EXPERIMENT 04 · CLUSTER EXPERIMENTS

Train. Compare. Explore.

Configure a Hamiltonian and examine its learned quantum dynamics.

MixFunn · optional FNO · QuTiP

Configure a new training experimentHamiltonian, network architecture and temporal continuation

Hamiltonian and initial state

The couplings define an exchange-symmetric two-qubit Hamiltonian with ℏ = 1.

hxhyhzJxxJyyJzzJxyJxzJyzInitial state

Training and evaluation time

Training horizonEvaluation horizonTraining pointsEvaluation points

Both trajectories start at t = 0. The viewing interval can be selected independently after the results are loaded.

State network

Hidden layers are evaluated in the order shown, followed by a linear output of six real amplitude components.

OptimizerOuter epochsLearning rateRandom seed

Include supervised QuTiP observable loss

With supervision disabled, the state network uses the Schrödinger residual, initial-condition and norm losses. QuTiP remains the reference for evaluation. FNO input, target and seed trajectories are generated by MixFunn.

Fourier neural operatorEnable FNO temporal continuation

Latent channelsFourier layersRetained modesEpochsBatch sizeLearning rateWindow sizeShift sizeWindow stride

Download training code + Slurm script

Run the experiment on the cluster

Place train_experiment.py, mixfunn.py, run_training.sh and the downloaded experiment.json in the same project directory.
Set PROJECT_DIR and PYTHON_BIN in the Slurm script to your project folder and Python environment, then submit:

sbatch run_training.sh experiment.json

After completion, open runs/slurm_JOBID/curves.json using the results control below. The site does not submit jobs, monitor Slurm or upload results automatically.

The JSON contains the complete configuration, including physical loss weights and LBFGS settings. These additional options can be edited in the JSON before submission.

RUN

Explore training results

Included example

Open the curves.json produced by a completed training run, or explore the included experiment without supervised data.

Open curves.json

Imported results stay in this browser tab and are cleared on reload.

Loading the included experiment…

Mean fidelity · MixFunn—

Minimum fidelity · MixFunn—

Mean fidelity · MixFunn + FNO—

Time evolution

Observable

ExactMixFunnMixFunn + FNO

Curves and colors

Exact solution

MixFunn

MixFunn + FNO

Shaded region: beyond the training intervalℏ = 1 · time in model units

State fidelity

F = |⟨ψ_exact|ψ_model⟩|²

F = 1 denotes identical normalized states up to a global phase.

State snapshot

t = 5.00Training domain

020

F · MixFunn

F · FNO

Norm² · unnormalized MixFunn

Fidelity in the selected interval

Summary statistics are evaluated on 801 uniformly spaced samples of the selected viewing interval. The exact reference remains unchanged when its curve is hidden. Imported trajectories are interpolated between stored samples.


Method	Mean fidelity	Minimum fidelity	Maximum error 1 − F
MixFunn
MixFunn + FNO

Methods and numerical evaluation

Hamiltonian-specific neural dynamics

Each experiment approximates the evolution of an exchange-symmetric two-qubit system for a fixed Hamiltonian and initial state. The state network combines MixFunn layers [1] and optional dense layers to represent three complex triplet amplitudes as functions of time. Training minimizes the Schrödinger residual, initial-condition and normalization losses, with optional supervision by QuTiP observables.

Temporal continuation

The optional Fourier neural operator advances overlapping windows of the MixFunn trajectory. Its architecture follows the time-domain neural-operator approach of Shah et al. [2], adapted to a hybrid MixFunn–FNO pipeline. Evaluation against an independent QuTiP solution quantifies the accuracy within and beyond the training interval.

Visualization protocol

Probabilities, correlators and fidelities are linearly interpolated from stored evaluation samples. Each chart uses 801 points within the selected viewing interval; the table and summary cards use that same grid. The time-inspection control interpolates the trajectory at its selected time. Initial conditions remain anchored at t = 0 when the viewing interval changes. No additional evolution is inferred beyond the available evaluation horizon. The shaded region identifies times beyond the training interval and does not imply guaranteed extrapolation accuracy.

References

Farias, T. de S., de Lima, G. G., Maziero, J. & Villas-Boas, C. J. MixFunn: A Neural Network for Differential Equations with Improved Generalization and Interpretability. arXiv 2503.22528 (2025). doi:10.48550/arXiv.2503.22528
Shah, F., Patti, T. L., Berner, J., Tolooshams, B., Kossaifi, J. & Anandkumar, A. Fourier neural operators for learning dynamics in quantum spin systems. Communications Physics 9, 226 (2026). doi:10.1038/s42005-026-02644-1