Deep Mesoscopic Dynamics

Yue Zhao; Shiqi Wu; Yihang Ouyang; Sergey Grebenchuk; Magdalena Grzeszczyk; Xiaoxu Zhao; Kostya S. Novoselov; Maciej Koperski

cs.LGOct 13, 2026

Deep Mesoscopic Dynamics

Authors: Yue Zhao, Shiqi Wu, Yihang Ouyang, Sergey Grebenchuk, Magdalena Grzeszczyk, Xiaoxu Zhao, Kostya S. Novoselov, Maciej Koperski,

Organizations: Institute for Functional Intelligent Materials, National University of Singapore, Singapore 117544, Singapore · Department of Mathematics, National University of Singapore, Singapore 117543, Singapore · Department of Materials Science and Engineering, National University of Singapore, Singapore 117575, Singapore · Integrative Sciences and Engineering Programme, NUS Graduate School, National University of Singapore, 119077, Singapore ·

Abstract

Microscopic simulations resolve spatial patterns but are feasible only over small domains and short timescales, whereas bulk descriptions erase those patterns through spatial averaging. Mesoscopic fields preserve this spatial organisation, but their effective equations are known only for selected microscopic systems and scaling limits. We introduce Deep Mesoscopic Dynamics (DeepMesoD), which uses neural networks to learn an effective mesoscopic field equation from microscopic trajectories at a prescribed coarse-graining scale. For the nonconserved dissipative dynamics studied here, DeepMesoD learns the free-energy functional with a spatial interaction kernel and a state-dependent dissipation operator constrained by the fluctuation-dissipation relation. We first validate the construction in the Ising-Kac model, where the learned mesoscopic equation recovers the known hydrodynamic limit and rollouts remain stable far beyond the training horizon. We then treat two systems without closed-form mesoscopic descriptions. For a two-dimensional Heisenberg ferromagnet governed by atomistic spin dynamics, DeepMesoD learns a mesoscopic model whose predictions match atomistic references up to 1 ns, and a rollout with an unseen initial condition remains bounded up to 25 ns. For layered CrBr $_3$ , the learned kernel captures the suppression and sign reversal of interlayer coupling at a stacking fault whose location and strength are not supplied as inputs. Controlled changes to the learned interaction show how fault number and placement shape multistep magnetic responses, complementing our magneto-photoluminescence measurements. Across the two atomistic systems, DeepMesoD achieves a three-to-four-order-of-magnitude reduction in wall-clock time compared with atomistic spin dynamics.

Figures & tables

Figure 1: Workflow of DeepMesoD. Microscopic trajectories are coarse-grained into observations of a mesoscopic field at a prescribed spatial scale. DeepMesoD uses these trajectories to estimate the physical structures in a mesoscopic field equation within a prescribed dynamical class.

Figure 2: Validation of DeepMesoD in the Ising–Kac system with hydrodynamic limit. The DeepMesoD rollouts are noise-free with the Itô thermal drift retained. A,B. Field mean absolute error at each evaluation time for $T=0.75$ and $T=1.5$ . Green shading marks the training interval, and yellow shading marks the subsequent rollout-only interval. C,D. Mean magnetisation during the same rollouts. E,F. Homogeneous free-energy density $g(m)$ , compared with the hydrodynamic-limit reference. Each curve is shifted independently so that its minimum is zero. G. Conditional dissipation-operator response $\mathbb{E}\{[\mathcal{K}[\phi]\mathbf{1}](\mathbf{x})\mid\phi(\mathbf{x})=m\}$ , compared with the reference in ( 4 ) computed from the corresponding microscopic trajectories. H. Off-centre interaction weights, averaged over offsets with the same radial distance and compared with the Gaussian Ising–Kac kernel. For DeepMesoD and the CNN baseline, lines show the mean across three training seeds, and shaded bands indicate one standard deviation.

Figure 3: Long-term stability and thermodynamic-structure recovery in atomistic spin dynamics. A. Long-term rollout from a triangular-domain initial condition on the same lattice. The DeepMesoD prediction is noise-free with the Itô thermal drift retained. Training uses random-initial-state trajectories. The triangular-domain initial condition is used for geometric-initial-condition extrapolation, not for fitting these stages. The dashed boundary at

1\,\mathrm{ns}

separates the interval with an atomistic reference (“Observed”) from the interval without a reference (“Extrapolation”), ending at

25\,\mathrm{ns}

. Time labels are in ps. The atomistic reference permits accuracy comparisons over

0

–

1\,\mathrm{ns}

, but extending it to

25\,\mathrm{ns}

is computationally costly. The

1

–

25\,\mathrm{ns}

learned-model rollout assesses numerical stability and qualitative relaxation. B. Predicted magnetisation fields at the indicated times in ps. The triangular minority domain contracts towards a uniform magnetised state. C. Recovered Kernel. Left: spatial decay of the learned dissipation operator, quantified by the Frobenius norm of the block

K_{i,i+d}

with distance

d

relative to the on-site block. Middle: dissipation coupling constants

K_{\perp}

and

K_{\parallel}

as functions of centre spin

m_{z}

, exhibiting an even (symmetric) profile that peaks near the disordered state and reflecting the expected axial anisotropy between the two coupling modes. Right: learned interaction kernel, revealing a dominant nearest-neighbour structure consistent with short-range exchange coupling. D. Recovered free energy. Left: free-energy landscape

F(m_{z},Jm_{z})

exhibiting a double-well profile with minima at

m_{z}=\pm 1

, consistent with the underlying uniaxial anisotropy. Middle: thermodynamic force

\partial F/\partial(Jm_{z})

recovered by DeepMesoD, which is a general nonlinear function of both

m_{z}

and

Jm_{z}

and exhibits non-trivial contour structure. Right: the hydrodynamic limit

\partial F/\partial(Jm_{z})

, which assumes an additive dependence on

m_{z}

alone, yields a purely striped pattern independent of

Jm_{z}

, demonstrating that the additive assumption fails to capture the full free-energy structure of this system.

Model	Random	Domain Wall	Triangular domain
DeepMesoD	0.00%	0.00%	0.00%
CNN	37.59%	94.48%	94.44%
FNO	81.66%	96.66%	98.31%

Table 1: Physical constraint violations in learned-model predictions. Percentage of grid points where the predicted spin magnitude exceeds the unit sphere (

|\mathbf{m}|>1

) across three magnetic configurations, without any post-processing. The 0.00% entries for DeepMesoD indicate zero detected violations.

Figure S1 : Ising field evolution at $T=0.75$ . The DeepMesoD rollouts are noise-free with the Itô thermal drift retained. Rows show the microscopic reference, the hydrodynamic-limit equation, the Allen–Cahn approximation, the patch CNN, and the final finite-range DeepMesoD model. All methods start from the same evaluation field, and the microscopic reference is the average of three independent realisations initialised from that field. Green column labels mark saved times inside the training duration. Ochre labels mark later rollout times. Every panel uses the display range $[-1,1]$ . Values outside this range in the two late CNN panels appear saturated.

Figure S2 : Ising field evolution at $T=1.5$ . The DeepMesoD rollouts are noise-free with the Itô thermal drift retained. Rows show the microscopic reference, the hydrodynamic-limit equation, the Allen–Cahn approximation, the patch CNN, and the final finite-range DeepMesoD model. All methods start from the same evaluation field, and the microscopic reference is the average of three independent realisations initialised from that field. Green column labels mark saved times inside the training duration. Ochre labels mark later rollout times. Every panel uses the display range $[-1,1]$ .

Figure S3 : Training-horizon and larger-domain diagnostics for the Stage 3 drift fit. The DeepMesoD rollouts are noise-free with the Itô thermal drift retained. The first two panels compare the hydrodynamic-limit equation, the Allen–Cahn approximation, the patch CNN, and DeepMesoD when the initial 10% or 50% of the recorded fields in each training trajectory is used. The last two panels compare the same methods on $128\times 128$ , $256\times 256$ , and $512\times 512$ periodic fields. The models are not retrained for the larger domains, and the mesoscopic cell scale and $V_{c}$ remain fixed. Bars report the trajectory-averaged spatial MAE. Error bars for the learned models show one population standard deviation across three random seeds.

Figure S4 : Spatial structure of the learned Ising dissipation operator. The plots show $K_{\mathrm{RMS}}(\mathbf{r})$ , computed from the coordinate realisation $K_{N}(u)=B_{N}(u)B_{N}(u)^{\mathsf{T}}$ on evaluation fields. The RMS statistic reports coefficient magnitude without sign cancellation. The displayed $5\times 5$ window is the complete coordinate support of $K_{N}$ . The coefficient network uses a $15\times 15$ field patch to determine these entries.

Figure S5 : Validation of the effective Model-A description for the longitudinal magnetisation

m_{z}

in a two-dimensional ferromagnet with large damping. (a) The spectral power in both the high-frequency (

100\text{--}500\,\mathrm{GHz}

) and relaxational (

<10\,\mathrm{GHz}

) bands decreases with the coarse-graining volume

V_{c}

, with fitted exponents

-0.95

and

-0.81

, respectively, close to the

V_{c}^{-1}

scaling expected for incoherent fluctuations. (b) The circularly polarised spectrum reveals a coherent, single-handed precession about the

z

axis, with mean frequency

\bar{f}_{\mathrm{prec}}=+5.42\,\mathrm{GHz}

, corresponding to

B_{\mathrm{eff}}=193.3\pm 1.5\,\mathrm{mT}

. (c) The inferred reversible generator is dominated by

J_{xy}

, which describes rotation within the transverse

m_{x}

–

m_{y}

plane, while

J_{zx}

and

J_{zy}

remain consistent with zero (inset, with error bars denoting

\pm 2

SEM). The coherent transverse precession therefore produces no reversible motion in the

m_{z}

channel. Results are obtained from VAMPIRE simulations with

16\times 16

coarse-graining blocks,

V_{c}=256

, and 10 independent seeds.

Figure S6 : Random-initial-condition rollout for the two-dimensional ferromagnet. The DeepMesoD rollout is noise-free with the Itô thermal drift retained. Each column shows a saved time, with green labels marking times within the training-data interval and ochre labels marking extrapolated times. Rows show the atomistic reference, the finite-range DeepMesoD model, the Fourier neural operator (FNO), and the patch CNN. The reference is the mean of independent atomistic realisations initialised from the same coarse field. The right-hand panels show the rollout MAE and the corresponding absolute-error fields. All magnetisation fields use the display range $[-1,1]$ .

Figure S7 : Triangular-domain rollout for the two-dimensional ferromagnet. The DeepMesoD rollout is noise-free with the Itô thermal drift retained. The triangular minority domain is evolved from the same initial condition by the atomistic reference, DeepMesoD, the FNO, and the patch CNN. Green and ochre labels distinguish times before and after the $150\,\mathrm{ps}$ training horizon of the random-initial-state data. Neither colour indicates that triangular-domain data entered training. Comparisons with the atomistic reference are restricted to $0$ – $1000\,\mathrm{ps}$ . DeepMesoD reproduces the domain evolution over this reference interval, whereas the unconstrained baselines develop substantially larger errors. The right-hand panels show the rollout MAE and absolute-error fields only where atomistic reference data are available. All magnetisation fields use the display range $[-1,1]$ .

Figure S8 : Domain-wall rollout for the two-dimensional ferromagnet. The DeepMesoD rollout is noise-free with the Itô thermal drift retained. Time-label colours refer to the random-initial-state training horizon and do not indicate training on domain-wall data. The initial configuration contains a nearly planar domain wall separating oppositely magnetised regions. Rows compare the atomistic reference, DeepMesoD, the FNO, and the patch CNN over the observed and extrapolated time intervals. DeepMesoD maintains the position and morphology of the wall, while the baseline models accumulate larger deviations during extrapolation. The right-hand panels show the rollout MAE and absolute-error fields only where atomistic reference data are available. All magnetisation fields use the display range $[-1,1]$ .

Figure S9 : Validation of the effective Model-A description for the longitudinal magnetisation

m_{z}

in the layered CrBr 3 film. (a) The normalised high-frequency transverse spectral power (

100\text{--}250\,\mathrm{GHz}

) decreases with coarse-graining volume

V_{c}

with a fitted exponent of

-0.86

, close to the

V_{c}^{-1}

scaling expected for spatially incoherent fluctuations. In contrast, the low-frequency relaxational power (

<10\,\mathrm{GHz}

) decreases more slowly, with a fitted exponent of

-0.26

. (b) The circularly polarised spectrum identifies a coherent, single-handed precession about the

z

axis, with mean frequency

\bar{f}_{\mathrm{prec}}=+3.53\,\mathrm{GHz}

, corresponding to an effective field

B_{\mathrm{eff}}=125.9\pm 1.6\,\mathrm{mT}

. (c) The antisymmetric correlation rate is dominated by

J_{xy}

, which describes rotation in the transverse

m_{x}

–

m_{y}

plane, whereas

J_{zx}

and

J_{zy}

are consistent with zero (inset, with error bars denoting

\pm 2

SEM). The coherent transverse precession therefore generates no detectable reversible motion in the

m_{z}

channel. Results are obtained from VAMPIRE simulations of the 84-layer film using lattice-commensurate blocks with the training coarse-graining volume

V_{c}=480

and 10 independent seeds.

Figure S10 : Construction of multi-fault interlayer kernels at fixed film thickness. From left to right, the learned single-fault structure is used to assemble $28\times 28$ layer matrices containing 1, 3, 5, and 7 stacking faults. Dashed horizontal and vertical lines mark the prescribed fault interfaces. At every added interface, the local fault correction extracted from the learned single-fault $J_{z}$ is added to the background matrix defined above. All matrices are displayed on the same normalised colour scale. Only $J_{z}$ is modified. The film thickness, local free energy, in-plane interaction, dissipation operator, and network weights remain unchanged, and no retraining is performed.

Figure S11 : Effect of fault placement at fixed fault number. A. Four interlayer matrices assembled from the learned fault motif, each containing three stacking faults at different prescribed interfaces. The labels give the corresponding block thicknesses in coarse-slab units. B. Equilibrium mean fault-plane correlation $\xi_{\mathrm{sim}}$ as a function of external field for the same four geometries. The field dependence is calculated from the equilibrium sector weights described below. Points below the running maximum of $\xi_{\mathrm{sim}}$ as the field increases are omitted from the displayed switching traces. The complete unfiltered values are retained in the source data. The local free energy, in-plane interaction, dissipation operator, field calibration, and network weights are unchanged.

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