively slow. 859 3.4 Discussion860 Following the hypotheses and objectives outlined earlier, the following key points should 861 be considered when selecting an appropriate surrogate model: 862 • Minimize the number of training instances whenever possible. 863 • Ensure that the last training instance used is sufficiently far from the final prediction 864 instance.865 • Achieve high fitting metrics with the smallest possible number of samples. 866 • Select a model that balances training cost and fitting performance. 867 As observed in previous results, increasing the number of training instances leads to 868 higher training costs, particularly for neural networks. 869 Regarding the last training instance used, this choice carries significant weight. In- 870 stances closer to the prediction scenario tend to yield better fitting metrics. However, 871 when deploying the surrogate model, greater dependency on the physical model may 872 arise, potentially reducing the overall efficiency of the methodology. For example, in the 873 case of ID1, an efficiently trained surrogate can reduce up to 50% of the original model’s 874 computational time for predictions. For ID2, this saving reaches up to 37.5%, and for 875 ID3, only up to 12.5%.876 The number of samples used for training is critical in scenarios where the physical 877 model is computationally demanding. For instance, if one model requires 300 samples 878 32 to reach R 2 = 0.90 and another only 150, the sampling cost can be reduced by 50%. 879 When simulations require hours or days, achieving high accuracy with fewer samples is 880 particularly important.881 Finally, the balance between training cost and model accuracy must be maintained. 882 Even if all previous conditions are satisfied, the training process must remain efficient; 883 otherwise, the surrogate model will not offer meaningful computational savings relative 884 to the original model.885 Figure 13 illustrates the balance between the number of training instances, the last 886 training instance, and the trade-off between fitting accuracy and computational cost for 887 150, 500, and 700 samples. For adaptive sampling, results closest to these sample sizes are 888 shown. Total computational cost includes sampling, simulation, and training; simulation 889 cost is effectively constant for the same sample size across models. Thus, the cost reflects 890 the trade-off between sampling and training. 891 Models that achieve a good balance between accuracy and computational cost include 892 XGBoost with either classical or adaptive sampling, and convolutional neural networks 893 with classical sampling. For convolutional networks with adaptive sampling, computa- 894 tional cost increases significantly with sample size, but better results are obtained with 895 fewer samples compared to classical sampling, maintaining the trade-off. 896 Regarding training instances, ID3 experiments for XGBoost are unnecessary, as res- 897 ults are very similar to ID2, while prediction savings are much larger. Between ID1 898 and ID2, the choice should depend on simulation cost. For high-cost simulations, the 899 accuracy offered by ID1 may be sufficient considering prediction savings; if higher fitting 900 is prioritized, ID2 is preferable.901 For convolutional networks, adaptive sampling increases computational cost, but small 902 sample sizes (ID1 and ID2) achieve good results, especially when considering the predic- 903 tion cost of ID3. Specifically, with 150 samples and adaptive sampling for ID2, compu- 904 tational cost and fitting metrics are much better than those obtained with 300 samples 905 using classical sampling, as shown in Figures 13a and 13b. 906 Self-attentive convolutional networks are generally not suitable as surrogate models, 907 as the trade-off between computational cost and fitting accuracy is poor in all cases. 908 A main objective is to train a surrogate model that achieves high fitting metrics with 909 minimal sample size. Following literature findings, this work hypothesizes that adaptive 910 sampling allows training a more accurate model with fewer samples. 911 To test this hypothesis, a comparison was performed between the number of samples 912 and computational cost required to achieve the maximum R 2 with classical sampling 913 versus adaptive sampling.914 The results are summarized in Table 2 and visually in Figure 14. 915 As shown in Table 2 and Figure 14, except for ID3 experiments with XGBoost, ad- 916 aptive sampling requires fewer samples to reach the maximum classical R 2 , substantially 917 reducing simulation time. For example, in XGBoost with ID1, adaptive sampling achieves 918 33 250050007500100001250015000175002000022500 Total computational cost 0.4 0.5 0.6 0.7 0.8 0.9 R squared ID1 ID2 ID3 ID1 ID2 ID3 ID1 ID2ID3 ID1 ID2 ID3 ID1 ID2 ID3 ID1 ID2ID3 Model CNN CNN_MC_dropout CNN_MC_dropout_s CNN_s XGB XGB_stochastic (a) Total computational cost and R 2 for 150 samples 10000200003000040000500006000070000 Total computational cost 0.5 0.6 0.7 0.8 0.9 1.0 R squared ID1 ID2 ID3 ID1 ID2 ID3 ID1 ID2ID3 ID1 ID2 ID3 ID1 ID2 ID3 ID1 ID2ID3 Model CNN CNN_MC_dropout CNN_MC_dropout_s CNN_s XGB XGB_stochastic (b) Total computational cost and R 2 for 500 samples 20000400006000080000 Total computational cost 0.5 0.6 0.7 0.8 0.9 1.0 R squared ID1 ID2 ID3 ID1 ID2 ID3 ID1 ID2ID3 ID1 ID2 ID3 ID1 ID2 ID3 ID1 ID2ID3 Model CNN CNN_MC_dropout CNN_MC_dropout_s CNN_s XGB XGB_stochastic (c) Total computational cost and R 2 for 700 samples Figure 13: Trade-off between total computational cost and R 2 across training instance sets and models for different sample sizes. 34 XGB - ID1XGB - ID2XGB - ID3 CNN - ID1CNN - ID2CNN - ID3 CNN_s - ID1CNN_s - ID2CNN_s - ID3 0 2000 4000 6000 8000 10000 12000 14000 Time (s) Sampling Cl. Sampling Ad. (a) Sampling cost to reach maximum classical R 2 XGB - ID1XGB - ID2XGB - ID3 CNN - ID1CNN - ID2CNN - ID3 CNN_s - ID1CNN_s - ID2CNN_s - ID3 0 5000 10000 15000 20000 25000 Time (s) Classic Trainig Adaptive Training (b) Training cost to reach maximum classical R 2 XGB - ID1XGB - ID2XGB - ID3 CNN - ID1CNN - ID2CNN - ID3 CNN_s - ID1CNN_s - ID2CNN_s - ID3 0 100 200 300 400 500 600 700 Sampling size n classic n adaptive (c) Comparison of sample numbers to reach maximum classical R 2 Figure 14: Computational costs in sampling and training, and number of samples, required by each method to reach the maximum R 2 achieved with classical sampling. 35 ModelIDMax R 2 (classical)n (classical)Train t (classical)Sample t (classical)n (adaptive)Train t (adaptive)Sample t (adaptive) XGBID10.899500514.187479.67213850.772750.05 XGBID20.936500677.127479.672821426.403553.86 XGBID30.963500786.607479.676523434.857919.31 CNNID10.8337006726.7713877.101407981.122390.51 CNNID20.9065006311.467145.3714010237.302392.34 CNNID30.9575008131.227145.37701727.711139.70 CNN_agID10.5387001570713879.8012211618.602075.11 CNN_agID20.53370022119.4013879.8010610954.801788.75 CNN_agID30.49670026538.5013879.8010612092.201791.29 Table 2: Computational cost and number of samples required by each method to reach the maximum classical R 2 . a 57.4% saving in simulation cost. On average, across all experiments, 65.7% of simulation 919 time could be saved with adaptive sampling. 920 As previously noted, adaptive sampling is computationally more efficient than classical 921 sampling when considering the time to select samples. Combined with achieving the same 922 accuracy with fewer samples (Figure 14b), adaptive sampling provides significantly better 923 computational efficiency in most cases.924 Finally, although the cumulative computational cost of models trained with adaptive 925 sampling is higher than that of classical sampling, reaching the same fitting metrics with 926 fewer samples reduces the overall cost, as shown in Figure 14a. Except for self-attentive 927 convolutional networks, training costs are higher but more balanced across the remaining 928 models.929 4. Conclusions930 The main goal of this work has been to identify an appropriate surrogate modelling 931 strategy for the spatio-temporal nature of dendritic solidification simulations, with the 932 aim of reducing the computational cost of the original phase field model. To this end, 933 the study has examined how different modelling choices, the selection of temporal train- 934 ing instances, the sampling methodology, and the model, have influenced the number of 935 required samples, the computational cost and the final predictive performance of the sur- 936 rogate. Since each sample corresponds to an expensive phase field simulation, minimizing 937 the number of samples has been essential for reducing total computational effort. The 938 evaluation has been conducted through quantitative metrics capturing surrogate accur- 939 acy (R 2 ), computational time, training and sampling costs, and their associated energy 940 consumption and CO 2 emissions. Training time exhibited an almost perfectly linear 941 correlation with environmental indicators, thus, it has served as a consistent basis for as- 942 sessing the trade-off between sustainability, computational efficiency, and surrogate-model 943 performance.944 The comparison between the classical and adaptive sampling methods proposed re- 945 vealed clear differences. Classical sampling, based on an Optimal Latin Hypercube 946 Sampling conducted by the PSO algorithm, struggles to remain computationally efficient 947 as the number of samples increases, since the optimization is distance-based and its cost 948 grows accordingly, leading to higher sampling costs. For the proposed adaptive sampling 949 36 method, the cost associated with identifying new samples is lighter than in the classical 950 approach, because new points are added iteratively and the search process is designed 951 to minimise its computational impact. In addition, the method progressively refines the 952 design space and incorporates uncertainty—estimated through cross-validation—to guide 953 the selection of new samples. While this strategy increases training time, since the model 954 must be retrained at every iteration, it generally reduces the total number of samples 955 required to reach a given accuracy. As a result, adaptive sampling achieves better sample 956 efficiency and reduces the simulation cost in most experiments. 957 Regarding the comparison between the convolutional models—which automatically ex- 958 tract spatial features—and the XGB model—which relies on a domain-knowledge-based 959 feature transformation—the training cost of the CNNs is consistently higher, as they must 960 optimise a significantly larger number of parameters. Furthermore, as mentioned previ- 961 ously, the total training cost in the adaptive approach is also higher, since it accumulates 962 the training effort required at each iteration. This cost is further increased by the need to 963 estimate model uncertainty across the entire design space: in the current implementation, 964 this is achieved through cross-validation, which adds an additional computational burden 965 to every adaptive iteration.966 When comparing the different sets of temporal instances, it can be observed that—except967 for the self-supervised models—better predictive metrics are obtained when the final time 968 t f is closer to the target solidification state. This improvement, however, comes at the 969 expense of a slight increase in computational cost. Considering these results together with 970 the simulation time required when using the surrogate for prediction, we conclude that 971 the best balance between performance and computational efficiency is achieved with the 972 ID1 and ID2 configurations.973 Finally, in terms of the trade-off between training cost and predictive performance, the 974 XGB model achieves the best overall balance, largely due to the efficiency of the domain- 975 knowledge-based feature extraction employed. However, although adaptive sampling re- 976 duces the number of required samples for both modelling approaches, the reduction is 977 substantially more pronounced for the CNN. This is relevant because, unlike XGB—which 978 benefits heavily from the handcrafted feature transformation and thus depends on prior 979 knowledge of the problem—the CNN learns spatial representations directly from the data. 980 Consequently, when combined with the adaptive sampling strategy, the CNN can achieve 981 competitive performance with significantly fewer samples. This not only enhances its po- 982 tential for generalisation in scenarios where domain knowledge is limited, but also enables 983 a notable reduction in the number of phase field simulations required, and therefore in 984 the computational cost associated with generating the training data. 985 Future work would explore several directions aimed at further improving the modelling 986 and training efficiency of the proposed framework. First, incorporating architectures cap- 987 able of capturing both spatial and temporal correlations, such as recurrent–convolutional 988 hybrids or attention-based spatio-temporal models, would allow a more precise represent- 989 37 ation of the underlying space-time dynamics without discarding temporal dependencies 990 between frames. Second, strategies that reduce the reliance on the original phase field 991 model during inference would be investigated, for example through autoregressive sur- 992 rogate formulations that predict successive states from previous surrogate outputs while 993 controlling the accumulated error. Third, the methodology should be evaluated on more 994 computationally demanding phase field formulations, including higher-resolution 2D and 995 fully 3D cases. Fourth, developing more computationally efficient uncertainty-estimation 996 techniques would further reduce the overhead of the adaptive sampling procedure. Finally, 997 physics-informed neural network (PiNN) approaches could be considered as a means to 998 embed physical constraints directly into the surrogate, potentially improving generalisa- 999 tion and reducing the amount of training data required. 1000 Data Availability1001 Data is generated using the phase-field model described in Biner, 2017. 1002 Code Availability1003 The code used to generate the results in this study will be made publicly available in 1004 the future. In the meantime, the code may be shared upon reasonable request to the 1005 corresponding author.1006 Acknowledgments1007 This work has been supported by an Elkartek project funded by the Basque Government 1008 for basic research.1009 Author Contributions1010 Eider Garate-Perez: Conceptualization, Methodology, Software/Coding, Validation, Formal1011 Analysis, Investigation, Data Curation, Writing – Original Draft, Writing – Review & 1012 Editing, Visualization. Kerman López de Calle-Etxabe: Conceptualization, Methodo- 1013 logy, Resources, Writing – Review & Editing, Supervision, Funding Acquisition. Oihana 1014 Garcia: Software/Coding, Validation, Formal Analysis, Investigation, Writing – Original 1015 Draft, Writing – Review & Editing, Visualization. Borja Calvo: Conceptualization, Meth- 1016 odology, Writing – Review & Editing, Supervision. Meritxell Gómez-Omella: Conceptu- 1017 alization, Methodology, Software/Coding, Writing – Review & Editing. Jon Lambarri: 1018 Software/Coding, Writing – Review & Editing. 1019 38 Competing Interests1020 The authors declare no competing interests. 1021 Preprint notice1022 This manuscript is a preprint and has not yet been peer-reviewed. 1023 References1024 Afzal, A., Kim, K.-Y., & Seo, J.-w. (2017). Effects of Latin hypercube sampling on sur- 1025 rogate modeling and optimization. International Journal of Fluid Machinery and 1026 Systems, 10(3), 240–253. https://doi.org/10.5293/IJFMS.2017.10.3.240 1027 Ahmad, O., Kumar, N., Mukherjee, R., & Bhowmick, S. (2023). Accelerating micro- 1028 structure modeling via machine learning: A method combining Autoencoder and 1029 ConvLSTM. Physical Review Materials, 7(8), 083802. https://doi.org/10.1103/ 1030 PhysRevMaterials.7.0838021031 Alizadeh, R., Allen, J. K., & Mistree, F. (2020). Managing computational complexity 1032 using surrogate models: A critical review. Research in Engineering Design, 31(3), 1033 275–298. https://doi.org/10.1007/s00163-020-00336-7 1034 Azarhoosh, Z., & Ilchi Ghazaan, M. (2025). A Review of Recent Advances in Surrogate 1035 Models for Uncertainty Quantification of High-Dimensional Engineering Applic- 1036 ations. Computer Methods in Applied Mechanics and Engineering, 433, 117508. 1037 https://doi.org/10.1016/j.cma.2024.117508 1038 Aziz, M., & Tayarani- N., M. H. (2014). An adaptive memetic Particle Swarm Optimiza- 1039 tion algorithm for finding large-scale Latin hypercube designs. Engineering Applic- 1040 ations of Artificial Intelligence, 36, 222–237. https://doi.org/10.1016/j.engappai. 1041 2014.07.0211042 Bhagat, K., & Rudraraju, S. (2023). Modeling of dendritic solidification and numerical 1043 analysis of the phase-field approach to model complex morphologies in alloys. En- 1044 gineering with Computers, 39(4), 2345–2363. https://doi.org/10.1007/s00366-022- 1045 01767-71046 Biner, S. B. (2017). Programming Phase-Field Modeling. Springer International Publish- 1047 ing. https://doi.org/10.1007/978-3-319-41196-5 1048 Borisut, P., & Nuchitprasittichai, A. (2023). Adaptive Latin Hypercube Sampling for a 1049 Surrogate-Based Optimization with Artificial Neural Network. Processes, 11(11), 1050 3232. https://doi.org/10.3390/pr11113232 1051 Brophy, J., & Lowd, D. (2022). Instance-Based Uncertainty Estimation for Gradient- 1052 Boosted Regression Trees [Version Number: 2]. https://doi.org/10.48550/ARXIV. 1053 2205.114121054 39 Chen, X., Ding, M., Wang, X., Xin, Y., Mo, S., Wang, Y., Han, S., Luo, P., Zeng, G., & 1055 Wang, J. (2024). Context Autoencoder for Self-supervised Representation Learn- 1056 ing. International Journal of Computer Vision, 132(1), 208–223. https://doi.org/ 1057 10.1007/s11263-023-01852-41058 Choi, J. Y., Xue, T., Liao, S., & Cao, J. (2024). Accelerating phase-field simulation of 1059 three-dimensional microstructure evolution in laser powder bed fusion with com- 1060 posable machine learning predictions. Additive Manufacturing, 79, 103938. https: 1061 //doi.org/10.1016/j.addma.2023.103938 1062 Chollet, F., et al. (2015). Keras [Accessed: 2025-07-01]. 1063 Courty, B., Schmidt, V., Luccioni, S., & and, G.-K. (2024, May). Mlco2/codecarbon: 1064 V2.4.1 (Version v2.4.1). Zenodo. https://doi.org/10.5281/zenodo.11171501 1065 Dantzig, J. A., & Rappaz, M. (2009). Solidification. EPFL press CRC press. 1066 Di Fiore, F., Nardelli, M., & Mainini, L. (2024). Active Learning and Bayesian Optim- 1067 ization: A Unified Perspective to Learn with a Goal. Archives of Computational 1068 Methods in Engineering, 31(5), 2985–3013. https://doi.org/10.1007/s11831-024- 1069 10064-z1070 Dobravec, T., Mavrič, B., Zahoor, R., & Šarler, B. (2023). A coupled domain–boundary 1071 type meshless method for phase-field modelling of dendritic solidification with the 1072 fluid flow. International Journal of Numerical Methods for Heat & Fluid Flow, 1073 33(8), 2963–2981. https://doi.org/10.1108/HFF-03-2023-0131 1074 Duquesnoy, M., Liu, C., Dominguez, D. Z., Kumar, V., Ayerbe, E., & Franco, A. A. (2023). 1075 Machine learning-assisted multi-objective optimization of battery manufacturing 1076 from synthetic data generated by physics-based simulations. Energy Storage Ma- 1077 terials, 56, 50–61. https://doi.org/10.1016/j.ensm.2022.12.040 1078 Escobar-Cuevas, H., Cuevas, E., Avila, K., & Avalos, O. (2024). An advanced initialization 1079 technique for metaheuristic optimization: A fusion of Latin hypercube sampling 1080 and evolutionary behaviors. Computational and Applied Mathematics, 43(4), 234. 1081 https://doi.org/10.1007/s40314-024-02744-0 1082 Fakour, F., Mosleh, A., & Ramezani, R. (2024). A Structured Review of Literature on 1083 Uncertainty in Machine Learning & Deep Learning [Version Number: 1]. https: 1084 //doi.org/10.48550/ARXIV.2406.00332 1085 Fassi, H. F., & Aniyou, H. (2023). Towards an Optimal Motor Mounting Bracket Using 1086 Topology Optimization Combined with Sustainability and Manufacturing Cost 1087 Analysis. Management and Production Engineering Review. https://doi.org/10. 1088 24425/mper.2023.1472081089 Forrester, A. I., & Keane, A. J. (2009). Recent advances in surrogate-based optimization. 1090 Progress in Aerospace Sciences, 45(1-3), 50–79. https://doi.org/10.1016/j.paerosci. 1091 2008.11.0011092 40 Freitag, C., Berners-Lee, M., Widdicks, K., Knowles, B., Blair, G. S., & Friday, A. (2021). 1093 The real climate and transformative impact of ict: A critique of estimates, trends, 1094 and regulations. Patterns, 2(9). 1095 Fuhg, J. N. (2019). Adaptive surrogate models for parametric studies [Version Number: 1096 1]. https://doi.org/10.48550/ARXIV.1905.05345 1097 Fuhg, J. N., Fau, A., & Nackenhorst, U. (2021). State-of-the-Art and Comparative Review 1098 of Adaptive Sampling Methods for Kriging. Archives of Computational Methods in 1099 Engineering, 28(4), 2689–2747. https://doi.org/10.1007/s11831-020-09474-6 1100 Gal, Y., & Ghahramani, Z. (2015). Dropout as a Bayesian Approximation: Representing 1101 Model Uncertainty in Deep Learning [Version Number: 6]. https://doi.org/10. 1102 48550/ARXIV.1506.021421103 Goswami, S., Anitescu, C., Chakraborty, S., & Rabczuk, T. (2020). Transfer learning 1104 enhanced physics informed neural network for phase-field modeling of fracture. 1105 Theoretical and Applied Fracture Mechanics, 106, 102447. https://doi.org/10. 1106 1016/j.tafmec.2019.1024471107 Greenhouse gas emission intensity of electricity generation in europe. (2025, June 27). 1108 European Environment Agency. Retrieved October 14, 2025, from https://w. 1109 eea.europa.eu/en/analysis/indicators/greenhouse-gas-emission-intensity-of-1 1110 Heid, E., McGill, C. J., Vermeire, F. H., & Green, W. H. (2023). Characterizing Uncer- 1111 tainty in Machine Learning for Chemistry. Journal of Chemical Information and 1112 Modeling, 63(13), 4012–4029. https://doi.org/10.1021/acs.jcim.3c00373 1113 Herman, E., Stewart, J. A., & Dingreville, R. (2020). A data-driven surrogate model 1114 to rapidly predict microstructure morphology during physical vapor deposition. 1115 Applied Mathematical Modelling, 88, 589–603. https://doi.org/10.1016/j.apm. 1116 2020.06.0461117 Herten, J. V. D., Steenkiste, T. V., Couckuyt, I., & Dhaene, T. (2017, June). Surrogate 1118 Modelling with Sequential Design for Expensive Simulation Applications. In D. 1119 Cvetkovic (Ed.), Computer Simulation. InTech. https://doi.org/10.5772/67739 1120 Hoffmann, L., Fortmeier, I., & Elster, C. (2021). Uncertainty quantification by ensemble 1121 learning for computational optical form measurements. Machine Learning: Science 1122 and Technology, 2(3), 035030. https://doi.org/10.1088/2632-2153/ac0495 1123 Hüllermeier, E., & Waegeman, W. (2021). Aleatoric and epistemic uncertainty in machine 1124 learning: An introduction to concepts and methods. Machine Learning, 110(3), 1125 457–506. https://doi.org/10.1007/s10994-021-05946-3 1126 Jarfors, A. E. W., Bogdanoff, T., & Lattanzi, L. (2024). Functionally integrated castings 1127 (Giga-castings) for body in white applications: Consequences for materials use and 1128 mix in automotive manufacturing. Matériaux & Techniques, 112(6), 605. https: 1129 //doi.org/10.1051/mattech/2025005 1130 Javaid, M., Haleem, A., Singh, R. P., Suman, R., & Rab, S. (2021). Role of additive man- 1131 ufacturing applications towards environmental sustainability. Advanced Industrial 1132 41 and Engineering Polymer Research, 4(4), 312–322. https://doi.org/10.1016/j. 1133 aiepr.2021.07.0051134 Ji, K., Zhong, M., & Karma, A. (2025). Phase-field model of alloy solidification far from 1135 chemical equilibrium at the solid-liquid interface. Physical Review Research, 7(3), 1136 033128. https://doi.org/10.1103/vyd6-nj4h 1137 Jiang, J., Xiong, Y., Zhang, Z., & Rosen, D. W. (2022). Machine learning integrated 1138 design for additive manufacturing. Journal of Intelligent Manufacturing, 33(4), 1139 1073–1086. https://doi.org/10.1007/s10845-020-01715-6 1140 Kaminsky, A. L., Wang, Y., & Pant, K. (2021). An Efficient Batch K-Fold Cross-Validation 1141 Voronoi Adaptive Sampling Technique for Global Surrogate Modeling. Journal of 1142 Mechanical Design, 143(1), 011706. https://doi.org/10.1115/1.4047155 1143 Khatouri, H., Benamara, T., Breitkopf, P., & Demange, J. (2022). Metamodeling tech- 1144 niques for CPU-intensive simulation-based design optimization: A survey. Advanced 1145 Modeling and Simulation in Engineering Sciences, 9(1), 1. https://doi.org/10. 1146 1186/s40323-022-00214-y1147 Kobayashi, R. (1993). Modeling and numerical simulations of dendritic crystal growth. 1148 Physica D: Nonlinear Phenomena, 63(3-4), 410–423. https://doi.org/10.1016/ 1149 0167-2789(93)90120-P1150 Kurz, W., Rappaz, M., & Trivedi, R. (2021). Progress in modelling solidification micro- 1151 structures in metals and alloys. Part I: Dendrites from 2001 to 2018. International 1152 Materials Reviews, 66(1), 30–76. https://doi.org/10.1080/09506608.2020.1757894 1153 Lavinas, Y., Haut, N., Punch, W., Banzhaf, W., & Cussat-Blanc, S. (2024). Adaptive 1154 Sampling of Biomedical Images with Cartesian Genetic Programming. In M. Af- 1155 fenzeller, S. M. Winkler, A. V. Kononova, H. Trautmann, T. Tušar, P. Machado 1156 & T. Bäck (Eds.), Parallel Problem Solving from Nature – PPSN XVIII (p. 256– 1157 272, Vol. 15148). Springer Nature Switzerland. https://doi.org/10.1007/978-3- 1158 031-70055-2_161159 Lee, H., & Kim, D. (2023). Recent Computational Approaches for Accelerating Dendrite 1160 Growth Prediction: A Short Review. Multiscale Science and Engineering, 5(3-4), 1161 119–125. https://doi.org/10.1007/s42493-024-00098-7 1162 Li, W., Fang, R., Jiao, J., Vassilakis, G. N., & Zhu, J. (2024). Tutorials: Physics-informed 1163 machine learning methods of computing 1D phase-field models. APL Machine 1164 Learning, 2(3), 031101. https://doi.org/10.1063/5.0205159 1165 Liu, H., Ong, Y.-S., & Cai, J. (2018). A survey of adaptive sampling for global metamod- 1166 eling in support of simulation-based complex engineering design. Structural and 1167 Multidisciplinary Optimization, 57(1), 393–416. https://doi.org/10.1007/s00158- 1168 017-1739-81169 Liu, Z., Ma, Q., Zhang, T., Zhao, S., Gao, X., Sun, T., & Dai, Y. (2025). Quantitative 1170 modeling and uncertainty estimation for small-sample LIBS using Gaussian neg- 1171 42 ative log-likelihood and monte carlo dropout methods. Optics & Laser Technology, 1172 181, 111720. https://doi.org/10.1016/j.optlastec.2024.111720 1173 Lookman, T., Balachandran, P. V., Xue, D., & Yuan, R. (2019). Active learning in mater- 1174 ials science with emphasis on adaptive sampling using uncertainties for targeted 1175 design. npj Computational Materials, 5(1), 21. https://doi.org/10.1038/s41524- 1176 019-0153-81177 Luo, J., Ma, X., Ji, Y., Li, X., Song, Z., & Lu, W. (2023). Review of machine learning-based 1178 surrogate models of groundwater contaminant modeling. Environmental Research, 1179 238, 117268. https://doi.org/10.1016/j.envres.2023.117268 1180 Malinin, A., Prokhorenkova, L., & Ustimenko, A. (2021). Uncertainty in gradient boosting 1181 via ensembles [Poster presentation]. International Conference on Learning Repres- 1182 entations (ICLR).1183 Mao, S., Cao, Y., Chen, W., & Sun, D. (2024). An anisotropic lattice Boltzmann - phase 1184 field model for dendrite growth and movement in rapid solidification of binary 1185 alloys. npj Computational Materials, 10(1), 63. https://doi.org/10.1038/s41524- 1186 024-01245-21187 Mao, W., Chen, J., Liu, J., & Liang, X. (2023). Self-Supervised Deep Domain-Adversarial 1188 Regression Adaptation for Online Remaining Useful Life Prediction of Rolling 1189 Bearing Under Unknown Working Condition. IEEE Transactions on Industrial 1190 Informatics, 19(2), 1227–1237. https://doi.org/10.1109/TII.2022.3172704 1191 McKay, M. D., Beckman, R. J., & Conover, W. J. (1979). A Comparison of Three Methods 1192 for Selecting Values of Input Variables in the Analysis of Output from a Computer 1193 Code. Technometrics, 21(2), 239. https://doi.org/10.2307/1268522 1194 Migdady, A., Khamayseh, Y., AlZoubi, O., & Yassein, M. B. (2025). An Adaptive Query 1195 Approach for Extracting Medical Images for Disease Detection Applications. Ara- 1196 bian Journal for Science and Engineering, 50(2), 1127–1142. https://doi.org/10. 1197 1007/s13369-024-09152-w1198 Montes De Oca Zapiain, D., Stewart, J. A., & Dingreville, R. (2021). Accelerating phase- 1199 field-based microstructure evolution predictions via surrogate models trained by 1200 machine learning methods. npj Computational Materials, 7(1), 3. https://doi.org/ 1201 10.1038/s41524-020-00471-81202 Nyamekye, P., Lakshmanan, R., Tepponen, V., & Westman, S. (2024). Sustainability as- 1203 pects of additive manufacturing: Leveraging resource efficiency via product design 1204 optimization and laser powder bed fusion. Heliyon, 10(1), e23152. https://doi.org/ 1205 10.1016/j.heliyon.2023.e231521206 Padarian, J., Minasny, B., & McBratney, A. (2022). Assessing the uncertainty of deep 1207 learning soil spectral models using Monte Carlo dropout. Geoderma, 425, 116063. 1208 https://doi.org/10.1016/j.geoderma.2022.116063 1209 Palmer, G., Du, S., Politowicz, A., Emory, J. P., Yang, X., Gautam, A., Gupta, G., 1210 Li, Z., Jacobs, R., & Morgan, D. (2022). Calibration after bootstrap for accurate 1211 43 uncertainty quantification in regression models. npj Computational Materials, 8(1), 1212 115. https://doi.org/10.1038/s41524-022-00794-8 1213 Panagiotopoulou, V. C., Stavropoulos, P., & Chryssolouris, G. (2022). A critical review 1214 on the environmental impact of manufacturing: A holistic perspective. The Inter- 1215 national Journal of Advanced Manufacturing Technology, 118(1-2), 603–625. https: 1216 //doi.org/10.1007/s00170-021-07980-w 1217 Park, J., Ryu, I., Ryu, D., & Lee, Y. (2025). An efficient near-optimal Latin hypercube 1218 design for large datasets. Journal of Mechanical Science and Technology, 39(1), 1219 315–328. https://doi.org/10.1007/s12206-024-1229-z 1220 Peivaste, I., Siboni, N. H., Alahyarizadeh, G., Ghaderi, R., Svendsen, B., Raabe, D., & 1221 Mianroodi, J. R. (2022). Machine-learning-based surrogate modeling of microstruc- 1222 ture evolution using phase-field. Computational Materials Science, 214, 111750. 1223 https://doi.org/10.1016/j.commatsci.2022.111750 1224 Rahaman, R., & thiery alexandre, a. (2021). Uncertainty quantification and deep en- 1225 sembles. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang & J. W. Vaughan 1226 (Eds.), Advances in neural information processing systems (p. 20063–20075, Vol. 34).1227 Curran Associates, Inc.1228 Ravutla, S., Bai, A., Realff, M. J., & Boukouvala, F. (2025). Effects of Surrogate Hybrid- 1229 ization and Adaptive Sampling for Simulation-Based Optimization. Industrial & 1230 Engineering Chemistry Research, 64(18), 9228–9251. https://doi.org/10.1021/acs. 1231 iecr.4c033031232 Samadian, D., Muhit, I. B., & Dawood, N. (2025). Application of Data-Driven Surrogate 1233 Models in Structural Engineering: A Literature Review. Archives of Computational 1234 Methods in Engineering, 32(2), 735–784. https://doi.org/10.1007/s11831-024- 1235 10152-01236 Seiz, M., Kellner, M., & Nestler, B. (2023). Simulation of dendritic–eutectic growth with 1237 the phase-field method. Acta Materialia, 254, 118965. https://doi.org/10.1016/j. 1238 actamat.2023.1189651239 Seoh, R. (2020). Qualitative Analysis of Monte Carlo Dropout [Version Number: 1]. https: 1240 //doi.org/10.48550/ARXIV.2007.01720 1241 Tang, C., & Du, H. (2022). Phase Field Modelling of Dendritic Solidification Under Ad- 1242 ditive Manufacturing Conditions. JOM, 74(8), 2996–3009. https://doi.org/10. 1243 1007/s11837-022-05310-31244 Tenne, Y. (2015). Initial sampling methods in metamodel-assisted optimization. Engineer- 1245 ing with Computers, 31(4), 661–680. https://doi.org/10.1007/s00366-014-0372-z 1246 Tourret, D., Liu, H., & LLorca, J. (2022). Phase-field modeling of microstructure evolu- 1247 tion: Recent applications, perspectives and challenges. Progress in Materials Sci- 1248 ence, 123, 100810. https://doi.org/10.1016/j.pmatsci.2021.100810 1249 44 Tyralis, H., & Papacharalampous, G. (2024). A review of predictive uncertainty estimation 1250 with machine learning. Artificial Intelligence Review, 57(4), 94. https://doi.org/ 1251 10.1007/s10462-023-10698-81252 Usman, M., Ahmad, I., Ahsan, M., & Caliskan, H. (2024). Prediction and optimization 1253 of emissions in cement manufacturing plant under uncertainty by using artificial 1254 intelligence-based surrogate modeling. Environment, Development and Sustainab- 1255 ility. https://doi.org/10.1007/s10668-024-05068-5 1256 Vasylenko, A., Gamon, J., Duff, B. B., Gusev, V. V., Daniels, L. M., Zanella, M., Shin, 1257 J. F., Sharp, P. M., Morscher, A., Chen, R., Neale, A. R., Hardwick, L. J., Clar- 1258 idge, J. B., Blanc, F., Gaultois, M. W., Dyer, M. S., & Rosseinsky, M. J. (2021). 1259 Element selection for crystalline inorganic solid discovery guided by unsupervised 1260 machine learning of experimentally explored chemistry. Nature Communications, 1261 12(1), 5561. https://doi.org/10.1038/s41467-021-25343-7 1262 Viana, F. A. C., Gogu, C., & Goel, T. (2021). Surrogate modeling: Tricks that endured 1263 the test of time and some recent developments. Structural and Multidisciplinary 1264 Optimization, 64(5), 2881–2908. https://doi.org/10.1007/s00158-021-03001-2 1265 Viardin, A., Nöth, K., Pickmann, C., & Sturz, L. (2025). Automatic Detection of Dendritic 1266 Microstructure Using Computer Vision Deep Learning Models Trained with Phase 1267 Field Simulations. Integrating Materials and Manufacturing Innovation, 14(1), 89– 1268 105. https://doi.org/10.1007/s40192-025-00392-8 1269 Wang, J., Li, Y., Gao, R. X., & Zhang, F. (2022). Hybrid physics-based and data-driven 1270 models for smart manufacturing: Modelling, simulation, and explainability. Journal 1271 of Manufacturing Systems, 63, 381–391. https://doi.org/10.1016/j.jmsy.2022.04. 1272 0041273 Wang, Q., Li, Y., & Li, R. (2024). Ecological footprints, carbon emissions, and energy 1274 transitions: The impact of artificial intelligence (ai). Humanities and Social Sci- 1275 ences Communications, 11(1), 1–18. 1276 Wang, S., Lv, L., Du, L., & Song, X. (2019). An improved lhs approach for constrained 1277 design space based on successive local enumeration algorithm. 2019 IEEE 9th An- 1278 nual International Conference on CYBER Technology in Automation, Control, and 1279 Intelligent Systems (CYBER), 896–899. https://doi.org/10.1109/CYBER46603. 1280 2019.90666771281 Würth, T., Krauß, C., Zimmerling, C., & Kärger, L. (2023). Physics-informed neural 1282 networks for data-free surrogate modelling and engineering optimization – An ex- 1283 ample from composite manufacturing. Materials & Design, 231, 112034. https: 1284 //doi.org/10.1016/j.matdes.2023.112034 1285 Xiao, Q., Wang, J., Jiang, B., Yang, W., & Yang, X. (2022). Multi-objective optimization 1286 of squirrel cage fan for range hood based on Kriging model. Proceedings of the 1287 Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering 1288 Science, 236(1), 219–232. https://doi.org/10.1177/0954406221995863 1289 45 Xue, D., Li, Y., Zhang, H., Tong, X., Gao, B., & Yu, J. (2024). Reliability-based robust 1290 optimization design for tolerance of aerospace thin-walled components based on 1291 surrogate model. Advanced Engineering Informatics, 62, 102754. https://doi.org/ 1292 10.1016/j.aei.2024.1027541293 Yan, H., Yu, H., Zhu, S., Yin, Y., & Guo, L. (2024). Machine learning based framework 1294 for rapid forecasting of the crack propagation. Engineering Fracture Mechanics, 1295 307, 110278. https://doi.org/10.1016/j.engfracmech.2024.110278 1296 Yu, Y., Wang, L., Zhou, J., Li, H., Li, Y., Yan, W., & Lin, F. (2022). Impact of fluid flow 1297 on the dendrite growth and the formation of new grains in additive manufacturing. 1298 Additive Manufacturing, 55, 102832. https://doi.org/10.1016/j.addma.2022.102832 1299 Zeevi, T., Venkataraman, R., Staib, L. H., & Onofrey, J. A. (2024). Monte-Carlo Fre- 1300 quency Dropout for Predictive Uncertainty Estimation in Deep Learning. 2024 1301 IEEE International Symposium on Biomedical Imaging (ISBI), 1–5. https://doi. 1302 org/10.1109/ISBI56570.2024.10635511 1303 Zeng, H.-b., Ai, X.-g., Chen, M., Guan, R., Chao, Y.-f., & Zhang, J.-c. (2024). Phase- 1304 field simulation study on dendritic growth behavior during bilateral directional 1305 solidification. Materials Today Communications, 40, 109618. https://doi.org/10. 1306 1016/j.mtcomm.2024.1096181307 46