Neural surrogates for crystal growth dynamics with variable supersaturation: explicit vs. implicit conditioning

Daniele Lanzoni; Francesco Montalenti; Matteo Rigoni; Roberto Bergamaschini

arxiv: 2604.21753 · v1 · submitted 2026-04-23 · ❄️ cond-mat.mtrl-sci · cond-mat.mes-hall· cs.CE· cs.LG

Neural surrogates for crystal growth dynamics with variable supersaturation: explicit vs. implicit conditioning

Matteo Rigoni , Daniele Lanzoni , Francesco Montalenti , Roberto Bergamaschini This is my paper

Pith reviewed 2026-05-09 21:45 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.mes-hallcs.CEcs.LG

keywords neural surrogatescrystal growthAllen-Cahn dynamicssupersaturation conditioningconvolutional recurrent networksexplicit vs implicitphase-field modelingfaceting

0 comments

The pith

Explicit conditioning on supersaturation produces higher-fidelity neural surrogates for crystal growth than implicit inference from short input sequences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors train convolutional recurrent neural networks on sequences from Allen-Cahn simulations that include kinetic anisotropy to produce faceted crystals. One architecture receives the supersaturation value directly as an extra input channel together with a single starting frame; the other receives only a short run of prior frames and must infer the driving parameter internally before continuing the sequence. Systematic tests on datasets of different sizes show that the explicit-input version matches the reference shapes and growth speeds more closely, while the implicit version reaches similar accuracy only after the training set is enlarged. Both versions remain stable when the spatial domain is enlarged by a factor of 256 and when the predicted sequence is extended more than tenfold beyond the training length.

Core claim

Convolutional recurrent networks can act as accurate surrogates for Allen-Cahn crystal-growth dynamics when supersaturation is supplied explicitly as an input parameter, reproducing both global growth rates and local faceted morphology with high fidelity; the same networks can also infer the parameter from a short input sequence, but require substantially more training data to reach comparable accuracy.

What carries the argument

Convolutional recurrent neural networks that either accept supersaturation as an explicit scalar input or infer it implicitly by processing a short input mini-sequence before predicting the continuation.

If this is right

Explicit conditioning reproduces ground-truth crystal profiles with high fidelity across the tested supersaturation range even when the training set is small.
Implicit mini-sequence conditioning reaches comparable accuracy only after the training dataset is enlarged.
Trained models remain accurate on spatial domains 256 times larger than those used in training.
Prediction error stays limited when the generated sequences are extended more than ten times beyond the training length.
Both architectures consistently capture the global effect of supersaturation on growth speed and its local effect on facet formation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same explicit-conditioning strategy could be applied to other phase-field models to create fast surrogates for different material systems or driving forces.
Hybrid networks that combine a short implicit check with explicit parameter input might remain robust when the true supersaturation is only known approximately.
Because error accumulates slowly, the surrogates could support long-time optimization loops in which supersaturation is varied to achieve desired crystal shapes.

Load-bearing premise

The numerical Allen-Cahn solutions with kinetic anisotropy form a representative, noise-free distribution that lets the networks learn dynamics capable of generalizing to new supersaturation values, domain sizes, and sequence lengths.

What would settle it

Running a fresh numerical integration of the Allen-Cahn equation at a supersaturation value held out from training and finding that the explicit model's predicted interface positions deviate by more than a few percent from that reference solution.

Figures

Figures reproduced from arXiv: 2604.21753 by Daniele Lanzoni, Francesco Montalenti, Matteo Rigoni, Roberto Bergamaschini.

**Figure 2.** Figure 2: FIG. 2. (a) Schematics of the used convolutional recurrent [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Training and validation losses during training for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Evolution sequence for a representative case (∆ [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Evolution sequence for a representative case (∆ [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Distribution of the maximum of MAE for the predic [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. (a) Dependency of the maximum of MAE distribu [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Evolution sequence for crystal growth under the same conditions of training set but on a 2048 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Evolution sequence for crystal growth starting from a low coverage configuration of [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

read the original abstract

Simulations of crystal growth are performed by using Convolutional Recurrent Neural Network surrogate models, trained on a dataset of time sequences computed by numerical integration of Allen-Cahn dynamics including faceting via kinetic anisotropy. Two network architectures are developed to take into account the effects of a variable supersaturation value. The first infers it implicitly by processing an input mini-sequence of a few evolution frames and then returns a consistent continuation of the evolution. The second takes the supersaturation parameter as an explicit input along with a single initial frame and predicts the entire sequence. The two models are systematically tested to establish strengths and weaknesses, comparing the prediction performance for models trained on datasets of different size and, in the first architecture, different lengths of input mini-sequence. The analysis of point-wise and mean absolute errors shows how the explicit parameter conditioning guarantees the best results, reproducing with high-fidelity the ground-truth profiles. Comparable results are achievable by the mini-sequence approach only when using larger training datasets. The trained models show strong conditioning by the supersaturation parameter, consistently reproducing its overall impact on growth rates as well as its local effect on the faceted morphology. Moreover, they are perfectly scalable even on 256 times larger domains and can be successfully extended to more than 10 times longer sequences with limited error accumulation. The analysis highlights the potential and limits of these approaches in view of their general exploitation for crystal growth simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Explicit conditioning on supersaturation outperforms implicit inference from mini-sequences in these CRNN surrogates, with better data efficiency and clear scalability, though details on parameter sampling remain thin.

read the letter

The main point is that giving the supersaturation value directly to the network produces lower errors and works with smaller training sets than forcing the model to infer it from a short sequence of frames. This holds in their tests on faceted crystal growth modeled by the Allen-Cahn equation with kinetic anisotropy. The explicit model takes the parameter plus one initial frame and outputs the full trajectory, while the implicit version processes a mini-sequence and continues the evolution. Across dataset sizes the explicit approach matches ground-truth profiles more closely on both point-wise and mean absolute error. The implicit model needs noticeably more data to reach comparable performance. Both versions reproduce the expected changes in growth rate and local facet shape when supersaturation varies. They also run without major issues on domains 256 times larger and sequences more than 10 times longer, with only limited error buildup over time. The head-to-head comparison on data efficiency and the scalability checks are the concrete contributions here. Prior work on neural surrogates for phase-field models exists, but this specific test of explicit versus implicit conditioning for a continuously varying physical input in a convolutional recurrent setup is new enough to be worth noting. The experiments are empirical and grounded in independent numerical simulations, with no circular fitting issues. The main soft spot is the missing information on how supersaturation values were chosen and split between training and test sets. Without that, it is hard to tell whether the reported fidelity reflects genuine extrapolation or mostly interpolation within a limited range of morphologies. The Allen-Cahn trajectories with fixed anisotropy may not span all relevant long-time behaviors or domain effects, so claims of broad applicability rest on an assumption that the training distribution is representative. Error accumulation is described as limited, but the paper does not quantify how that scales with even longer runs or different anisotropy strengths. This paper is aimed at computational materials researchers who run many parameter sweeps on crystal growth or similar phase-field problems and want faster surrogates. A reader working on machine learning for parameterized PDEs will get the most from the explicit-implicit comparison and the dataset-size ablations. It has enough systematic tests and practical results to deserve a serious referee, even if the generalization story needs tighter evidence on sampling and out-of-distribution cases. I would send it for peer review rather than desk reject.

Referee Report

3 major / 2 minor

Summary. The paper develops two convolutional recurrent neural network (CRNN) surrogate models for crystal growth dynamics governed by the Allen-Cahn equation with kinetic anisotropy and variable supersaturation. One architecture conditions implicitly by ingesting a short input mini-sequence of frames before predicting continuation; the other conditions explicitly by receiving the supersaturation value together with a single initial frame and predicting the full sequence. Both are trained on numerical simulation trajectories and compared via point-wise and mean-absolute-error metrics across training-set sizes, with additional tests of scalability to 256-fold larger domains and >10-fold longer sequences.

Significance. If the reported error reductions and scalability hold under rigorous out-of-distribution testing, the explicit-conditioning architecture would supply a practical, high-fidelity surrogate that captures both global growth-rate dependence and local faceting morphology, offering substantial computational savings for materials-science simulations that must explore many supersaturation values.

major comments (3)

[Results (error analysis and dataset-size experiments)] The central claim that explicit conditioning 'guarantees the best results' and reproduces ground-truth profiles with high fidelity rests on error comparisons across dataset sizes, yet the manuscript supplies no information on the discrete or continuous sampling of supersaturation values, the precise train/test partition with respect to those parameter values, or quantitative out-of-distribution metrics. Without these details the observed advantage could be in-distribution interpolation rather than true generalization.
[Scalability and long-sequence extension tests] The scalability assertions—perfect extrapolation to 256× domain size and >10× sequence length with only limited error accumulation—are presented without tabulated MAE values, error-growth curves, or direct comparison against the implicit model on the same extrapolated regimes. These numbers are load-bearing for the claim that the surrogates are 'perfectly scalable'.
[Methods (dataset generation) and Discussion] The weakest-assumption concern is not addressed: the training trajectories are generated from a finite set of Allen-Cahn solutions with kinetic anisotropy; if supersaturation values are sparsely or correlatedly sampled, low in-distribution MAE does not guarantee correct local faceting response or growth-rate scaling for truly unseen supersaturations. No ablation on supersaturation density or morphology diversity is reported.

minor comments (2)

[Model architectures] Notation for the two architectures is introduced clearly in the abstract but would benefit from a short table in the main text that lists input dimensionality, conditioning mechanism, and output horizon for each model.
[Figures] Figure captions for error heat-maps or morphology snapshots should explicitly state the supersaturation value(s) shown and whether they lie inside or outside the training range.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our results. We address each major comment below and indicate the revisions that will be incorporated into the next version of the manuscript.

read point-by-point responses

Referee: [Results (error analysis and dataset-size experiments)] The central claim that explicit conditioning 'guarantees the best results' and reproduces ground-truth profiles with high fidelity rests on error comparisons across dataset sizes, yet the manuscript supplies no information on the discrete or continuous sampling of supersaturation values, the precise train/test partition with respect to those parameter values, or quantitative out-of-distribution metrics. Without these details the observed advantage could be in-distribution interpolation rather than true generalization.

Authors: We agree that these details are necessary to substantiate the generalization claim. The revised manuscript will include a complete description of the supersaturation sampling procedure used to generate the training trajectories, the exact train/test partitioning with respect to supersaturation values, and quantitative out-of-distribution error metrics for supersaturation values held out from training. These additions will demonstrate that the explicit-conditioning advantage is not limited to in-distribution interpolation. revision: yes
Referee: [Scalability and long-sequence extension tests] The scalability assertions—perfect extrapolation to 256× domain size and >10× sequence length with only limited error accumulation—are presented without tabulated MAE values, error-growth curves, or direct comparison against the implicit model on the same extrapolated regimes. These numbers are load-bearing for the claim that the surrogates are 'perfectly scalable'.

Authors: We acknowledge that the current presentation relies on qualitative statements. The revised manuscript will add tabulated MAE values for the 256-fold domain scaling and >10-fold sequence extension experiments, together with error-growth curves over time. Direct comparisons between the explicit and implicit models on these extrapolated regimes will also be included to quantify the limited error accumulation. revision: yes
Referee: [Methods (dataset generation) and Discussion] The weakest-assumption concern is not addressed: the training trajectories are generated from a finite set of Allen-Cahn solutions with kinetic anisotropy; if supersaturation values are sparsely or correlatedly sampled, low in-distribution MAE does not guarantee correct local faceting response or growth-rate scaling for truly unseen supersaturations. No ablation on supersaturation density or morphology diversity is reported.

Authors: This concern is valid and will be addressed. The revised Methods section will expand on the dataset generation process, and we will add an ablation study that varies supersaturation sampling density while measuring both in-distribution and out-of-distribution performance on faceting morphology and growth-rate scaling. The Discussion will be updated to interpret these results in light of the finite training set. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical ML surrogate evaluation on independent simulations

full rationale

The paper trains and evaluates convolutional recurrent neural networks on time sequences generated by numerical integration of the Allen-Cahn equation with kinetic anisotropy. Explicit vs. implicit supersaturation conditioning is compared via point-wise and mean absolute errors on held-out ground-truth trajectories, with additional tests for scalability to larger domains and longer sequences. No load-bearing steps reduce to self-definition, fitted inputs renamed as predictions, or self-citation chains; results follow directly from standard train/test splits on external simulation data without any derivation that equates outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central performance claims rest on the representativeness of the Allen-Cahn simulation data and on standard neural-network generalization assumptions; no new physical entities or ad-hoc constants are introduced.

axioms (1)

domain assumption Allen-Cahn dynamics with kinetic anisotropy accurately capture the essential physics of faceted crystal growth under variable supersaturation
This supplies all ground-truth training and test sequences.

pith-pipeline@v0.9.0 · 5572 in / 1212 out tokens · 49850 ms · 2026-05-09T21:45:50.683800+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

[1]

J. Wei, X. Chu, X.-Y. Sun, K. Xu, H.-X. Deng, J. Chen, Z. Wei, and M. Lei, Machine learning in materials science, InfoMat1, 338 (2019)

work page 2019
[2]

Choudhary, B

K. Choudhary, B. DeCost, C. Chen, A. Jain, F. Tavazza, R. Cohn, C. W. Park, A. Choudhary, A. Agrawal, S. J. Billinge,et al., Recent advances and applications of deep learning methods in materials science, npj Computa- tional Materials8, 59 (2022)

work page 2022
[3]

Schmidt, M

J. Schmidt, M. R. Marques, S. Botti, and M. A. Marques, Recent advances and applications of machine learning in solid-state materials science, npj computational materi- als5, 83 (2019)

work page 2019
[4]

Peivaste, S

I. Peivaste, S. Belouettar, F. Mercuri, N. Fantuzzi, H. Dehghani, R. Izadi, H. Ibrahim, J. Lengiewicz, M. Belouettar-Mathis, K. Bendine,et al., Artificial in- telligence in materials science and engineering: Current landscape, key challenges, and future trajectories, Com- posite Structures372, 119419 (2025)

work page 2025
[5]

Lyngby and K

P. Lyngby and K. S. Thygesen, Data-driven discovery of 2D materials by deep generative models, npj Computa- tional Materials8, 232 (2022)

work page 2022
[6]

Y. Zhao, E. M. D. Siriwardane, Z. Wu, N. Fu, M. Al- Fahdi, M. Hu, and J. Hu, Physics guided deep learning for generative design of crystal materials with symmetry constraints, npj Computational Materials9, 38 (2023)

work page 2023
[7]

E. T. Chenebuah, M. Nganbe, and A. B. Tchagang, A deep generative modeling architecture for designing lattice-constrained perovskite materials, npj Computa- tional Materials10, 198 (2024)

work page 2024
[8]

Karpovich, E

C. Karpovich, E. Pan, and E. A. Olivetti, Deep reinforce- ment learning for inverse inorganic materials design, npj Computational Materials10, 287 (2024)

work page 2024
[9]

Provatas and K

N. Provatas and K. Elder,Phase-Field Methods in Ma- terials Science and Engineering, 1st ed. (Wiley, 2010)

work page 2010
[10]

B. Li, J. Lowengrub, A. Ratz, and A. Voigt, Geomet- ric Evolution Laws for Thin Crystalline Films: Mod- eling and Numerics, Communications in Computational Physics6, 433 (2009)

work page 2009
[11]

Montes De Oca Zapiain, J

D. Montes De Oca Zapiain, J. A. Stewart, and R. Dingre- ville, Accelerating phase-field-based microstructure evo- lution predictions via surrogate models trained by ma- chine learning methods, npj Computational Materials7, 3 (2021)

work page 2021
[12]

C. Hu, S. Martin, and R. Dingreville, Accelerating phase- field predictions via recurrent neural networks learning the microstructure evolution in latent space, Computer Methods in Applied Mechanics and Engineering397, 115128 (2022)

work page 2022
[13]

Oommen, K

V. Oommen, K. Shukla, S. Goswami, R. Dingreville, and G. E. Karniadakis, Learning two-phase microstructure evolution using neural operators and autoencoder archi- tectures, npj Computational Materials8, 190 (2022)

work page 2022
[14]

Tep and M

P. Tep and M. Bernacki, High-fidelity grain growth mod- eling: Leveraging deep learning for fast computations, Acta Materialia301, 121486 (2025)

work page 2025
[15]

Alhada–Lahbabi, D

K. Alhada–Lahbabi, D. Deleruyelle, and B. Gautier, Ma- chine learning surrogate for 3D phase-field modeling of ferroelectric tip-induced electrical switching, npj Com- putational Materials10, 197 (2024)

work page 2024
[16]

Bonneville, N

C. Bonneville, N. Bieberdorf, P. Robbe, M. Asta, H. Najm, L. Capolungo, and C. Safta, Towards spatio- temporal extrapolation of phase-field simulations with convolution-only neural networks (2026), 2601.04510

work page arXiv 2026
[17]

Peivaste, A

I. Peivaste, A. Makradi, and S. Belouettar, Teaching ar- tificial intelligence to perform rapid, resolution-invariant grain growth modeling via fourier neural operator, Com- puter Methods in Applied Mechanics and Engineering 440, 117945 (2025)

work page 2025
[18]

S. Fan, A. L. Hitt, M. Tang, B. Sadigh, and F. Zhou, Ac- celerate microstructure evolution simulation using graph neural networks with adaptive spatiotemporal resolution, Machine Learning: Science and Technology5, 025027 (2024)

work page 2024
[19]

Lanzoni, F

D. Lanzoni, F. Montalenti, and R. Bergamaschini, Deep learning for simulating the evolution of condensed matter systems at the continuum scale: Methods and applica- tions, Journal of Physics: Condensed Matter37, 403003 (2025)

work page 2025
[20]

X. Shi, Z. Chen, H. Wang, D.-Y. Yeung, W.-k. Wong, and W.-c. Woo, Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting (2015), 1506.04214

work page Pith review arXiv 2015
[21]

Delving Deeper into Convolutional Networks for Learning Video Representations

N. Ballas, L. Yao, C. Pal, and A. Courville, Delving Deeper into Convolutional Networks for Learning Video Representations (2016), arXiv:1511.06432

work page Pith review arXiv 2016
[22]

K. Yang, Y. Cao, Y. Zhang, S. Fan, M. Tang, D. Aberg, B. Sadigh, and F. Zhou, Self-supervised learning and pre- diction of microstructure evolution with convolutional re- current neural networks, Patterns2, 100243 (2021)

work page 2021
[23]

P. Wu, A. S. Iquebal, and K. Ankit, Emulating mi- crostructural evolution during spinodal decomposition using a tensor decomposed convolutional and recurrent neural network, Computational Materials Science224, 112187 (2023)

work page 2023
[24]

Lanzoni, A

D. Lanzoni, A. Fantasia, R. Bergamaschini, O. Pierre- Louis, and F. Montalenti, Extreme time extrapolation capabilities and thermodynamic consistency of physics- inspired neural networks for the 3D microstructure evolu- tion of materials via Cahn–Hilliard flow, Machine Learn- ing: Science and Technology5, 045017 (2024)

work page 2024
[25]

Fantasia, D

A. Fantasia, D. Lanzoni, N. Di Eugenio, A. Monteleone, R. Bergamaschini, and F. Montalenti, A parametrically- Conditioned Deep Learning Surrogate for Coherent Spin- odal Decomposition, Advanced Theory and Simulations 9, e02144 (2026)

work page 2026
[26]

A. A. K. Farizhandi, O. Betancourt, and M. Mamivand, Deep learning approach for chemistry and processing his- tory prediction from materials microstructure, Scientific Reports12, 4552 (2022)

work page 2022
[27]

Lanzoni, M

D. Lanzoni, M. Albani, R. Bergamaschini, and F. Mon- talenti, Morphological evolution via surface diffusion learned by convolutional, recurrent neural networks: Ex- trapolation and prediction uncertainty, Physical Review Materials6, 103801 (2022)

work page 2022
[28]

Kutsukake, Review of machine learning applications for crystal growth research, Journal of Crystal Growth 630, 127598 (2024)

K. Kutsukake, Review of machine learning applications for crystal growth research, Journal of Crystal Growth 630, 127598 (2024)

work page 2024
[29]

Petkovic, L

M. Petkovic, L. Vieira, and N. Dropka, Machine learn- ing in crystal growth: A review of methods, data, and applications, Progress in Crystal Growth and Character- ization of Materials71, 100689 (2025). 13

work page 2025
[30]

M. Lu, S. Rao, H. Yue, J. Han, and J. Wang, Recent advances in the application of machine learning to crys- tal behavior and crystallization process control, Crystal Growth & Design24, 5374 (2024)

work page 2024
[31]

Y. Lu, X. Yang, B. Niu, T. Li, Z. Zheng, L. Jia, D. Chen, H. Qi, K. H. L. Zhang, M. Zhu, H. Zhang, and X. Lu, Achievement of high-quality gallium oxide epitaxial growth via machine learning, Advanced Func- tional Materials36, e19854 (2026)

work page 2026
[32]

S. M. Allen and J. W. Cahn, A microscopic theory for an- tiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica27, 1085 (1979)

work page 1979
[33]

D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization (2017), arXiv:1412.6980

work page internal anchor Pith review Pith/arXiv arXiv 2017
[34]

Bengio, J

Y. Bengio, J. Louradour, R. Collobert, and J. Weston, Curriculum learning, inProceedings of the 26th Annual International Conference on Machine Learning, ICML ’09 (2009) pp. 41–48. SUPPLEMENTARY MATERIAL Neural surrogates for crystal growth dynamics with variable supersaturation: explicit vs. implicit conditioning Matteo Rigoni, 1 Daniele Lanzoni, 1, 2 Fr...

work page 2009

[1] [1]

J. Wei, X. Chu, X.-Y. Sun, K. Xu, H.-X. Deng, J. Chen, Z. Wei, and M. Lei, Machine learning in materials science, InfoMat1, 338 (2019)

work page 2019

[2] [2]

Choudhary, B

K. Choudhary, B. DeCost, C. Chen, A. Jain, F. Tavazza, R. Cohn, C. W. Park, A. Choudhary, A. Agrawal, S. J. Billinge,et al., Recent advances and applications of deep learning methods in materials science, npj Computa- tional Materials8, 59 (2022)

work page 2022

[3] [3]

Schmidt, M

J. Schmidt, M. R. Marques, S. Botti, and M. A. Marques, Recent advances and applications of machine learning in solid-state materials science, npj computational materi- als5, 83 (2019)

work page 2019

[4] [4]

Peivaste, S

I. Peivaste, S. Belouettar, F. Mercuri, N. Fantuzzi, H. Dehghani, R. Izadi, H. Ibrahim, J. Lengiewicz, M. Belouettar-Mathis, K. Bendine,et al., Artificial in- telligence in materials science and engineering: Current landscape, key challenges, and future trajectories, Com- posite Structures372, 119419 (2025)

work page 2025

[5] [5]

Lyngby and K

P. Lyngby and K. S. Thygesen, Data-driven discovery of 2D materials by deep generative models, npj Computa- tional Materials8, 232 (2022)

work page 2022

[6] [6]

Y. Zhao, E. M. D. Siriwardane, Z. Wu, N. Fu, M. Al- Fahdi, M. Hu, and J. Hu, Physics guided deep learning for generative design of crystal materials with symmetry constraints, npj Computational Materials9, 38 (2023)

work page 2023

[7] [7]

E. T. Chenebuah, M. Nganbe, and A. B. Tchagang, A deep generative modeling architecture for designing lattice-constrained perovskite materials, npj Computa- tional Materials10, 198 (2024)

work page 2024

[8] [8]

Karpovich, E

C. Karpovich, E. Pan, and E. A. Olivetti, Deep reinforce- ment learning for inverse inorganic materials design, npj Computational Materials10, 287 (2024)

work page 2024

[9] [9]

Provatas and K

N. Provatas and K. Elder,Phase-Field Methods in Ma- terials Science and Engineering, 1st ed. (Wiley, 2010)

work page 2010

[10] [10]

B. Li, J. Lowengrub, A. Ratz, and A. Voigt, Geomet- ric Evolution Laws for Thin Crystalline Films: Mod- eling and Numerics, Communications in Computational Physics6, 433 (2009)

work page 2009

[11] [11]

Montes De Oca Zapiain, J

D. Montes De Oca Zapiain, J. A. Stewart, and R. Dingre- ville, Accelerating phase-field-based microstructure evo- lution predictions via surrogate models trained by ma- chine learning methods, npj Computational Materials7, 3 (2021)

work page 2021

[12] [12]

C. Hu, S. Martin, and R. Dingreville, Accelerating phase- field predictions via recurrent neural networks learning the microstructure evolution in latent space, Computer Methods in Applied Mechanics and Engineering397, 115128 (2022)

work page 2022

[13] [13]

Oommen, K

V. Oommen, K. Shukla, S. Goswami, R. Dingreville, and G. E. Karniadakis, Learning two-phase microstructure evolution using neural operators and autoencoder archi- tectures, npj Computational Materials8, 190 (2022)

work page 2022

[14] [14]

Tep and M

P. Tep and M. Bernacki, High-fidelity grain growth mod- eling: Leveraging deep learning for fast computations, Acta Materialia301, 121486 (2025)

work page 2025

[15] [15]

Alhada–Lahbabi, D

K. Alhada–Lahbabi, D. Deleruyelle, and B. Gautier, Ma- chine learning surrogate for 3D phase-field modeling of ferroelectric tip-induced electrical switching, npj Com- putational Materials10, 197 (2024)

work page 2024

[16] [16]

Bonneville, N

C. Bonneville, N. Bieberdorf, P. Robbe, M. Asta, H. Najm, L. Capolungo, and C. Safta, Towards spatio- temporal extrapolation of phase-field simulations with convolution-only neural networks (2026), 2601.04510

work page arXiv 2026

[17] [17]

Peivaste, A

I. Peivaste, A. Makradi, and S. Belouettar, Teaching ar- tificial intelligence to perform rapid, resolution-invariant grain growth modeling via fourier neural operator, Com- puter Methods in Applied Mechanics and Engineering 440, 117945 (2025)

work page 2025

[18] [18]

S. Fan, A. L. Hitt, M. Tang, B. Sadigh, and F. Zhou, Ac- celerate microstructure evolution simulation using graph neural networks with adaptive spatiotemporal resolution, Machine Learning: Science and Technology5, 025027 (2024)

work page 2024

[19] [19]

Lanzoni, F

D. Lanzoni, F. Montalenti, and R. Bergamaschini, Deep learning for simulating the evolution of condensed matter systems at the continuum scale: Methods and applica- tions, Journal of Physics: Condensed Matter37, 403003 (2025)

work page 2025

[20] [20]

X. Shi, Z. Chen, H. Wang, D.-Y. Yeung, W.-k. Wong, and W.-c. Woo, Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting (2015), 1506.04214

work page Pith review arXiv 2015

[21] [21]

Delving Deeper into Convolutional Networks for Learning Video Representations

N. Ballas, L. Yao, C. Pal, and A. Courville, Delving Deeper into Convolutional Networks for Learning Video Representations (2016), arXiv:1511.06432

work page Pith review arXiv 2016

[22] [22]

K. Yang, Y. Cao, Y. Zhang, S. Fan, M. Tang, D. Aberg, B. Sadigh, and F. Zhou, Self-supervised learning and pre- diction of microstructure evolution with convolutional re- current neural networks, Patterns2, 100243 (2021)

work page 2021

[23] [23]

P. Wu, A. S. Iquebal, and K. Ankit, Emulating mi- crostructural evolution during spinodal decomposition using a tensor decomposed convolutional and recurrent neural network, Computational Materials Science224, 112187 (2023)

work page 2023

[24] [24]

Lanzoni, A

D. Lanzoni, A. Fantasia, R. Bergamaschini, O. Pierre- Louis, and F. Montalenti, Extreme time extrapolation capabilities and thermodynamic consistency of physics- inspired neural networks for the 3D microstructure evolu- tion of materials via Cahn–Hilliard flow, Machine Learn- ing: Science and Technology5, 045017 (2024)

work page 2024

[25] [25]

Fantasia, D

A. Fantasia, D. Lanzoni, N. Di Eugenio, A. Monteleone, R. Bergamaschini, and F. Montalenti, A parametrically- Conditioned Deep Learning Surrogate for Coherent Spin- odal Decomposition, Advanced Theory and Simulations 9, e02144 (2026)

work page 2026

[26] [26]

A. A. K. Farizhandi, O. Betancourt, and M. Mamivand, Deep learning approach for chemistry and processing his- tory prediction from materials microstructure, Scientific Reports12, 4552 (2022)

work page 2022

[27] [27]

Lanzoni, M

D. Lanzoni, M. Albani, R. Bergamaschini, and F. Mon- talenti, Morphological evolution via surface diffusion learned by convolutional, recurrent neural networks: Ex- trapolation and prediction uncertainty, Physical Review Materials6, 103801 (2022)

work page 2022

[28] [28]

Kutsukake, Review of machine learning applications for crystal growth research, Journal of Crystal Growth 630, 127598 (2024)

K. Kutsukake, Review of machine learning applications for crystal growth research, Journal of Crystal Growth 630, 127598 (2024)

work page 2024

[29] [29]

Petkovic, L

M. Petkovic, L. Vieira, and N. Dropka, Machine learn- ing in crystal growth: A review of methods, data, and applications, Progress in Crystal Growth and Character- ization of Materials71, 100689 (2025). 13

work page 2025

[30] [30]

M. Lu, S. Rao, H. Yue, J. Han, and J. Wang, Recent advances in the application of machine learning to crys- tal behavior and crystallization process control, Crystal Growth & Design24, 5374 (2024)

work page 2024

[31] [31]

Y. Lu, X. Yang, B. Niu, T. Li, Z. Zheng, L. Jia, D. Chen, H. Qi, K. H. L. Zhang, M. Zhu, H. Zhang, and X. Lu, Achievement of high-quality gallium oxide epitaxial growth via machine learning, Advanced Func- tional Materials36, e19854 (2026)

work page 2026

[32] [32]

S. M. Allen and J. W. Cahn, A microscopic theory for an- tiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica27, 1085 (1979)

work page 1979

[33] [33]

D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization (2017), arXiv:1412.6980

work page internal anchor Pith review Pith/arXiv arXiv 2017

[34] [34]

Bengio, J

Y. Bengio, J. Louradour, R. Collobert, and J. Weston, Curriculum learning, inProceedings of the 26th Annual International Conference on Machine Learning, ICML ’09 (2009) pp. 41–48. SUPPLEMENTARY MATERIAL Neural surrogates for crystal growth dynamics with variable supersaturation: explicit vs. implicit conditioning Matteo Rigoni, 1 Daniele Lanzoni, 1, 2 Fr...

work page 2009