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 ( ±0.85 ) and spinodal ( ±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
c^i(t+Δt)=ci(t)−∑jFij
same
same
same
c^i(t+Δt)=ci(t)+Φθ,i(c(t))
Fij eval.
M^ij(μ^i−μ^j)Δt+B^ij2Δtξ
same but B^=0
same as FE-MV
same but B^=0
N/A
Chemical potential
autograd μ^=∂E^θ2/∂c
same
direct F^θ2(c,T)
same
N/A
Stochastic
Yes
No
Yes
No
No
Table 1: Comparison of surrogate model variants.
Figure 3: Results of 1D CHE initialized with c∼0+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.
Figure 4: Example 3D SPDE trajectories with 163 grid of 50:50 mixture (⟨c⟩=0) with ε=0.2 starting from the same initial condition.
Figure 5: Example 3D SPDE trajectories of 70:30 mixture ( ⟨c⟩=0.4 ) on a 643 grid with ε=0.2 , starting from the same initial condition with slight perturbation U(−0.01,0.01) . (a) Evolution of the number of precipitate clusters (voxels with c≤−0.8 ) and the precipitate volume fraction. (b) Precipitate structure evolution for ε=0 and (c) ε=0.2 .
Figure 6: Generalization test of learned surrogate models with ⟨c⟩=0.4 on a 643 grid for a long rollout. (a) Ground truth SPDE trajectory with ε=0.2 , (b) non-flux predictions, (c) FE-MV , (d) FE-M , (e) nonFE-MV , and (f) nonFE-M .
Figure 7: The standard deviation of voxel values of ML models. Free energy based models FE-MV , FE-M for (a) ⟨c⟩=0 and (b) ⟨c⟩=0.4 , and non free energy nonFE-MV , nonFE-M for (c) ⟨c⟩=0 and (d) ⟨c⟩=0.4 . The mean-only models show substantial delay in demixing.
Figure 8: Machine learned mobility M of different models compared to the ground truth from 100 snapshots up to t=10 with ⟨c⟩=0 on the 163 grid.
Figure 9: Extrapolated agglomeration test for clusters and volume fraction evolution of ⟨c⟩=0.4 on a 643 grid
Figure 10: Total free energy evolution for the same tests.
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).
Figure 12: Comparison for extrapolation trajectory ⟨c⟩=−0.51 . (a) Ground truth value ε=0.2 . (b) FE-MV model.
Figure 13: Distribution of precipitation time tp over 40 independent runs, FE-MV VS stochastic PDE simulation.
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Supplementary Figure S1 : Bulk free energy and solubility limit at different temperature.
Supplementary Figure S2 : Results for a 322 grid with ⟨c⟩=0 (TOP) The standard deviation voxel (1st MIDDLE) value frames (2nd MIDDLE), Fourier spectrum (BOTTOM) normalized Fourier spectrum.
Supplementary Figure S3 : 32 meshgrid with c0=0.4+U(−0.01,0.01) (TOP) The standard deviation voxel (1st MIDDLE) value frames (2nd MIDDLE), Fourier spectrum (BOTTOM) normalized Fourier spectrum.
Supplementary Figure S4 : 256 meshgrid with c0=0.4+U(−0.01,0.01) (TOP) The standard deviation voxel (1st MIDDLE) value frames (2nd MIDDLE), Fourier spectrum (BOTTOM) normalized Fourier spectrum.
Supplementary Figure S5 : Top The standard deviation of voxel values from 10 independent 3D simulations at ⟨c⟩=0 with different noise levels ε= 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 ( ε=0 ) solved with different time step size. Bottom Same for for SPDE simulations ( ε=0.2 ). Left column: time step of 10−3 , right: time step of 10−4 .
Supplementary Figure S6 : Top: The evolution of mean ± standard deviation of voxel values of 10 random SPDE samples with respect to different stochastic noise levels, ε=0 and ε=0.2 . Middle: The corresponding histogram of voxel values. Bottom: The concentration frames from SPDE ε=0 and ε=0.2 Left: Initial concentration field c0=U(−0.01,0.01) . Right: Initial concentration field c0=0.4+U(−0.01,0.01) .
Supplementary Figure S7 : The standard deviation of voxel value of the non-flux model averaged over 100 validation trajectories on a 163 grid and the corresponding c value histogram at t=10 . Left: c0=U(−0.01,0.01) Right: c0=0.4+U(−0.01,0.01) .
Supplementary Figure S8 : Prediction and GT (no noise and noise ε=0.2 ) histograms at t=10s on mesh grid 16×16×16 . Top row: c0=U(−0.01,0.01) Bottom row: c0=0.4+U(−0.01,0.01)
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.
Vijay Yadav, Pallvi Pandey, Madhu Priya +2
Department of Physics, Indian Institute of Technology Jodhpur, Jodhpur, Rajasthan 342 030, India · Department of Physics, Birla Institute of Technology Mesra, Ranchi, Jharkhand 835 215, India · Department of Physics, Central University of Rajasthan, Ajmer, Rajasthan 305 817, India
The Schrödinger Bridge Problem connects an initial distribution to a terminal one along a minimum-energy stochastic process. Its mean-field extension, the Mean-Field Schrödinger Bridge, governs interacting populations whose dynamics and costs depend on the collective distribution. When these interactions are nonlocal, their direct evaluation scales quadratically with the population size, making large ensembles intractable within FBSDE-based solvers. We replace these terms with neural surrogates in state and time, trained on empirical interaction values along sampled trajectories and embedded in a four-stage alternating scheme that updates the forward and backward potentials and the surrogates in turn, while preserving forward--backward consistency and the prescribed endpoint marginals. We derive Grönwall-type stability bounds quantifying how surrogate errors propagate to the generated trajectories under a small-gain condition. On crowd-navigation and high-dimensional opinion-dynamics benchmarks, the surrogates reproduce the trajectories obtained with exact evaluation at reduced training cost. The advantage is most significant when the interaction is a nonlinear functional of the measure, such as the normalized bounded-confidence drift, for which random-batch subsampling is biased and unstable whereas the learned surrogate remains accurate.
Daisuke Inoue, Dante Kalise, Mathieu Laurière
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom. · Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning, NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai, Shanghai 200126, People’s Republic of China.
Exact stochastic equations for non-equilibrium dynamics are rarely accessible. We show that the long-time evolution of stochastic dynamics can be predicted from configuration pairs at a fixed short time lag, without knowledge of the equation of motion. Generative diffusion models learn the finite-time transition kernel from these pairs, and iterating it propagates the dynamics far beyond the training lag. For two-dimensional Model B, the diffusive dynamics of a conserved order parameter, the learned kernels reproduce dynamic critical scaling and self-similar t1/3 coarsening. Agreement with direct simulations persists on lattices twice the largest training size and for initial ensembles absent from training. For driven colloids in a periodic optical potential, ten minutes of measured trajectories suffice to predict the particle current and mean passage time over the next twenty minutes within experimental uncertainty. Short-time observations thus contain the information needed to predict emergent non-equilibrium dynamics at much longer times.
Yang-yang Tan, Jinyang Li, Lingxiao Wang
Institute for Physics of Intelligence, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan · RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), Wako, Saitama 351-0198, Japan · KEK Theory Center, Institute of Particle and Nuclear Studies +1