matthewju/ferroelectric-tdgl-bench

Ferroelectric TDGL Phase-Field Benchmark

This is an open benchmark for learning ferroelectric phase-field dynamics. It contains 1,475 3D simulations of the out-of-plane polarization $P_z(\mathbf r, t)$ in a hafnia-like ferroelectric capacitor. They were produced with the GPU phase-field solver FerroX by solving time-dependent Ginzburg–Landau (TDGL) gradient flow coupled to the electrostatics of the stack. The data and splits are the ones used in:

M. Ju, Y. Tang, K. Wang, H. Sha. Duhamel Neural Operator for Learning Phase-Field Dynamics. AI for Accelerated Materials Design (AI4Mat) Workshop, NeurIPS 2026. Code: https://github.com/matthew-ju/Duhamel-Neural-Operator Dataset DOI: 10.57967/hf/11169

Each trajectory starts from small random noise, about 0.5% of the saturated polarization. In the first stage (incubation), a band of Fourier modes of that noise grows exponentially. Domains then nucleate and saturate. The final domain pattern is decided by which noise modes win during incubation. Every trajectory has its own noise seed. No initial field is shared between training and test data, so a model cannot score well by memorizing seeds.

Contents

folder	trajectories	per trajectory	varied	size
d1/	1001	(401, 10, 64, 64) float32	Landau coefficients α, β, γ (Latin hypercube in |α|, P_S, r)	66 GB
d2/	300	(401, 10, 64, 64) float32	α, β, γ drawn independently in five regions, partly outside D1's range	20 GB
hysteresis/	174	(T, 10, 64, 64) float32, T = 440–2765, plus voltage	α, β, γ and peak voltage V_max	34 GB
simulation/		FerroX input templates and job lists		<1 MB

Array axes are (frame, depth layer, y, x). The polarization unit is C/m². The 10 depth layers span the 5 nm ferroelectric film, bottom (next to the dielectric) to top. The dielectric layers of the simulated stack carry no polarization and are omitted.

Physical system

The polarization relaxes by gradient flow of a Ginzburg–Landau free energy plus electrostatics:

$${\mathrm{\partial }}_t{P}_z=-\mathrm{\Gamma }\frac{\delta F}{\delta P_z},F=\int [\frac{\alpha }{2}{P}_z^2+\frac{G_{11}}{2}\parallel \mathrm{\nabla }{P}_z{\parallel }^2+\frac{\beta }{4}{P}_z^4+\frac{\gamma }{6}{P}_z^6]d\mathbf{r}+U_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}[P_z]\backslash partial\_t\; P\_z\; =\; -\backslash Gamma\backslash ,\backslash frac\{\backslash delta\; F\}\{\backslash delta\; P\_z\},\backslash qquad\; F\; =\; \backslash int\; \backslash Big[\backslash tfrac\{\backslash alpha\}\{2\}P\_z^2\; +\; \backslash tfrac\{G\_\{11\}\}\{2\}\backslash lVert\backslash nabla\; P\_z\backslash rVert^2\; +\; \backslash tfrac\{\backslash beta\}\{4\}P\_z^4\; +\; \backslash tfrac\{\backslash gamma\}\{6\}P\_z^6\backslash Big]\backslash ,d\backslash mathbf\; r\; +\; U\_\{\backslash mathrm\{elec\}\}[P\_z]$$

• Device: a metal / 5 nm ferroelectric (ε_fe = 24) / 4 nm dielectric (ε_de = 10) / metal stack, with shorted electrodes and periodic boundaries in-plane.
• Grid: 64 × 64 in-plane, 10 layers in the film.
• Constants: G₁₁ = 10⁻⁹ V m³/C and Γ = 100 (Ω m)⁻¹.
• Initial field: uniform random in ±2×10⁻³ C/m² in every cell, set by the integer seed.
• Frames: 401 frames (frame 0 is the initial noise). The frame interval dt_frame is chosen per material so the frames cover about 1.25× its time to saturation.
• Hysteresis: ε_de = 100. The top electrode follows a quasi-static sweep 0 → +V_max → −V_max → +V_max in 0.125 V increments, each held until the Poisson solve converges.

Metadata (<dataset>/metadata.csv, one row per trajectory)

column	meaning
traj_id, file	trajectory name and path relative to the dataset folder
split	split used in the paper (below)
seed	FerroX random seed of the initial field
alpha, beta, gamma	Landau coefficients (V m/C, V m⁵/C³, V m⁹/C⁵)
P_s, r	design coordinates: saturated polarization (C/m²) and well shape. r is the share of the restoring force from the sextic term; r > 1 means β < 0, a first-order well
dt, nsteps, plot_int, dt_frame	solver time step (s), number of solver steps, steps per saved frame, frame interval (s)
region (D2)	A_small_Ps, B_units (P_S up to 2.2× the D1 maximum), B_alpha (large |α|, small P_S), C_first_order (r = 1.15–2), D_all
material, eps_de, V_max, paper_eval (hysteresis)	D1 material tag, dielectric permittivity, peak voltage (V), and whether the loop is in the paper's evaluation set

Hysteresis .npz files also hold step (solver step of each frame) and V (the applied voltage in force at each frame, in volts). inc_step and inc_V give the times and values of the voltage increments. The paper uses the frames at every 20th solver step (ferro_bench.subsample_hysteresis).

Splits and protocol

dataset	train	val	test
D1	train (682)	val (63, every 12th training material)	test_interior (82, inside the sampled box), test_edge (174, at its edges, including every well with r ≥ 1.15)
D2	–	–	test (all 300), zero-shot
Hysteresis	train (90 loops)	val (10)	test_material (30, held-out materials), test_noise (34, new noise), test_closed (10, independent closed loops)

Protocol used in the paper:

1. Roll out every prediction from the frame-0 field alone, conditioned on (α, β, γ) and dt_frame. Never feed in a solver frame.
2. Report the pattern correlation (Pearson correlation over all voxels of one frame) and the relative L2 error at frame 1 and frame 400, averaged over trajectories.
3. Select hyperparameters on val only.

For hysteresis, the voltage sequence V is an input and the loop error is reported. The loop error is the mean absolute deviation of the polarization–voltage curve, in units of P_S.

Usage

import ferro_bench as fb                          # numpy only
train = fb.metadata("d1", "train")
P = fb.load("d1", train[0])                       # (401, 10, 64, 64), memory-mapped
test = fb.metadata("d1", ["test_interior", "test_edge"])
scores = fb.score(my_rollout, fb.load("d1", test[0]))   # {'corr@1', 'rel@1', 'corr@400', 'rel@400'}

To download only part of the data:

hf download matthewju/ferroelectric-tdgl-bench --repo-type dataset --include "d2/*" --local-dir ./ferro

Regenerating the data

simulation/ holds the FerroX input templates (inputs_tdgl, inputs_hysteresis) and one job line per trajectory:

• TDGL: name alpha beta gamma nsteps plot_int dt seed phi_hi
• Hysteresis: name alpha beta gamma dt eps_de V_max seed

With FerroX (github.com/AMReX-Microelectronics/FerroX), these reproduce every trajectory from its seed. The code repository has the driver scripts. SHA256SUMS lists a checksum for every file.

Limitations

• All data come from one simulator. It is a uniaxial (P_z only) model without elastic coupling, with one electrode and dielectric configuration.
• The data are compared with simulation, not with experiment.
• Each material is simulated once.

License and citation

Data: CC BY 4.0. ferro_bench.py: MIT. If you use the benchmark, please cite the paper and the dataset:

@inproceedings{ju2026duhamel,
  title     = {Duhamel Neural Operator for Learning Phase-Field Dynamics},
  author    = {Ju, Matthew and Tang, Yingheng and Wang, Kang'an and Sha, Haozhi},
  booktitle = {AI for Accelerated Materials Design (AI4Mat) Workshop, NeurIPS},
  year      = {2026}
}

@misc{ju2026ferrobench,
  title     = {Ferroelectric {TDGL} Phase-Field Benchmark},
  author    = {Ju, Matthew and Tang, Yingheng and Wang, Kang'an and Sha, Haozhi},
  year      = {2026},
  publisher = {Hugging Face},
  doi       = {10.57967/hf/11169},
  url       = {https://huggingface.co/datasets/matthewju/ferroelectric-tdgl-bench}
}

Simulations used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility.