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

physics.comp-phFeb 17, 2026

Adaptive Uncertainty-Guided Surrogates for Efficient phase field Modeling of Dendritic Solidification

Authors: Eider Garate-Perez, Kerman López de Calle-Etxabe, Oihana Garcia, Borja Calvo, Meritxell Gómez-Omella, Jon Lambarri

Organizations: Intelligent Information Systems, Tekniker, Iñaki Goenaga 5, Eibar (20600) Spain · Faculty of Informatics UPV-EHU, Paseo Manuel de Lardizabal 1, Donostia (20700) Spain · Department of Computer Sciences and Artificial Intelligence, UPV-EHU, Donostia (20700) Spain · Advanced Manufacturing Technologies, Tekniker, Iñaki Goenaga 5, Eibar (20600) Spain

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.

Figures & tables

Parameter	Definition	Min value	Max value
$\tau$	Characteristic time scale	$1.72\cdotp 10^{-4}$	$6\cdotp 10^{-4}$
$\varepsilon$	Interface thickness	$0.005$	$1.95\cdotp 10^{-2}$
$\kappa$	Diffusion constant	1	3.6
$\delta$	Related to anisotropy	0.001	0.04
j	Dendritic symmetry	4	6
$\alpha$	Constant related to thermodynamic driving force	0.4	0.9

Table 1: Minimum and maximum feasible values for each physical model in the phase-field model.

Figure 1: Extracted spatial features for the XGBoost model and for $(i,j)$ pixel.

Figure 2: Architecture of the convolutional neuronal network. $t_{0}$ , $t_{1}$ , …, $t_{n}$ are the training temporal instances.

Figure 3: Architecture of the Variational Auto Encoder used in the self-supervised CNN

Figure 4:

7

points LHS sampling for the

[0,1]\times[0,1]

space.

Figure 5: Transformation toward the local and global optimum. The example shows the local optimum case; the global case is symmetric.

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

U^{(i)}_{\text{mixed}}=w_{\mathrm{liq}}\,\bar{U}^{(i)}_{\mathrm{liq}}+w_{\mathrm{int}}\,\bar{U}^{(i)}_{\mathrm{int}}+w_{\mathrm{sol}}\,\bar{U}^{(i)}_{\mathrm{sol}}

Algorithm 2 Adaptive Sampling Based on Mixed Uncertainty

Panel of Figure 7 — Figure 7: CodeCarbon metrics with respect to duration

Figure 8: Density distribution of the time required to simulate a single sample by the phase field model.

Panel of Figure 9 — Figure 9: Computational cost of sampling for each group of training instances, across all models and both sampling methods.

Panel of Figure 10 — Figure 10: Computational cost of training surrogate models for each group of training instances, for both sampling methods.

Panel of Figure 11 — Figure 11: Training computational cost of surrogate models across different training instance sets.

Panel of Figure 12 — Figure 12: Fitting metrics obtained for each model across training instance sets and both sampling methods.

Panel of Figure 13 — Figure 13: Trade-off between total computational cost and $R^{2}$ across training instance sets and models for different sample sizes.

Model	ID	Max $R^{2}$ (classical)	n (classical)	Train t (classical)	Sample t (classical)	n (adaptive)	Train t (adaptive)	Sample t (adaptive)
XGB	ID1	0.899	500	514.18	7479.67	213	850.77	2750.05
XGB	ID2	0.936	500	677.12	7479.67	282	1426.40	3553.86
XGB	ID3	0.963	500	786.60	7479.67	652	3434.85	7919.31
CNN	ID1	0.833	700	6726.77	13877.10	140	7981.12	2390.51
CNN	ID2	0.906	500	6311.46	7145.37	140	10237.30	2392.34
CNN	ID3	0.957	500	8131.22	7145.37	70	1727.71	1139.70

Table 2: Computational cost and number of samples required by each method to reach the maximum classical

R^{2}

Panel of Figure 14 — 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.

Surrogate Modeling

Surrogate modeling builds faster or more interpretable approximations of expensive simulations and black-box models.

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A Hybrid GNN-FEM Framework for Phase-Field Fracture Simulation. Physics-Preserving Hybridization for Generalizable Surrogate Modeling

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Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, 34141, Republic of Korea · KAIST InnoCORE PRISM-AI Center, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, 34141, Republic of Korea · Department of AX, College of AI, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, 34141, Republic of Korea

Structural Mechanics PDE Surrogate Modeling Physics-Informed ML Computational Materials Science

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Scientific machine learning (SciML) has emerged as a promising approach for accelerating simulations of complex physical systems, yet achieving physically consistent and generalizable predictions for nonlinear, history-dependent problems remains a central challenge. In this study, we propose a hybrid GNN--FEM framework for efficient and generalizable phase-field fracture modeling. While phase-field approaches provide a robust variational framework for simulating complex crack evolution, their high computational cost limits practical applications because they require solving coupled, nonlinear, and history-dependent systems within an incremental finite element procedure. To address this challenge, a graph neural network surrogate is integrated into the conventional staggered scheme, replacing the phase-field update at each load increment while retaining the FEM-based displacement solver to enforce mechanical equilibrium and boundary conditions. By preserving the incremental solution structure, the framework remains consistent with history-dependent fracture evolution without requiring the surrogate to approximate the full solution trajectory. This selective surrogate strategy emphasizes the identification of a physically meaningful and incrementally structured learning target, rather than relying on brute-force data generation to learn the full fracture process. The proposed framework achieves strong generalization across varying geometries, loading conditions, material properties, and discretizations through dimensionless feature design, a graph-based formulation on mesh-based domains, and a physics-informed loss derived from the governing phase-field equation. Numerical experiments demonstrate that the hybrid approach reduces computational cost while maintaining accuracy compared with conventional FEM, and exhibits robust predictive performance across diverse problem settings.

Abstract

Surrogate Modeling

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