arxiv: 2605.07279 · v1 · submitted 2026-05-08 · ❄️ cond-mat.mtrl-sci · cond-mat.dis-nn

Recognition: 3 theorem links

· Lean Theorem

Physics-informed operator learning for transferable energy-dissipative microstructure dynamics

Jie Xiong , Yue Wu , Xuewei Zhou , Peishuo Zhao , Jiaming Zhu

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:45 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.dis-nn

keywords physics-informed neural operatorphase field modelmicrostructure evolutionCahn-Hilliardoperator learningtransfer learningenergy dissipative systems

0 comments

The pith

A physics-informed neural operator learns to predict microstructure evolution across different material parameters without retraining.

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

Phase-field simulations model how microstructures in materials change over time but require heavy computation for long periods. The paper develops PFNet to learn the evolution operator directly, incorporating physical constraints like energy dissipation and boundary conditions. It demonstrates accurate short-term predictions and long-term stable simulations for Cahn-Hilliard coarsening under varying conditions. The same model applies to martensitic transformations without changes, suggesting broader use for energy-dissipative systems.

Core claim

PFNet combines a diffusion-inspired U-Net with periodic padding, entropy-based state conditioning and thermodynamic-parameter modulation to learn conditional evolution operators for phase-field dynamics. It achieves accurate one-step prediction and stable autoregressive rollouts for Cahn-Hilliard coarsening across composition, gradient-energy coefficient, coarsening stage and morphology class, and extends to a four-channel martensitic-transformation benchmark without specific redesign.

What carries the argument

PFNet: a physics-informed neural operator that uses a diffusion-inspired U-Net architecture with periodic padding to enforce boundary consistency, entropy-based conditioning to capture instantaneous ordering state, and thermodynamic-parameter modulation to account for changes in the free-energy landscape.

Load-bearing premise

The combination of U-Net, periodic padding, entropy conditioning, and parameter modulation ensures stability and transferability without hidden overfitting or need for retraining.

What would settle it

A test showing divergence in long-term autoregressive predictions when changing the gradient-energy coefficient or morphology class beyond training data would disprove the transferability.

read the original abstract

Phase-field simulations provide mechanistic descriptions of microstructure evolution, but repeated high-fidelity integration over long horizons and broad parameter spaces remains computationally expensive. We present PFNet, a physics-informed neural operator framework that advances microstructural states by learning conditional evolution operators rather than direct correlations. PFNet combines a diffusion-inspired U-Net with periodic padding, entropy-based state conditioning and thermodynamic-parameter modulation to encode boundary consistency, instantaneous ordering state and changes in the free-energy landscape. For Cahn-Hilliard coarsening, PFNet achieves accurate one-step prediction and stable autoregressive rollouts across composition, gradient-energy coefficient, coarsening stage and morphology class, with errors concentrated near diffuse interfaces and topology-changing regions. The same framework extends to a four-channel martensitic-transformation benchmark without martensite-specific redesign. These results indicate that physics-informed operator learning can provide transferable surrogates for phase-field dynamics and broader energy-dissipative dynamical systems.

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

PFNet gives a workable U-Net operator for phase-field rollouts with decent transfer across parameters and to martensite, but the thermodynamic consistency of long autoregressive trajectories is not convincingly shown.

read the letter

The paper's main contribution is PFNet: a conditional neural operator built around a diffusion-style U-Net, periodic padding, entropy conditioning, and thermodynamic-parameter modulation. It learns one-step advances for Cahn-Hilliard coarsening and then rolls them out autoregressively across changes in composition, gradient energy, coarsening stage, and morphology. The same weights are reused on a four-channel martensitic benchmark without architecture changes. That transferability is the concrete new piece; prior neural-operator work on phase fields has not shown this level of cross-problem reuse with these conditioning tricks. Visually the rollouts look stable and errors stay near interfaces, which is useful for people who need fast surrogates inside larger design loops. The architecture choices are motivated and the training is supervised on external simulation data, so there is no obvious circularity. The soft spot is the missing check on long-horizon physics. The central claim is that the learned operator respects energy dissipation, yet the reported evidence is limited to pointwise state errors and visual stability. There is no mention of tracking the time derivative of total free energy or the L2 norm of chemical potential on the rolled-out fields, nor any comparison to the ground-truth solver over hundreds of steps. Small per-step interface errors can accumulate into non-monotonic free-energy trajectories that still look locally plausible. Without that diagnostic, or at least quantitative rollout metrics and ablations on the conditioning terms, it is hard to know how far the transferability actually extends. This work is aimed at computational materials groups that already run phase-field codes and want faster exploration of parameter spaces. It is coherent enough on its own terms to deserve a serious referee who can ask for the missing dissipation diagnostics and error tables.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces PFNet, a physics-informed neural operator framework for microstructure evolution in energy-dissipative systems. It uses a diffusion-inspired U-Net with periodic padding, entropy-based state conditioning, and thermodynamic-parameter modulation to learn conditional evolution operators from phase-field simulation data. The authors report accurate one-step predictions and stable autoregressive rollouts for Cahn-Hilliard coarsening across composition, gradient-energy coefficient, coarsening stage, and morphology class, with errors localized near interfaces. The same architecture is applied without redesign to a four-channel martensitic-transformation benchmark. The central claim is that physics-informed operator learning yields transferable surrogates for phase-field dynamics and similar systems.

Significance. If the central claims are substantiated, the work would provide a meaningful contribution to computational materials science by offering fast, parameter-transferable surrogates that bypass repeated high-fidelity phase-field integrations. The explicit incorporation of entropy conditioning and thermodynamic modulation to encode variational structure is a positive design choice that distinguishes it from purely data-driven baselines. Reproducible code or machine-checked physical consistency checks would further elevate its impact, but the current evidence base limits immediate adoption for production microstructure modeling.

major comments (2)

[§4.2] §4.2 (autoregressive rollout experiments): The reported evaluation relies on pointwise state errors and visual inspection of rolled-out fields, but contains no explicit computation of the time derivative of the total free energy (or L2 norm of the chemical potential) on long-horizon trajectories. Without this check, it remains unverified whether small per-step interface errors accumulate into non-monotonic or non-variational dynamics, which directly undermines the claim of stable, energy-dissipative surrogates across parameter transfers.
[§3.3] §3.3 (thermodynamic modulation and conditioning): The architecture description motivates entropy-based conditioning and thermodynamic-parameter modulation, yet no ablation study isolates their contribution to transferability. A controlled comparison against a baseline U-Net lacking these modules on the same held-out morphology and parameter sets is needed to establish that these physics-informed components are load-bearing for the observed generalization.

minor comments (3)

[Abstract] The abstract asserts 'accurate' one-step predictions without quoting any error metric (e.g., relative L2 norm or interface-specific error); adding a single quantitative statement would improve precision.
[Figure captions] Figure captions for the martensitic benchmark (e.g., Figure 8) omit the precise values of the four-channel parameters and the number of training trajectories; this reduces reproducibility.
[§3.1] Notation for the entropy conditioning term is introduced without an explicit equation reference in the main text; cross-referencing to the supplementary derivation would aid clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the potential contribution of PFNet to computational materials science. We address each major comment below and describe the revisions we will implement.

read point-by-point responses

Referee: [§4.2] §4.2 (autoregressive rollout experiments): The reported evaluation relies on pointwise state errors and visual inspection of rolled-out fields, but contains no explicit computation of the time derivative of the total free energy (or L2 norm of the chemical potential) on long-horizon trajectories. Without this check, it remains unverified whether small per-step interface errors accumulate into non-monotonic or non-variational dynamics, which directly undermines the claim of stable, energy-dissipative surrogates across parameter transfers.

Authors: We agree that verifying monotonic free-energy dissipation and consistency with the variational structure on long-horizon trajectories is essential to substantiate the claim of stable, energy-dissipative surrogates. In the revised manuscript we will compute and report the time derivative of the total free energy (dF/dt) together with the L2 norm of the chemical potential for all autoregressive rollouts in the Cahn-Hilliard experiments, across the tested compositions, gradient-energy coefficients, coarsening stages and morphology classes. These quantities will be added to an expanded §4.2, with quantitative plots demonstrating that the surrogate trajectories remain dissipative and free of non-physical accumulation of errors. revision: yes
Referee: [§3.3] §3.3 (thermodynamic modulation and conditioning): The architecture description motivates entropy-based conditioning and thermodynamic-parameter modulation, yet no ablation study isolates their contribution to transferability. A controlled comparison against a baseline U-Net lacking these modules on the same held-out morphology and parameter sets is needed to establish that these physics-informed components are load-bearing for the observed generalization.

Authors: We acknowledge that an ablation study is required to isolate the contribution of the entropy-based conditioning and thermodynamic-parameter modulation to the observed transferability. In the revision we will train and evaluate a baseline U-Net that omits both modules on the identical held-out morphology and parameter sets used for the Cahn-Hilliard and martensitic-transformation benchmarks. Performance metrics (one-step and multi-step errors) will be reported side-by-side in an updated §3.3, thereby quantifying the improvement attributable to the physics-informed components. revision: yes

Circularity Check

0 steps flagged

No circularity; supervised operator learning on external phase-field data

full rationale

The paper trains PFNet as a conditional neural operator on simulation trajectories generated by an independent phase-field solver. Architectural elements (diffusion-inspired U-Net, periodic padding, entropy conditioning, thermodynamic modulation) are chosen to respect known physics but do not define the output operator in terms of itself or rename fitted quantities as predictions. No load-bearing self-citation, uniqueness theorem, or ansatz smuggling appears in the provided text; the central claim rests on empirical one-step accuracy and autoregressive stability measured against held-out ground-truth rollouts. The derivation chain therefore remains non-circular: inputs are external data, outputs are learned mappings whose fidelity is externally falsifiable.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard phase-field modeling assumptions and neural-network training assumptions; no new physical entities are introduced.

free parameters (2)

neural-network weights and biases
Learned during training on simulation data; exact values and regularization choices not specified in abstract.
entropy and thermodynamic modulation coefficients
Hand-designed or tuned conditioning mechanisms whose precise functional forms are not detailed.

axioms (2)

domain assumption Phase-field models (Cahn-Hilliard and martensitic transformation equations) faithfully represent the target microstructure dynamics
The training data and evaluation benchmarks are generated from these models.
domain assumption A neural operator conditioned on instantaneous state and thermodynamic parameters can learn a stable evolution operator
Core premise enabling autoregressive rollout and transferability.

pith-pipeline@v0.9.0 · 5468 in / 1365 out tokens · 48359 ms · 2026-05-11T01:45:51.377050+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean Jcost_pos_of_ne_one / washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Periodic boundary conditions yield dℱ/dt = −⟨δℱ/δϕ, ℳ δℱ/δϕ⟩ ≤ 0 … This inequality makes explicit the dissipative gradient-flow structure shared by AC and CH dynamics.
IndisputableMonolith/Foundation/ArrowOfTime.lean z_monotone_absolute / entropy_from_berry echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

PFNet … learns conditional evolution operators … entropy-based state conditioning and thermodynamic-parameter modulation to encode … changes in the free-energy landscape.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection refines

?

refines
Relation between the paper passage and the cited Recognition theorem.

The same framework extends to a four-channel martensitic-transformation benchmark without martensite-specific redesign … transferable surrogates for … energy-dissipative dynamical systems.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

[1]

Small positional shifts of diffuse interfaces therefore produce disproportionately large pointwise residuals

These locations coincide with regions of high local curvature and steep chemical- potential gradients. Small positional shifts of diffuse interfaces therefore produce disproportionately large pointwise residuals. The resulting degradation remains local rather than global. PFNet still preserves the dominant morphology class and the overall topological coar...

work page
[2]

The predicted interface positions maintain alignment with the reference solutions, keeping the error maps structurally weak and spatially sparse

PFNet captures broad and smooth compositional channels associated with thicker interfaces, as well as finer droplet-like features associated with sharper interfaces. The predicted interface positions maintain alignment with the reference solutions, keeping the error maps structurally weak and spatially sparse. Since changing κ alters the underlying Gibbs ...

work page
[3]

Hu, Y .Z

X. Hu, Y .Z. Ji, T.W . Heo, L.Q. Chen, X.Y . Cui, Phase-field model of deformation twin-grain boundary interactions in hexagonal systems, Acta Mater 200 (2020) 821-834

work page 2020
[4]

Cissé, M.A

C. Cissé, M.A. Zaeem, A phase-field model for non-isothermal phase transformation and plasticity in polycrystalline yttria-stabilized tetragonal zirconia, Acta Mater 191 (2020) 111-123

work page 2020
[5]

Kavousi, M.A

S. Kavousi, M.A. Zaeem, Quantitative phase-field modeling of solute trapping in rapid solidification, Acta Mater 205 (2021)

work page 2021
[6]

M.Y. Gong, J. Graham, V . Taupin, L. Capolungo, The effects of stress, temperature and facet structure on growth of {10(1)over-bar2} twins in Mg: A molecular dynamics and phase field study , Acta Mater 208 (2021)

work page 2021
[7]

Laxmipathy , F

V .P . Laxmipathy , F. Wang, M. Selzer, B. Nestler, Phase-field simulations of grain boundary grooving under diffusive-convective conditions, Acta Mater 204 (2021)

work page 2021
[8]

T.F. Ma, H. Kim, N. Mathew, D.J. Luscher, L. Cao, A. Hunter, Dislocation transmission across Σ3{112} incoherent twin boundary: a combined atomistic and phase-field study , Acta Mater 223 (2022)

work page 2022
[9]

Y .H. Zhao, H. Xing, L.J. Zhang, H.B. Huang, D.K. Sun, X.L. Dong, Y .X. Shen, J.C. Wang, Development of Phase-Field Modeling in Materials Science in China: A Review, Acta Metall Sin-Engl 36(11) (2023) 1749-1775

work page 2023
[10]

Mancias, B

J. Mancias, B. Vela, J. Flórez-Coronel, R. Tavakoli, D. Allaire, R. Arróyave, D. Tourret, Mapping of microstructure transitions during rapid alloy solidification using Bayesian -guided phase -field simulations, Acta Mater 297 (2025)

work page 2025
[11]

Liu, D.Z

C. Liu, D.Z. Cheng, J.H. Qu, D.H. Li, Y . Zhao, J. Zhang, Spatially heterogeneous evolution of helium bubbles in He-irradiated Inconel 617: Experimental observation and anisotropic phase-field simulation, J Nucl Mater 622 (2026)

work page 2026
[12]

Zecevic, B

M. Zecevic, B. Zuanetti, A.F.T. Leong, C.S. Meredith, C.A. Bolme, K.J. Ramos, M.J. Cawkwell, Phase-field modeling and experiments of dynamic fracture in single crystal quartz, Acta Mater 305 (2026)

work page 2026
[13]

Chen, Phase-field method of phase transitions/domain structures in ferroelectric thin films: A review, J Am Ceram Soc 91(6) (2008) 1835-1844

L.Q. Chen, Phase-field method of phase transitions/domain structures in ferroelectric thin films: A review, J Am Ceram Soc 91(6) (2008) 1835-1844

work page 2008
[14]

Gránásy , G.I

L. Gránásy , G.I. Tóth, J.A. Warren, F. Podmaniczky , G. Tegze, L. Rátkai, T. Pusztai, Phase-field modeling of crystal nucleation in undercooled liquids - A review, Prog Mater Sci 106 (2019)

work page 2019
[15]

Zhang, Z.P

A. Zhang, Z.P . Guo, B. Jiang, S.M. Xiong, F.S. Pan, Numerical solution to phase -field model of solidification: A review, Computational Materials Science 228 (2023)

work page 2023
[16]

Chen, Phase-field models for microstructure evolution, Annu Rev Mater Res 32 (2002) 113- 140

L.Q. Chen, Phase-field models for microstructure evolution, Annu Rev Mater Res 32 (2002) 113- 140

work page 2002
[17]

Chen, Y .H

L.Q. Chen, Y .H. Zhao, From classical thermodynamics to phase-field method, Prog Mater Sci 124 (2022)

work page 2022
[18]

C.H. Chen, E. Bouchbinder, A. Karma, Instability in dynamic fracture and the failure of the classical theory of cracks, Nat Phys 13(12) (2017) 1186-+

work page 2017
[19]

Yang, T.D

Y .Y .W . Yang, T.D. Oyedeji, P . Kühn, B.X. Xu, Investigation on temperature-gradient-driven effects in unconventional sintering via non -isothermal phase-field simulation, Scripta Mater 186 (2020) 152- 157

work page 2020
[20]

Emdadi, M.A

A. Emdadi, M.A. Zaeem, Phase-field modeling of crack propagation in polycrystalline materials, Computational Materials Science 186 (2021)

work page 2021
[21]

Tonks, L.K

M.R. Tonks, L.K. Aagesen, The Phase Field Method: Mesoscale Simulation Aiding Material Discovery, Annu Rev Mater Res 49 (2019) 79-102

work page 2019
[22]

Tourret, H

D. Tourret, H. Liu, J. LLorca, Phase-field modeling of microstructure evolution: Recent applications, perspectives and challenges, Prog Mater Sci 123 (2022)

work page 2022
[23]

Provatas, N

N. Provatas, N. Goldenfeld, J. Dantzig, Efficient computation of de ndritic microstructures using adaptive mesh refinement, Physical Review Letters 80(15) (1998) 3308-3311

work page 1998
[24]

Amirian, K

B. Amirian, K. Inal, A thermodynamically consistent machine learning-based finite element solver for phase-field approach, Acta Mater 277 (2024)

work page 2024
[25]

Takahashi, S

Y . Takahashi, S. Sakane, T. Takaki, Accelerated multi -phase-field simulation of microstructure evolution in powder bed fusion under scanning strategy variations via a moving frame algorithm, Computational Materials Science 261 (2026)

work page 2026
[26]

Collins, M.S

C. Collins, M.S. Dyer, M.J. Pitcher, G.F.S. Whitehead, M. Zanella, P . Mandal, J.B. Claridge, G.R. Darling, M.J. Rosseinsky, Accelerated discovery of two crystal structure types in a complex inorganic phase field, Nature 546(7657) (2017) 280-+

work page 2017
[27]

Nestler, A 3D parallel simulator for crystal growth and solidification in complex alloy systems, J Cryst Growth 275(1-2) (2005) E273-E278

B. Nestler, A 3D parallel simulator for crystal growth and solidification in complex alloy systems, J Cryst Growth 275(1-2) (2005) E273-E278

work page 2005
[28]

Greenwood, K.N

M. Greenwood, K.N. Shampur, N. Ofori -Opoku, T. Pinomaa, L. Wang, S. Gurevich, N. Provatas, Quantitative 3D phase field modelling of solidification using next-generation adaptive mesh refinement, Computational Materials Science 142 (2018) 153-171

work page 2018
[29]

Rosam, P .K

J. Rosam, P .K. Jimack, A.M. Mullis, An adaptive, fully implicit multigrid phase-field model for the quantitative simulation of non -isothermal binary alloy solidification, Acta Mater 56(17) (2008) 4559- 4569

work page 2008
[30]

L.Q. Chen, J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Computer Physics Communications 108(2-3) (1998) 147-158

work page 1998
[31]

W.M. Feng, P . Y u, S.Y . Hu, Z.K. Liu, Q. Du, L.Q. Chen, Spectral implementation of an adaptive moving mesh method for phase-field equations, J Comput Phys 220(1) (2006) 498-510

work page 2006
[32]

Y amanaka, T

A. Y amanaka, T. Aoki, S. Ogawa, T. Takaki, GPU-accelerated phase-field simulation of dendritic solidification in a binary alloy , J Cryst Growth 318(1) (2011) 40-45

work page 2011
[33]

Lan, Y .C

C.W. Lan, Y .C. Chang, C.J. Shih, Adaptive phase field simulation of non-isothermal free dendritic growth of a binary alloy , Acta Mater 51(7) (2003) 1857-1869

work page 2003
[34]

Boccardo, M

A.D. Boccardo, M. Tong, S.B. Leen, D. Tourret, J. Segurado, Efficiency and accuracy of GPU - parallelized Fourier spectral methods for solving phase-field models, Computational Materials Science 228 (2023)

work page 2023
[35]

Yang, Y.F

K.Q. Yang, Y.F. Cao, Y .T. Zhang, S.X. Fan, M. Tang, D. Aberg, B. Sadigh, F. Zhou, Self-supervised learning and prediction of microstructure evolution with convolutional recurrent neural networks, Patterns 2(5) (2021)

work page 2021
[36]

P .C. Wu, A.S. Iquebal, K. Ankit, Emulating microstructural evolution during spinodal decomposition using a tensor decomposed convolutional and recurrent neural network, Computational Materials Science 224 (2023)

work page 2023
[37]

C. Hu, S. Martin, R. Dingreville, Accelerating phase-field predictions via recurrent neural networks learning the microstructure evolution in latent space, Computer Methods in Applied Mechanics and Engineering 397 (2022)

work page 2022
[38]

Bonneville, N

C. Bonneville, N. Bieberdorf, A. Hegde, M. Asta, H.N. Najm, L. Capolungo, C. Safta, Accelerating phase field simulations through a hybrid adaptive Fourier neural operator with U -net backbone, Npj Computational Materials 11(1) (2025)

work page 2025
[39]

Zeng, Z.J

Y .H. Zeng, Z.J. He, X.Y . Ma, G.Z. Zhu, X. Zhu, Predicting ultrafast nonlinear dynamics in fiber optics with the enhanced Fourier neural operator, Opt Laser Technol 194 (2026)

work page 2026
[40]

L.L. He, F. Chen, Y .X. Qin, A novel data -driven method for predicting heat flux of hypersonic aircraft based on Fourier neural operator, Aerosp Sci Technol 169 (2026)

work page 2026
[41]

W. Li, M.Z. Bazant, J.E. Zhu, Phase -Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionals, Computer Methods in Applied Mechanics and Engineering 416 (2023)

work page 2023
[42]

Lim, G.Y

D.G. Lim, G.Y . Lee, Y .H. Park, Application of DeepONet to predict transient drop motion of the control rod in real-time, Nucl Eng Technol 57(9) (2025)

work page 2025
[43]

Wang, P .C

X.Q. Wang, P .C. Li, D.T. Lu, Phase -field hydraulic fracturing operator network based on En - DeepONet with integrated physics-informed mechanisms, Computer Methods in Applied Mechanics and Engineering 437 (2025)

work page 2025
[44]

Yang, DeepONet for solving nonlinear partial differential equations with physics -informed training, Neural Networks 197 (2026)

Y.H. Yang, DeepONet for solving nonlinear partial differential equations with physics -informed training, Neural Networks 197 (2026)

work page 2026
[45]

W ang, B.L

P .Y . W ang, B.L. Pan, Z. Liu, L.J. Gao, FDPI-DeepONet: A novel integration for 3D airfoil flow field computation, Acta Mech Sinica-Prc 42(7) (2026)

work page 2026
[46]

S.P . Wang, H. Zhang, X.Y . Jiang, Fractional physics-informed neural networks for time-fractional phase field models, Nonlinear Dynam 110(3) (2022) 2715-2739

work page 2022
[47]

Zang, P .S

Y .H. Zang, P .S. Koutsourelakis, Design-GenNO: A physics-informed generative model with neural operators for inverse microstructure design, Computer Methods in Applied Mechanics and Engineering 450 (2026)

work page 2026
[48]

Zhu, H.H

J.M. Zhu, H.H. Wu, D. W ang, Y .P . Gao, H.L. Wang, Y .L. Hao, R. Y ang, T.Y . Zhang, Y .Z. W ang, Crystallographic analysis and phase field simulation of transformation plasticity in a multifunctional β- Ti alloy, Int J Plasticity 89 (2017) 110-129

work page 2017
[49]

Wang, Z.H

K.Y . Wang, Z.H. Tian, H.H. Wu, J.M. Zhu, S.Z. Wang, G.L. Wu, J.H. Gao, H.T. Zhao, C.L. Zhang, X.P. Mao, Microstructure evolution of austenite -to-martensite transformation in low -alloy steel via thermodynamically assisted phase-field method, J Mater Res Technol 36 (2025) 1683-1689

work page 2025
[50]

Basak, V

A. Basak, V . Levitas, Matrix -precipitate interface-induced martensitic transformation within nanoscale phase field approach: Effect of energy and dimensionless interface width, A cta Mater 189 (2020) 255-265

work page 2020