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Deep Mesoscopic Dynamics

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

arXiv:2610.17885v1cs.LGcond-mat.stat-mechphysics.comp-ph

TL;DR

Microscopic simulations are costly at large spatial and temporal scales, and mesoscopic equations are not known in closed form for some magnetic systems. DeepMesoD learns mesoscopic field equations from microscopic trajectories, matching atomistic references over extended rollouts while substantially reducing simulation time.

Abstract

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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.

DeepMesoD Methodology

This section presents DeepMesoD, a method for learning thermodynamically constrained mesoscopic field equations from coarse-grained microscopic trajectories, and describes its learning procedure, dynamical scope, and computational performance. It validates the approach against the Ising–Kac hydrodynamic limit and applies it to atomistic magnetic systems, including analysis of CrBr₃ stacking-fault interactions and controlled magnetic responses.

DeepMesoD Methodology

  • DeepMesoD learns free energy and spatial interactions jointly, while factorising dissipation as K = BB∗ to enforce the fluctuation–dissipation relation.The free-energy density allows non-additive bulk and interfacial interactions; the dissipation factor is constrained by symmetry and locality, with a shared local rule on homogeneous periodic domains.Source: DeepMesoD Methodology · pp. 4–5

Validation against hydrodynamic limit

  • DeepMesoD maintains low field mean absolute error over 5,000 evaluation steps at T = 0.75 and 1,000 at T = 1.5, while tracking reference mean magnetisation.The learned homogeneous free-energy density reproduces the temperature-dependent double-well and single-well profiles, and learned dissipation response and radial interaction profile agree with their references.Source: Validation against hydrodynamic limit · p. 7, Figure 2 · p. 8

Discovery of mesoscopic structure in atomistic systems

  • 0.00% unphysical predictions for DeepMesoD across three tested initial conditions compares with >37% for CNN and >81% for FNO.For a triangular-domain initial condition, the learned rollout remains bounded through 25 ns; atomistic reference data are available over 0–1 ns, while CNN and FNO become unstable during long rollouts.Source: Table 1 · p. 10, Discovery of mesoscopic structure in… · p. 10
  • The learned atomistic free energy has minima at mz ≈ ±1 and a non-separable thermodynamic force depending on both mz and Jmz.This nonlinear structure extends beyond the additive dependence assumed in the Ising–Kac hydrodynamic limit.Source: Discovery of mesoscopic structure in… · p. 10, Figure 3 · p. 11
  • The recovered atomistic dissipation is strongly localised, with coupling largest on-site, while the learned interaction kernel has dominant nearest-neighbour structure.The on-site dissipation is approximately diagonal, and its in-plane and out-of-plane coupling strengths peak near the disordered state mz ≈ 0.Source: Discovery of mesoscopic structure in… · pp. 11–12

Microscopic data generation

  • CrBr3 trajectories come from a 51.4 nm film with one midplane antiferromagnetic fault; coarse-graining averages three atomic layers per slab, and magneto-PL measurements show abrupt intensity changes associated with switching.The measurements reveal switching but do not resolve the spatial magnetisation dynamics connecting a fault to the macroscopic response.Source: Discovery of mesoscopic structure in… · p. 12, Magneto-photoluminescence (PL)… · p. 20

CrBr3 implementation and equilibrium calculations

Discovery of mesoscopic structure in atomistic systems

  • DeepMesoD learns a sign-reversed interlayer coupling at the CrBr3 stacking fault and a short-range isotropic in-plane kernel, producing four free-energy minima rather than two.The fault location is not supplied as a model input; the four minima represent parallel or antiparallel magnetisations in blocks above and below the fault.Source: Discovery of mesoscopic structure in… · p. 12, Figure 4 · p. 13
  • Increasing CrBr3 fault number changes the sequence of field-response plateaus and extends the response to higher fields; with three faults, transition fields vary approximately linearly with inverse reversing-block thickness.The plateau changes reflect block alignment across fault interfaces and offer a magnetic interpretation of the magneto-PL steps; thicker reversing blocks switch at lower fields.Source: Figure 4 · p. 13, Discovery of mesoscopic structure in… · p. 15

Relation to other data-driven methods

  • Unlike methods that fit finite-dimensional dynamics or omit finite-volume thermal drift, DeepMesoD learns mesoscopic field equations with fluctuation–dissipation constraints and includes the drift term needed for Gibbs detailed balance.The same recorded mesoscopic trajectories provide increments for both mean evolution and fluctuation statistics, avoiding additional microscopic replica ensembles for prescribed profiles.Source: Relation to other data-driven methods · p. 16

Dynamical assumptions and scope

  • The model assumes Markovian Model-A-type dynamics for a nonconserved coarse field; it is suited to the compared longitudinal dynamics, not systems with conserved fields or appreciable reversible couplings.Spatial averaging suppresses incoherent transverse precession at the chosen scale, while unresolved variables may introduce memory and coloured noise beyond this description.Source: Dynamical assumptions and scope · p. 17

Common coarse-graining and learning procedure

  • The learning procedure represents free energy with a neural energy density and interaction kernel, and constructs a state-dependent positive-semidefinite dissipation operator as K = BBᵀ.Training fits an auxiliary drift, fits the dissipation factor to residual increments, then holds that factor fixed while jointly fitting free energy and interaction to field changes.Source: Common coarse-graining and learning… · p. 18

Simulation timing

  • 16.2 s per 1 ps with VAMPIRE versus 3.96 ms with DeepMesoD for the two-dimensional ferromagnet; for CrBr3, 190.05 s per 2 ps versus 4.8 ms.The VAMPIRE timings use 128 CPU cores; DeepMesoD uses one NVIDIA L40S GPU.Source: Simulation timing · p. 19

1 Theoretical and methodological details

The methodological details connect the free-energy derivative and finite-scale stochastic dynamics to a stationary density, then estimate the model from observed field increments. The estimation procedure separates drift, dissipation, and free-energy fitting.

A.1 Free-energy derivative

  • The free-energy derivative is defined through the L2 inner product, with derivatives of the energy density evaluated at the field and its interaction input.The functional is assumed Fréchet differentiable on the fields considered.Source: §A.1 · p. 24

B Finite-scale stochastic equation

  • The finite-scale equation interprets white noise through a Wiener-process integral, with conditions on the factor kernel and trajectory ensuring that the stochastic integral is well defined.The factor kernel defines a Hilbert–Schmidt operator; predictability and finite-time integrability are also assumed.Source: B Finite-scale stochastic equation · p. 24

C Finite-coordinate stationary density

  • The field observations are represented as coarse-cell coordinates, and the numerical free-energy gradient corresponds to the L2 derivative under unit spatial integration weights.A different numerical basis requires mass matrices in the derivative, factor-adjoint, and noise-covariance representations; the kernel and factor are parameterized by cell coefficients and local field patches.Source: C Finite-coordinate stationary density · pp. 24–25
  • For a nonexplosive finite-coordinate equation, the complete drift yields the stationary density πeq(u) ∝ exp[−βVcFN(u)] under normalisability and zero probability flux.Omitting the state-divergence term leaves an uncancelled derivative of the state-dependent dissipation operator.Source: C Finite-coordinate stationary density · p. 25

D Estimation from field increments

  • Short-time conditional moments motivate a Gaussian quasi-likelihood, using conditional drift for the mean and dissipation for the leading covariance of increments.The first two stages estimate drift and residual covariance, respectively, before the free-energy and interaction fit.Source: D Estimation from field increments · pp. 25–26
  • Stage 1 estimates conditional drift from finite differences, Stage 2 fits dissipation from residual-increment covariance, and Stage 3 fits free energy and interaction to observed field changes.The covariance fit uses a numerical floor; the fitted operator remains state dependent during Stage 3.Source: D Estimation from field increments · p. 26

2 Ising–Kac implementation and experiments

The Ising–Kac implementation coarse-grains Glauber spin dynamics into block magnetisation fields and compares learned dynamics with independent hydrodynamic and finite-block covariance references. The covariance reference is instantaneous, whereas finite-interval sampled increments can also reflect deterministic evolution.

A Microscopic dynamics and coarse graining

  • Block magnetisation is formed by averaging spins in each cell, and the microscopic reference uses heat-bath Glauber spin flips with a normalized interaction field.The simulations use a 4096 × 4096 lattice and 1024-spin blocks to produce a 128 × 128 mesoscopic field.Source: A Microscopic dynamics and coarse… · pp. 26–27

B Independent theoretical references

  • Under Kac scaling, block averages converge to a deterministic hydrodynamic evolution, whose drift can be expressed as dissipation acting against the reference free-energy derivative.The reference covariance instead describes fluctuations in finite averaging cells.Source: B Independent theoretical references · pp. 27–28

C Finite-block covariance reference

  • The finite-block instantaneous covariance is derived from spin-flip quadratic variation and is diagonal across nonoverlapping cells in the infinitesimal-time limit.At finite sampling intervals, sampled increments may include correlations induced by deterministic evolution.Source: C Finite-block covariance reference · p. 28

D Ising implementation

The Ising implementation estimates drift, dissipation, then free energy and interaction, and compares DeepMesoD with hydrodynamic, Allen–Cahn, and patch-CNN baselines. The supplied implementation details describe the fitting and evaluation setup rather than comparative outcome values.

Stage 1 conditional drift.

Stage 2 dissipation factor.

Stage 3 free energy and interaction.

  • Stage 3 jointly fits an even homogeneous free-energy density and a symmetric, unit-sum interaction kernel against observed drift.The learned kernel is not constrained to radial form during training.Source: Stage 3 free energy and interaction · p. 29

E Baselines and displayed quantities

  • The comparison includes a nonlocal hydrodynamic baseline, a second-moment Allen–Cahn approximation, and a patch CNN trained on the same trajectories.Learned and reference dissipation responses are evaluated on the same nonuniform fields and magnetisation bins; radial averaging of interaction coefficients is for visualization only.Source: E Baselines and displayed quantities · p. 30

F Evaluation field rollouts

Evaluation rollouts begin from a separately generated coarse field and compare deterministic DeepMesoD evolution with the mean of three microscopic realizations. The rollout sets the Wiener increment to zero but retains the state-divergence drift term.

F Evaluation field rollouts

G Training-horizon and larger-domain evaluations

At the fixed coarse-graining scale, DeepMesoD retains comparable rollout errors on larger periodic domains without retraining. Increasing the training portion slightly reduces its error at T = 0.75 and leaves it nearly unchanged at T = 1.5.

G Training-horizon and larger-domain evaluations

  • Figure S1 compares the microscopic reference, hydrodynamic-limit equation, Allen–Cahn approximation, patch CNN, and finite-range DeepMesoD across saved times.Green labels mark times within training; ochre labels mark later rollouts, with all panels displayed over [−1, 1].Source: Figure S1 · p. 31
  • DeepMesoD retains comparable rollout errors on larger periodic domains without retraining at the fixed coarse-graining scale, when trained on 128 × 128 fields; errors are full-rollout mean spatial-field MAE against microscopic references.Source: G Training-horizon and larger-domain… · pp. 30–31
  • Increasing the training portion from 10% to 50% slightly reduces DeepMesoD error at T = 0.75 and leaves it nearly unchanged at T = 1.5.The comparison uses the initial portion of the recorded fields in each training trajectory.Source: G Training-horizon and larger-domain… · pp. 30–31

H Spatial structure of the dissipation operator

The learned Ising dissipation operator is evaluated through the RMS magnitudes of its coordinate coefficients. At both temperatures, the on-site coefficient dominates and represented off-site coefficients are small.

H Spatial structure of the dissipation operator

  • At both temperatures, the RMS profile is dominated by the on-site coefficient, while represented off-site offsets are small on the common colour scale.The profile averages three factor seeds, three evaluation trajectories, 24 sampled times per trajectory, and all periodic cell indices.Source: H Spatial structure of the dissipation… · p. 31, Figure S2 · p. 32

A Microscopic dynamics and coarse graining

The study uses atomistic spin dynamics and block magnetisation to construct and test a coarse-grained description. For the longitudinal mz channel, coarse-graining suppresses incoherent fluctuations while coherent transverse precession remains uncoupled from mz.

A Microscopic dynamics and coarse graining

  • The two-dimensional reference is a 1024 × 1024 monolayer sampled into 64 × 64 mesoscopic fields using nonoverlapping 16 × 16 blocks.Ten trajectories are evolved for 250 ps; saved frames from 75–150 ps estimate drift and free energy, while later segments are reserved for evaluation.Source: A Microscopic dynamics and coarse… · p. 34

B Assessment of reversible dynamics under coarse graining

  • The Model-A representation is restricted to easy-axis magnetisation mz and assumes reversible precession contributes negligibly to its coarse-grained evolution.The proposed basis is that block averaging suppresses spatially incoherent transverse precession, while coherent rotation about the easy axis changes mx and my but not mz.Source: B Assessment of reversible dynamics… · p. 34

Z 500 GHz

  • Coarse-graining reduces high-frequency and relaxational spectral powers approximately inversely with block volume, while coherent z-axis precession remains at 5.42 GHz.Jxy is finite, whereas Jzx and Jzy are statistically consistent with zero, supporting Model-A dynamics for mz despite transverse precession.Source: Z 500 GHz · p. 35

C.1 Stage 1: conditional drift

Stage 1 estimates the conditional mean of coarse-field increments with a learned convolutional kernel and MLP. This fitted mean is subtracted before Stage 2, but the Stage 1 network is not part of the final mesoscopic equation.

C.1 Stage 1: conditional drift

  • Stage 1 estimates the conditional mean of coarse-field increments using a learned 13 × 13 circular-convolution kernel and an MLP.The same spatial kernel is applied componentwise to mx, my, and mz; the MLP has three hidden layers of width 64 with tanh activations.Source: §C.1 · p. 35
  • The convolution and MLP are trained jointly by mean-squared error between observed increments and the predicted conditional mean.No additional regularisation term is used.Source: §C.1 · p. 36
  • Stage 1 only estimates the conditional mean subtracted from increments in Stage 2; its MLP and auxiliary kernel do not enter the final mesoscopic equation..Source: §C.1 · p. 36

C.2 Stage 2: dissipation factor

Stage 2 learns a local, state-dependent dissipation operator from covariance observations. Stage 3 then fits the free energy and interaction kernel with the dissipation factor fixed.

C.2 Stage 2: dissipation factor

  • The dissipation operator is constructed from a learned factor, making it symmetric and positive semidefinite, with radius-one factor and maximum radius-two operator.A three-layer circular CNN predicts matrix-valued coefficients for nine offsets from the local magnetisation environment.Source: §C.2 · pp. 36–37
  • At one-cell offset, the learned dissipation magnitude is approximately 4.2%, while represented couplings at larger distances fall below 0.5%.This distance-binned magnitude is measured relative to the on-site block.Source: §C.2 · p. 37

C.3 Stage 3: free energy and interaction

  • Stage 3 freezes the fitted dissipation factor while jointly learning the free-energy density and final interaction kernel.The thermodynamic force differentiates the complete free energy, including neighbouring energy densities coupled through the convolution.Source: §C.3 · pp. 37–38

C.4 Optimisation schedule

  • The three stages run sequentially for 3000, 1000, and 3000 epochs, freezing each preceding checkpoint before fitting the next stage.The Stage 1 auxiliary kernel and Stage 3 interaction kernel are initialized and trained independently.Source: §C.4 · p. 38

D Evaluation field rollouts

DeepMesoD remains accurate beyond its training interval and on unseen domain geometries, where the unconstrained baselines accumulate larger errors. Comparisons with atomistic references are limited to intervals for which those references are available.

D Evaluation field rollouts

  • DeepMesoD remains accurate to 250 ps after training on data through 150 ps, while both baselines develop substantial extrapolation errors and the FNO destabilizes fastest.The drift and free-energy components use the 75–150 ps interval for training.Source: D Evaluation field rollouts · pp. 38–39
  • The rollout tests assess temporal extrapolation and geometric generalization using random, triangular-domain, and domain-wall initial conditions.Atomistic-reference errors are reported only over intervals with available reference data; later segments assess stability, boundedness, and qualitative relaxation.Source: D Evaluation field rollouts · p. 39, Figure S6 · p. 39
  • DeepMesoD matches the atomistic reference through 1000 ps for triangular-domain relaxation, while the FNO and patch CNN accumulate substantially larger errors.Reference-based triangular-domain comparisons cover 0–1000 ps; the triangular configuration was not included in training.Source: D Evaluation field rollouts · p. 39, Figure S7 · p. 40

A Microscopic dynamics and coarse graining

The CrBr3 coarse-graining analysis tests whether averaging suppresses incoherent transverse fluctuations while retaining the slower longitudinal dynamics. Coherent precession remains in the transverse plane, with no detectable reversible motion in mz.

A Microscopic dynamics and coarse graining

  • CrBr3 trajectories use stochastic LLG dynamics and save atomistic configurations and cell-averaged fields every 2 ps over 500 ps.Ten independent trajectories begin from random configurations and are quenched to 50 K at zero applied field.Source: A Microscopic dynamics and coarse… · pp. 40–41
  • CrBr3 simulations represent an 84-layer film with a single midplane antiferromagnetic fault and coarse-grain moments into three-layer slabs.The 200 × 200 × 51.4 nm3 reference film contains approximately 2 × 10^7 microscopic spins; learning uses Vc = 480 and a 40 × 37 × 28 mesoscopic field.Source: A Microscopic dynamics and coarse… · pp. 40–41
  • In CrBr3, high-frequency transverse power decreases nearly inversely with block volume, whereas low-frequency relaxational power decreases more slowly, with a fitted exponent of −0.26 below 10 GHz.Source: A Microscopic dynamics and coarse… · pp. 41–42
  • At 3.53 GHz, coherent single-handed precession about z persists, but its reversible coupling is confined to the transverse mx–my plane.Jzx and Jzy are statistically consistent with zero, while Jxy remains finite; the effective field is 125.9±1.6 mT.Source: A Microscopic dynamics and coarse… · p. 42

B Implementation

The layered implementation separates interlayer free-energy interactions from within-slab dissipation and fits the model in stages. Its learned single-fault interaction can be modified to test prescribed fault geometries without retraining.

B Implementation

  • The model uses a three-stage fit: estimate conditional drift, learn a factorized dissipation operator, then learn free energy and interaction with the factor fixed.Interlayer coupling is represented through the free-energy interaction, while the dissipation factor acts independently within each slab.Source: B Implementation · p. 42

B.1 Factorised three-dimensional interaction kernel

  • The interaction kernel factorizes into a layer-pair matrix and a shared in-plane profile, allowing layer-dependent interactions including surfaces and the midplane fault.Neither the fault location nor its coupling is supplied during training.Source: §B.1 · p. 43

B.2 Stage 1: conditional drift

  • Stage 1 fits a shared MLP to the conditional drift using each slab’s magnetisation and its learned interaction field.This drift estimate defines the conditional mean for Stage 2 and does not enter the final mesoscopic equation.Source: §B.2 · p. 43

B.3 Stage 2: dissipation factor

  • Stage 2 learns a finite-range factor separately within each slab, so dissipation does not couple distinct slabs.The factor has radius one in-plane, and its induced operator has maximum in-plane radius two.Source: §B.3 · pp. 43–44

B.4 Stage 3: free energy and interaction

  • Stage 3 holds the fitted factor fixed and learns the free energy using an independently trained interaction kernel and a local density network.Automatic differentiation of the complete field retains contributions from every energy density coupled through the interaction kernel.Source: §B.4 · p. 44

B.5 Zeeman calibration

  • The Zeeman scale is calibrated after fitting because zero-field training data do not identify it.Paired positive- and negative-field increments at 25 and 50 mT give a coefficient of 0.5581 T−1, with statistical uncertainty 0.0089 on αh.Source: §B.5 · p. 45

C Prediction dynamics

  • Noise-free rollouts retain the Itō thermal drift while setting the Wiener increment to zero; equilibrium field responses are computed from sector weights.The sector weights are estimated using annealed importance sampling.Source: C Prediction dynamics · p. 45

D Inference with multiple stacking faults

  • Prescribed fault motifs can be added to construct multi-fault matrices while leaving the learned free energy, in-plane kernel, dissipation, and network weights unchanged.For each geometry, equilibrium mean fault-plane correlation is evaluated against external field; the switching traces omit points below their running maximum.Source: D Inference with multiple stacking… · p. 45, Figure S10 · p. 46, Figure S11 · p. 47

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