Equilibrium flow: From Snapshots to Dynamics

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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Quick Start

Low-dimensional dynamical systems

Step 1: Train diffusion model

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.

Step 2: Train dynamics model

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.

Step 3: Load model

importtorchfrommodelsimportFlow, FlowKernelmodel_id='lorenz'score_model, dataset=load_model(model_id)
# Load trained dynamic modelv=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 $v=dx/dt$.

Cite our paper

@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}, }