How does snapshot distribution constrain the possible dynamics? When we see a pattern, how confidently can we say "This is the underlying dynamics" without seeing the time evolution? How does artificial life relate to real biological life? To answer these fundamental questions, we propose Equilibrium flow: by learning the distribution-preserving dynamics, we can find possible dynamics to preserve the given data distribution without time information.

For 2D systems, our method finds interesting non-trivial dynamics that preserve them. For Lorenz system, a dynamical system with chaotic behavior, the recovered dynamics also exhibit chaotic behavior with positive Lyapunov exponents. For Turing patterns, we propose a training-free method, which has a limited solution space, but is much faster. The resulting dynamics are also highly aligned to the ground-truth.

Beyond these, we also explore the design capability with our method on Artificial Life. With given manually designed patterns, our method not only finds the dynamics / neural cellular automata that preserve the pattern, but also reveals collective behaviors.

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Run the following command:

python train_diffusion.py --model lorenz [or two_peaks, ring, two_moons]

This will train a diffusion model on the Lorenz system. The trained model is saved in ./results/lorenz/diffusion_model.pth.

Run:

python train_dynamics.py --model lorenz --num_experiments 1
# Use the same model as the diffusion model
# You can set num_experiments to the desired number of experiments if you want multiple results

The trained dynamics model is saved in ./results/lorenz/models/dynamics_models_<id>.pth.

import torch
from models import Flow, FlowKernel


model_id = 'lorenz'
score_model, dataset = load_model(model_id)

# Load trained dynamic model
v = FlowKernel(dim=dataset.dim)
v.load_state_dict(torch.load(f'./results/lorenz/models/dynamics_model_{model_id}.pth'))

This v model takes a torch tensor and return $v=dx/dt$.

For Turing patterns, you can generate dataset by running:

cd ./2D_continous/turing_pattern/
python generate_dataset.py --preset life maze waves spirals --cuda --normalize

For Artificial Life, you can generate dataset by running:

cd ./2D_continous/alifes/
python random_images.py --num_sample 8192 --image_path [your_image_path.png]

cd ./2D_continous/
bash train_alifes_diffusion.sh
bash train_turing_diffusion.sh

cd ./2D_continous/
bash train_dynamics.sh

All the experiments can be found in ./2D_continous/training_free_turing.ipynb and training_free_alife.ipynb.

@misc{zhang2025equilibriumflowsnapshotsdynamics,
      title={Equilibrium flow: From Snapshots to Dynamics}, 
      author={Yanbo Zhang and Michael Levin},
      year={2025},
      eprint={2509.17990},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2509.17990}, 
}