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Structure-preserving Fourier Neural Operators for Long-time Cahn-Hilliard Dynamics under Coarse Temporal Supervision

Yingli Li, Y. C. Zhou

arXiv:2609.39053v1math.NAcs.NAphysics.comp-ph

TL;DR

Long-time Cahn–Hilliard prediction is challenging because high-fidelity simulations are costly and one-step accuracy does not ensure reliable rollouts or capture late-stage coarsening. The paper develops a two-stage FNO that combines soft amplitude control with structure-function supervision; the latter improves normalized structure functions and associated coarsening diagnostics in tested cases.

  • Problem

    High-fidelity Cahn–Hilliard simulations are costly, and many SciML evaluations omit late-stage coarsening and its characteristic-length growth and scaling collapse.

    Source: §1 · pp. 2–3

  • Method

    A two-stage FNO first uses a soft bound-conforming objective and then retains it while adding structure-function supervision.

    Source: §1 · p. 4

  • Results

    Structure-function regularization brought normalized structure functions closer to reference in representative cases, with fitted L^3(t) slopes and k1(t) exponents also moving closer to reference.

    Source: §6 · p. 26

  • Takeaways & Limitations

    The results support controlling amplitude excursions before interpreting coarsening statistics, while statistical supervision provides targeted correction beyond amplitude control alone.

    Source: §1 · p. 4, §6 · pp. 26–27

    The statistical correction is formulated for late-time coarsening; extending it across the distinct evolution regimes remains future work.

    Source: §6 · p. 27

Abstract

from arXiv · show

Accurate long-time prediction of Cahn-Hilliard dynamics, particularly in the late-stage coarsening regime, is computationally demanding because it requires simulations over extended physical time. Various machine learning approaches have been developed to speed up the simulations through efficient surrogate evaluations. While the learned map can be accurate over a single learned interval, small prediction error will accumulate when the map is repeatedly applied in a long-time prediction, leading to large numerical errors. In this work, we identify the structure function for the coarsening dynamics of the Cahn-Hilliard equation and propose a two-stage structure-preserving Fourier Neural Operator (FNO) learning framework. Stage~I introduces a bound-conforming FNO (bcFNO) with a soft, magnitude-dependent amplitude penalty. Stage~II retains the bound-conforming objective and supplements it with structure-function regularization, yielding a structure-preserving FNO (spFNO) that corrects late-stage coarsening statistics. Numerical experiments on the various spatial resolutions and reference-solver time steps suggest that the bcFNO stabilizes the long-time prediction by suppressing large amplitude excursions, without changing the low online cost of the FNO. Furthermore, the structure-function term consistently reduces discrepancies in the normalized structure function and provides further correction in associated late-stage coarsening quantities, including the $L^3(t)$ growth trend and the scaling collapse. These results indicate that our multi-stage implementation of physical principles in operator learning could be applied to other multiscale systems with statistical scaling, such as the functionalized Cahn--Hilliard equations and turbulence, provided that the state constraints and statistical observables are adapted to the governing dynamics.

1. Introduction

Long-time Cahn–Hilliard simulations are costly, while many existing learning studies emphasize simplified or short-horizon tests that do not capture late-stage coarsening. The paper proposes a two-stage FNO that first controls phase-field excursions and then corrects coarsening statistics.

Motivation

  • High-fidelity simulations are costly, and many existing SciML tests omit late-stage coarsening, characteristic-length growth, and structure-function scaling collapse.The fourth-order operator, fine spatial grids, and long physical times contribute to the simulation cost.Source: §1 · pp. 2–3
  • Accurate one-step FNO predictions do not ensure reliable long-time rollouts, where local errors can accumulate into amplitude excursions or nonphysical free-energy behavior.The introduction motivates physically informed training for recursive inference.Source: §1 · p. 3

Contribution

  • The nominal phase-field range [−1, 1] motivates a soft, magnitude-dependent bcFNO penalty, not a hard projection or universal invariant.The penalty discourages excursions while allowing small departures from the reference range.Source: §1 · p. 3

Motivation

  • Late-stage coarsening is the target regime because its growing domains and dynamic scaling unfold over long times, and the spFNO retains bcFNO while supervising the structure function.The introduction also points to statistical observables tailored to other multiscale systems, including turbulence.Source: §1 · pp. 3–4

Contribution

  • The two-stage framework pairs bcFNO amplitude control with spFNO structure-function supervision to stabilize long-time predictions and improve late-stage coarsening statistics.The second stage retains the bound-conforming objective while adding structure-function supervision.Source: §1 · p. 4

2. Cahn–Hilliard Equation and Dynamic Scaling

The section connects Cahn–Hilliard’s conserved, energy-dissipating phase-field evolution to late-stage coarsening statistics. It defines continuous and discrete spectral diagnostics for characteristic-length growth and dynamic scaling.

2.1. Phase-Field Formulation and Cahn–Hilliard Equation

  • The Cahn–Hilliard equation is an H−1 gradient flow for a conserved order parameter, with monotonically decreasing free energy.Its double-well bulk potential has minima at ϕ = ±1, the coexisting stable bulk phases.Source: §2.1 · p. 5

2.2. Correlation Functions and Dynamic Scaling

  • In late-stage coarsening, statistical properties depend on time only through L(t), and L^3(t) is expected to be approximately affine in time, independently of dimensionality for d ≥ 2.The first spectral moment estimates the characteristic inverse length scale.Source: §2.2 · pp. 6–8

2.3. Discrete Spectral Representation of the Scaling Framework

  • The discrete framework uses FFT-based shell-averaged spectra to derive characteristic length, fit late-stage L^3(t), and test collapse; the reference spectra shift toward lower wavenumbers and approximately collapse.The zero mode is isolated and interpolation is restricted to available positive-wavenumber samples.Source: §2.3 · pp. 9–10

3. Numerical Methods for Data Generation and Training

The numerical trajectories use a semi-implicit Fourier pseudospectral solver, with the linear biharmonic term treated implicitly and the nonlinear term explicitly. Coarse supervision uses saved states at a common interval, which an FNO learns to map through Fourier layers.

3.1. Numerical Solver

  • The solver treats the linear biharmonic term implicitly and the nonlinear term explicitly, then evaluates the Fourier-space update using the fast Fourier transform.Training and evaluation trajectories use a first-order semi-implicit Fourier pseudospectral discretization with periodic boundary conditions.Source: §3.1 · pp. 11–12

3.2. Coarse-Supervision Data Generation

  • All models use ΔT = 20 between predicted states, while reference trajectories vary the internal solver time step and provide six training trajectories plus one held-out rollout.The held-out trajectory’s initial condition and states are excluded from training.Source: §3.2 · p. 12

3.3. Fourier Neural Operator

  • The FNO learns the coarse evolution map using Fourier layers that process retained low-frequency modes alongside pointwise linear transforms, then project the latent state to the next solution.A lifting operator first maps input channels to a higher-dimensional latent representation.Source: §3.3 · pp. 12–13, Figure 3 · p. 13

4. Structure-Preserving Fourier Neural Operator

The method trains a bound-conforming FNO first, then continues training with structure-function regularization to target late-stage coarsening statistics. Its objective combines state and bound terms across samples with a structure-function term restricted to coarsening-regime samples.

4. Structure-Preserving Fourier Neural Operator

  • The two-stage procedure trains a bound-conforming FNO, then adds structure-function regularization to correct late-stage coarsening statistics.Stage II continues from the trained Stage I model.Source: §4 · p. 13

4.1. Bound-Conforming Penalty

  • The nominal range |ϕ| ≤ 1 is based on the double-well potential’s minima, not a universal invariant of Cahn–Hilliard dynamics.The bound penalty is soft, and its fixed experimental weight is β = 5.Source: §4.1 · p. 13

4.2. Structure-Function Regularization

  • Structure-function regularization compares predicted and reference normalized structure functions alongside the state loss; the statistic retains neither Fourier phase nor directional spectral information.The structure-function term is applied to samples in the coarsening regime.Source: §4.2 · p. 14

4.2.1. Spectral-shape term

  • The spectral-shape term uses predicted and reference structure functions normalized by their respective peak values, with stop-gradient applied to the predicted maximum.The supplied passages describe the normalization and stop-gradient operation but do not specify the complete shape-loss expression.Source: §4.2.1 · p. 14

4.2.2. Peak-bin and low-intensity-bin terms

  • The structure-function loss includes separately defined peak-bin and low-intensity-bin terms, evaluated per training sample and averaged over the mini-batch.The passages identify reference-based peak and low-intensity bin definitions but do not provide their complete formulas.Source: §4.2.2 · p. 14

4.3. Overall Training Objective

  • State and bound losses apply to every sample, while structure-function loss applies only to coarsening samples and begins in the second training epoch.The coarsening indicator is used during training but is not required for long-time rollout.Source: §4.3 · p. 15, Table 1 · p. 15

5. Experiments

The experiments evaluate bound-conforming FNO (bcFNO) on a manufactured trajectory and source-free coarsening, then test whether structure-function regularization further improves late-stage statistics without added online cost.

5.2. Forced Cahn–Hilliard Equation with an Exact Solution

  • The manufactured exact-solution problem supplies a pointwise reference for accumulated field errors and tests the bound-conforming penalty, while its prescribed forcing does not represent spontaneous phase separation or emergent coarsening.It uses the same penalty without modifying its definition for the forced problem.Source: §5.1 · pp. 15–16, §5.2 · p. 17

5.1. Experimental Setup

  • The experimental framework uses the manufactured problem for Stage I fieldwise evaluation and source-free coarsening to evaluate the full framework, including Stage II structure-function regularization.Both problems use two-dimensional periodic fields; the source-free references come from the numerical solver.Source: §5.1 · p. 16, Figure 4 · p. 16

5.2. Forced Cahn–Hilliard Equation with an Exact Solution

  • At t = 30000, relative field errors are approximately 4.9 for FNO and 0.28 for bcFNO on the held-out manufactured trajectory.The bcFNO suppresses amplitude growth and remains closer to the exact trajectory, though coherent spatial displacement remains visible late in prediction.Source: §5.2 · p. 17, Figure 5 · p. 18

5.3. Source-Free Cahn–Hilliard Coarsening Experiments

  • The source-free coarsening experiment evaluates FNO, bcFNO, and spFNO across eight (N, δt) cases using independently held-out random initial conditions and long-time rollouts.Evaluation includes free energy, amplitude, characteristic wavenumber and length, normalized structure function, and mass-mean deviations; detailed representative results use (128, 0.2) and (256, 0.1).Source: §5.1 · p. 16, §5.3 · pp. 18–19

5.4. Sampling Sensitivity in FNO

  • Across both representative cases, Sampling A keeps FNO predictions bounded, whereas Sampling B produces large excursions beyond the nominal range and departures from reference trajectories.The sampling distributions are case-specific, so this does not compare a fixed set of weights across resolutions; temporal redistribution alone does not reliably control long-time amplitude growth.Source: §5.4 · pp. 19–20, Figure 6 · p. 20

5.5. Effect of the Bound-Conforming Penalty on Long-Time Prediction

  • With the same architecture, data, sampling, and optimization budget at (N, δt) = (128, 0.2), bcFNO has smaller amplitude departures and retains phase-separated patterns, unlike FNO's distorted late-stage domains.On an independent evaluation trajectory, bcFNO also keeps free energy closer to the reference; large FNO excursions recur across tested training seeds.Source: §5.5 · pp. 20–21, Figure 7 · p. 21, Figure 8 · p. 22

5.6. Structure-Function Correction of Coarsening Dynamics

  • For both representative cases, spFNO brings the fitted L3(t) slope and wavenumber exponent closer to the reference than bcFNO; collapse score rises in one case and is unchanged in the other.The structure-function term also reduces discrepancies near the low-wavenumber coarsening peak; spFNO is a spectral correction rather than uniformly stronger stabilization.Source: Table 4 · p. 24, §5.6 · p. 24, Figure 13 · p. 25

5.7. Online Cost of Trained Neural Operators

  • Each neural operator generates the N = 256 trajectory over 0 ≤ t ≤ 30000 in approximately 8.0 s, with bcFNO and spFNO retaining FNO's inference architecture and online cost.The comparison excludes the one-time costs of generating training data and optimizing the operators.Source: §5.7 · p. 26, Table 5 · p. 26

6. Conclusion and Future Work

The conclusion reports that bound-conforming training reduces large amplitude excursions, while structure-function regularization improves late-stage coarsening statistics. The statistical correction is formulated for late-time coarsening; extending it across the full evolution remains future work.

6. Conclusion and Future Work

  • bcFNO reduced large amplitude excursions without changing online evaluation cost, while adding structure-function regularization produced spFNO.Repeated unregularized FNO application had produced free-energy growth, amplitude excursions, and late-time coarsening errors.Source: §6 · p. 26
  • For (N = 128, δt = 0.2) and (N = 256, δt = 0.1), spFNO brought normalized structure functions and fitted L^3(t) slopes and k1(t) exponents closer to reference; collapse scores rose and stayed unchanged, respectively.The collapse score increased for (128, 0.2) and was unchanged for (256, 0.1).Source: §6 · p. 26
  • Amplitude control and physically consistent free-energy behavior alone do not guarantee statistical accuracy, and unreliable spectral quantities cannot characterize coarsening when excursions and free-energy growth occur.The authors frame amplitude control and statistical correction as distinct stages in a hierarchy.Source: §6 · pp. 26–27
  • The structure-function correction targets late-time coarsening; extending the approach across the full evolution remains future work because its regimes have distinct physical mechanisms.The authors suggest stage-aware physical representations and multiscale operator architectures for such an extension.Source: §6 · p. 27
  • The authors identify adapting physically meaningful constraints across equations and dynamical regimes as future work, including for nonlinear PDEs with statistical structure.They mention turbulent flows and other systems with self-similar evolution as examples.Source: §6 · p. 27

Appendix A.1. Training Setup

  • Table A.6 summarizes loss parameters for two representative source-free cases and the ranges used in the remaining experiments.Source: Appendix A.1 · p. 32

Appendix A.2. Reference-defined coarsening interval

  • The coarsening interval is defined from the numerical-reference trajectory and applied to the reference, bcFNO, and spFNO trajectories for each fixed (N, δt).Entry requires five consecutive candidate frames; the interval ends at a specified domain-length threshold or a persistent low-slope condition.Source: Appendix A.2 · p. 32

Appendix A.3. Specification of the Manufactured Forced Trajectory

The appendix specifies a manufactured forced trajectory and its evaluation setup. It also describes how online timing measurements were collected for the solver and neural operators.

Appendix A.3. Specification of the Manufactured Forced Trajectory

  • The manufactured trajectory uses q = 2π/m and representative Fourier modes with 2 ≤ |n| ≤ 8; its phases are drawn once per trajectory and fixed in time.Source: Appendix A.3 · p. 32
  • Table A.8 gives parameters for the held-out manufactured trajectory used in field-level prediction evaluation; its domain side length is m = 256.Training trajectories use different phase realizations and may use different envelope parameters.Source: Table A.8 · p. 33, Appendix A.3 · p. 33
  • The target structure function S⋆ is evaluated analytically from Eq. (5.2) for each prescribed trajectory.Source: Appendix A.3 · p. 33

Appendix A.4. Online Timing Setup

  • Online timing used complete trajectories over 0 ≤ t ≤ 30000 with outputs every ΔT = 20; reported times are medians of five timed repetitions.Neural-operator predictions received two untimed single-step warm-up passes, whereas the spectral-solver benchmark used none.Source: Appendix A.4 · p. 33

Appendix B. Supplementary Results

The supplementary-results section presents training-seed comparisons and energy, amplitude, and mass-conservation results for source-free cases not detailed in the main text.

Appendix B. Supplementary Results

  • The appendix reports training-seed comparisons and energy, amplitude, and mass-conservation results for source-free cases omitted from detailed main-text coverage.Source: Appendix B · p. 34

Appendix B.1. Sensitivity to Training Initialization

Across three training seeds at (N, δt) = (256, 0.1) under Sampling B, bcFNO's maximum field magnitude varied far less than FNO's on the held-out prediction.

Appendix B.1. Sensitivity to Training Initialization

  • 1.15–1.23 for bcFNO versus 2.53–27.6 for FNO: maximum field magnitude over held-out long-time rollouts across three seeds at (N, δt) = (256, 0.1), Sampling B.Source: Appendix B.1 · p. 34

Appendix B.2. Additional Energy and Amplitude Predictions

Across the additional discretizations, bcFNO reduces large amplitude excursions and non-dissipative energy behavior relative to FNO. A short initial energy increase can remain for the coarsest targets, but it differs from persistent late-time growth.

Evaluation scope

Energy predictions

  • bcFNO substantially reduces FNO’s departures from the reference energy’s dissipative trend, although a short initial increase may remain for the coarsest targets.The initial increase comes from bulk-energy growth exceeding the concurrent decrease in interfacial-gradient energy; both contributions later decrease.Source: Appendix B.2 · p. 34, Figure B.17 · p. 36

Amplitude predictions

Appendix B.3. Mass-Conservation Evaluations

Mass drift is concentrated mainly in cases generated with the coarsest internal time step. Neither model explicitly enforces spatial-mean conservation, and bcFNO does not reduce drift uniformly.

Evaluation measure

  • The mass-drift evaluation measures conservation within each predicted trajectory, not deviation from the reference trajectory.Figure B.21 compares reference and model mass-drift trajectories for N = 256.Source: Appendix B.3 · p. 38, Figure B.21 · p. 39

Observed drift

  • Appreciable mass drift occurs primarily in cases generated using the coarsest internal time step.The evaluations compare FNO and bcFNO across N = 256 cases and multiple solver time steps.Source: Appendix B.3 · p. 38, Figure B.21 · p. 39

Interpretation

  • Neither FNO nor bcFNO explicitly enforces the spatial mean, so any bcFNO improvement in mass drift is empirical rather than guaranteed.The mass-drift measure tracks change within each predicted trajectory; exact preservation corresponds to δM(t) = 0.Source: Appendix B.3 · p. 38

Appendix B.4. Results for Additional Discretizations

Across the additional discretizations, spFNO’s normalized structure functions are closer to reference, particularly near the low-wavenumber peak. Improvements in fitted coarsening and collapse quantities are mixed across cases.

Structure functions

Evaluation method

  • The evaluation fits a over the reference-defined coarsening interval, fits β from k1(t), and computes the collapse score from coarsening-interval spectra.The same reference-defined interval is used for reference, bcFNO, and spFNO trajectories; a larger collapse score indicates closer agreement with the reference master curve.Source: Appendix B.4 · p. 39

Finite-regime coarsening results

  • For (N, δt) = (128, 0.5), spFNO moves both a and β closer to reference and increases the collapse score; for (128, 0.1), a and β move closer but the score decreases.No simultaneous improvement in all three quantities is observed for (128, 1.0), (256, 1.0), or (256, 0.2).Source: Appendix B.4 · p. 39, Table B.10 · p. 41

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