Learning noisy phase transition dynamics from stochastic partial differential equations

Luning Sun; Van Hai Nguyen; Shusen Liu; John Klepeis; Fei Zhou

physics.comp-phMar 31, 2026

Learning noisy phase transition dynamics from stochastic partial differential equations

Authors: Luning Sun, Van Hai Nguyen, Shusen Liu, John Klepeis, Fei Zhou

Organizations: Lawrence Livermore National Laboratory, Livermore, CA, 94550,USA · The University of Texas at Austin, Austin, TX, 78712, USA

Abstract

The non-equilibrium dynamics of mesoscale phase transitions are fundamentally shaped by thermal fluctuations, which not only seed instabilities but actively control kinetic pathways, including rare barrier-crossing events such as nucleation that are entirely inaccessible to deterministic models. Machine-learning surrogates for such systems must therefore represent stochasticity explicitly, enforce conservation laws by construction, and expose physically interpretable structure. We develop physics-aware surrogate models for the stochastic Cahn-Hilliard equation in 3D that satisfy all three requirements simultaneously. The key innovation is to parameterize the surrogate at the level of inter-cell fluxes, decomposing each flux into a deterministic mobility-weighted chemical-potential gradient and a learnable noise amplitude. This design guarantees exact mass conservation at every step and adds physical fluctuations to inter-cell mass transport. A learnable free energy functional provides thermodynamic interpretability, validated by independent recovery of the bulk double-well landscape, interfacial excess energy, and curvature-independent interfacial tension. Tests demonstrate accurate reproduction of ensemble statistics and noise-accelerated coarsening, with generalization to spatial domains 64 times larger in volume and temporal horizons 160x longer than those seen during training. Critically, the stochastic surrogate captures thermally activated nucleation in the metastable regime, a qualitative capability that no deterministic surrogate can provide regardless of training, thus establishing flux-level stochasticity as an architectural necessity rather than an optional enhancement.

Figures & tables

Figure 1: Bulk free energy and chemical potential at

T=0.2

. The concentrations of solubility (

\pm 0.85

) and spinodal (

\pm 0.5

) are highlighted.

Figure 2: Flux-preserved training and sampling framework.

	FE-MV	FE-M	nonFE-MV	nonFE-M	non-flux
model	Free energy, mean+variance	Free energy, mean only	Non-FE, mean+variance	Non-FE, mean only	Non-flux black-box
Conserv.	Yes	Yes	Yes	Yes	No
Value update	$\hat{c}_{i}(t{+}\Delta t)=c_{i}(t)-\sum_{j}F_{ij}$	same	same	same	$\hat{c}_{i}(t{+}\Delta t)=c_{i}(t)+\Phi_{\theta,i}(c(t))$
$F_{ij}$ eval.	$\hat{M}_{ij}(\hat{\mu}_{i}{-}\hat{\mu}_{j})\Delta t+\hat{B}_{ij}\sqrt{2\Delta t}\,\xi$	same but $\hat{B}{=}0$	same as FE-MV	same but $\hat{B}{=}0$	N/A
Chemical potential	autograd $\hat{\mu}=\partial\hat{E}_{\theta_{2}}/\partial c$	same	direct $\hat{F}_{\theta_{2}}(c,T)$	same	N/A
Stochastic	Yes	No	Yes	No	No

Table 1: Comparison of surrogate model variants.

Panel of Figure 3 — Figure 3: Results of 1D CHE initialized with $c\sim 0+\mathcal{U}(-0.01,0.01)$ . (a) Time evolution of $c$ without and with noise. (b) Standard deviation of $c$ over time. (c) Representative frames without and with noise.

Panel of Figure 4 — Figure 4: Example 3D SPDE trajectories with $16^{3}$ grid of 50:50 mixture $(\langle c\rangle=0)$ with $\varepsilon=0.2$ starting from the same initial condition.

Panel of Figure 5 — Figure 5: Example 3D SPDE trajectories of 70:30 mixture ( $\langle c\rangle=0.4$ ) on a $64^{3}$ grid with $\varepsilon=0.2$ , starting from the same initial condition with slight perturbation $\mathcal{U}(-0.01,0.01)$ . (a) Evolution of the number of precipitate clusters (voxels with $c\leq-0.8$ ) and the precipitate volume fraction. (b) Precipitate structure evolution for $\varepsilon=0$ and (c) $\varepsilon=0.2$ .

Panel of Figure 6 — Figure 6: Generalization test of learned surrogate models with $\langle c\rangle=0.4$ on a $64^{3}$ grid for a long rollout. (a) Ground truth SPDE trajectory with $\varepsilon=0.2$ , (b) non-flux predictions, (c) FE-MV , (d) FE-M , (e) nonFE-MV , and (f) nonFE-M .

Panel of Figure 7 — Figure 7: The standard deviation of voxel values of ML models. Free energy based models FE-MV , FE-M for (a) $\langle c\rangle=0$ and (b) $\langle c\rangle=0.4$ , and non free energy nonFE-MV , nonFE-M for (c) $\langle c\rangle=0$ and (d) $\langle c\rangle=0.4$ . The mean-only models show substantial delay in demixing.

Panel of Figure 8 — Figure 8: Machine learned mobility $M$ of different models compared to the ground truth from 100 snapshots up to $t=10$ with $\langle c\rangle=0$ on the $16^{3}$ grid.

Panel of Figure 9 — Figure 9: Extrapolated agglomeration test for clusters and volume fraction evolution of $\langle c\rangle=0.4$ on a $64^{3}$ grid

Panel of Figure 10 — Figure 10: Total free energy evolution for the same tests.

Panel of Figure 11 — Figure 11: FE-MV learned free energy curves. (a) Bulk free energy landscape. (b) Free energy for flat interface along the $z$ axis. (c) Interfacial free energy per area vs radius (spherical interface).

Panel of Figure 12 — Figure 12: Comparison for extrapolation trajectory $\langle c\rangle=-0.51$ . (a) Ground truth value $\varepsilon=0.2$ . (b) FE-MV model.

Figure 13: Distribution of precipitation time $t_{p}$ over 40 independent runs, FE-MV VS stochastic PDE simulation.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 15 — Supplementary Figure S1 : Bulk free energy and solubility limit at different temperature.

Panel of Figure 16 — Supplementary Figure S2 : Results for a $32^{2}$ grid with $\langle c\rangle=0$ (TOP) The standard deviation voxel (1st MIDDLE) value frames (2nd MIDDLE), Fourier spectrum (BOTTOM) normalized Fourier spectrum.

Panel of Figure 17 — Supplementary Figure S3 : 32 meshgrid with $c_{0}=0.4+\mathcal{U}\left(-0.01,0.01\right)$ (TOP) The standard deviation voxel (1st MIDDLE) value frames (2nd MIDDLE), Fourier spectrum (BOTTOM) normalized Fourier spectrum.

Panel of Figure 18 — Supplementary Figure S4 : 256 meshgrid with $c_{0}=0.4+\mathcal{U}\left(-0.01,0.01\right)$ (TOP) The standard deviation voxel (1st MIDDLE) value frames (2nd MIDDLE), Fourier spectrum (BOTTOM) normalized Fourier spectrum.

Panel of Figure 19 — Supplementary Figure S5 : Top The standard deviation of voxel values from 10 independent 3D simulations at $\langle c\rangle=0$ with different noise levels $\varepsilon=$ 0, 0.02, 0.2 and integrated with time step size of $10^{-3}$ and $10^{-4}$ . Middle The histogram of voxel values from 10 PDE simulations ( $\varepsilon=0$ ) solved with different time step size. Bottom Same for for SPDE simulations ( $\varepsilon=0.2$ ). Left column: time step of $10^{-3}$ , right: time step of $10^{-4}$ .

Panel of Figure 20 — Supplementary Figure S6 : Top: The evolution of mean $\pm$ standard deviation of voxel values of 10 random SPDE samples with respect to different stochastic noise levels, $\varepsilon=0$ and $\varepsilon=0.2$ . Middle: The corresponding histogram of voxel values. Bottom: The concentration frames from SPDE $\varepsilon=0$ and $\varepsilon=0.2$ Left: Initial concentration field $c_{0}=\mathcal{U}\left(-0.01,0.01\right)$ . Right: Initial concentration field $c_{0}=0.4+\mathcal{U}\left(-0.01,0.01\right)$ .

Panel of Figure 21 — Supplementary Figure S7 : The standard deviation of voxel value of the non-flux model averaged over 100 validation trajectories on a $16^{3}$ grid and the corresponding $c$ value histogram at $t=10$ . Left: $c_{0}=\mathcal{U}\left(-0.01,0.01\right)$ Right: $c_{0}=0.4+\mathcal{U}\left(-0.01,0.01\right)$ .

Panel of Figure 22 — Supplementary Figure S8 : Prediction and GT (no noise and noise $\varepsilon=0.2$ ) histograms at $t=10s$ on mesh grid $16\times 16\times 16$ . Top row: $c_{0}=\mathcal{U}\left(-0.01,0.01\right)$ Bottom row: $c_{0}=0.4+\mathcal{U}\left(-0.01,0.01\right)$

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Dynamical Systems PDE Surrogate Modeling Physics-Informed ML

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The spatiotemporal evolution of many physical, chemical, and biological systems is described by nonlinear partial differential equations (PDEs). Recently, deep neural network-based surrogate models have emerged as efficient alternatives to computationally expensive numerical PDE solvers. In this work, we propose a physics-constrained deep neural network as a surrogate model to learn the microstructural evolution of a binary mixture, in which conservation of the order parameter is imposed directly on the network output as a hard constraint. We train the model to accurately predict the time-evolution of phase separation in binary mixtures governed by the Cahn-Hilliard equation. We show that predictions from our trained surrogate model remain stable and accurate over long-time rollouts for both critical and off-critical mixtures and preserve the mixture composition throughout evolution. In contrast, a variant in which conservation is enforced only via a penalty term in the loss function drifts away from the initial composition and significantly loses predictive accuracy over the same rollout. This establishes that the hard constraint is essential for long-time stability and order parameter conservation. We also show that our model accurately captures the growth of domain size and is consistent with the Lifshitz-Slyozov domain-growth law. These results demonstrate the effectiveness of the proposed framework for modeling systems with conserved kinetics, and the construction extends directly to other systems with conserved quantities.

Abstract

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