Equilibrium flow: From Snapshots to Dynamics

Abstract: Scientific data, from cellular snapshots in biology to celestial distributions in cosmology, often consists of static patterns from underlying dynamical systems. These snapshots, while lacking temporal ordering, implicitly encode the processes that preserve them. This work investigates how strongly such a distribution constrains its underlying dynamics and how to recover them. We introduce the Equilibrium flow method, a framework that learns continuous dynamics that preserve a given pattern distribution. Our method successfully identifies plausible dynamics for 2-D systems and recovers the signature chaotic behavior of the Lorenz attractor. For high-dimensional Turing patterns from the Gray-Scott model, we develop an efficient, training-free variant that achieves high fidelity to the ground truth, validated both quantitatively and qualitatively. Our analysis reveals the solution space is constrained not only by the data but also by the learning model's inductive biases. This capability extends beyond recovering known systems, enabling a new paradigm of inverse design for Artificial Life. By specifying a target pattern distribution, we can discover the local interaction rules that preserve it, leading to the spontaneous emergence of complex behaviors, such as life-like flocking, attraction, and repulsion patterns, from simple, user-defined snapshots.

• The paper introduces Equilibrium Flow, a method that infers continuous dynamics preserving static empirical distributions without time-series data.
• It leverages neural network parameterization and score estimation via diffusion models, with a training-free skew-symmetric approach for divergence-free flows.
• Empirical results validate its use across low- and high-dimensional systems, recovering chaotic behavior, Turing patterns, and enabling inverse design of emergent artificial life.

Equilibrium Flow: Inferring Dynamics from Static Pattern Distributions

Introduction and Motivation

The paper introduces the Equilibrium flow method, a framework for inferring continuous dynamics that preserve a given empirical pattern distribution, even when only static snapshot data is available. This approach addresses a central challenge in scientific inference: recovering the underlying dynamical rules from static observations, a problem that arises in fields ranging from biology (cellular morphologies) to cosmology (spatial distributions of celestial bodies). The method is designed to operate without temporal ordering or explicit time-series data, leveraging only the statistical structure of observed patterns.

Methodological Framework

The Equilibrium flow method formalizes the inverse problem as finding a vector field $f(x)$ such that the empirical distribution $p(x)$ is invariant under the induced flow. This is achieved by enforcing the local continuity equation:

$\nabla \cdot [p(x) f(x)] = 0$

which, after manipulation and score function estimation, leads to the pointwise constraint:

$\nabla \cdot f(x) + f(x)^\top s(x) = 0$

where $s(x) = \nabla \log p(x)$ is the score function, estimated via a pre-trained diffusion model. The divergence term is efficiently approximated using Hutchinson's Trace Estimator, enabling tractable optimization in moderate dimensions. The dynamics $f_\theta(x)$ are parameterized by a neural network, trained to minimize the squared violation of the continuity constraint over the data distribution.

Batch normalization is applied to the output of $f_\theta$ to enforce the physical constraint that the mean velocity over the distribution is zero, preventing trivial solutions and improving training stability.

Figure 1: The Equilibrium flow method infers distribution-preserving dynamics for 2D systems, chaotic Lorenz attractors, Turing patterns, and enables inverse design of artificial life forms.

Empirical Results: Recovery of Dynamics

Low-Dimensional Systems

The method is validated on synthetic 2D distributions (mixtures of Gaussians, rings, two-moons), where the learned dynamics exhibit vortical, rotational, or boundary-following flows that preserve the empirical distribution. When applied to samples from the Lorenz system, the recovered dynamics not only preserve the attractor's shape but also exhibit positive Lyapunov exponents, confirming the retention of chaotic behavior.

Figure 2: Learned dynamics preserve 2D distributions and recover chaos in the Lorenz system, as evidenced by divergence of initially close trajectories.

High-Dimensional Pattern Formation

For high-dimensional data, such as Turing patterns generated by the Gray-Scott model, the computational cost of the full neural approach is prohibitive. The authors introduce a training-free method, constructing dynamics as a skew-symmetric linear transformation of the score function, $f_S(x) = S \nabla \log p(x)$, where $S$ is skew-symmetric. This yields a divergence-free flow orthogonal to the score, representing the curl component in Helmholtz decomposition. To stabilize the dynamics and ensure ergodicity, a Langevin term is added, resulting in an irreversible SDE.

Figure 3: Training-free dynamics reproduce qualitative features of Gray-Scott patterns, including soliton movement, spiral and wave propagation, and static maze-like structures.

Uniqueness and Fidelity Analysis

The solution space of distribution-preserving dynamics is empirically characterized using high-dimensional representations of vector fields at sampled points. UMAP embeddings and cosine similarity analyses reveal that neural network-trained dynamics cluster tightly near the ground truth, exhibiting high self-similarity ($0.99 \pm 0.01$) and fidelity ($0.25 \pm 0.01$) to the Lorenz system. In contrast, training-free solutions are more dispersed, and random dynamics are scattered, indicating that neural inductive biases (smoothness, simplicity) strongly constrain the solution space.

Figure 4: UMAP embedding and cosine similarity analysis show that trained dynamics cluster near ground truth, while training-free and random solutions are more dispersed.

Quantitative results for Turing patterns confirm that the training-free method achieves the highest similarity to ground-truth dynamics across all pattern types, outperforming random baselines.

Inverse Design and Artificial Life

A notable application of the Equilibrium flow framework is inverse design: specifying a target pattern distribution and discovering local interaction rules that preserve it. By scattering target patterns in space and inferring dynamics, the method generates artificial life forms exhibiting emergent behaviors such as flocking, attraction, and repulsion, directly linking morphology to motile behavior.

Figure 5: Inverse design yields artificial life forms with emergent collective behaviors, including flocking and pattern-based attraction.

Score Function Estimation and Implementation Details

Score functions are estimated using diffusion models (U-Net architectures with circular padding for periodic boundary conditions), trained to predict either noise or velocity (v-prediction) for improved accuracy. The neural dynamics model employs MLPs with positional encoding and SiLU activations to capture high-frequency and nonlinear behaviors. For high-dimensional convolutional data, skew-symmetric kernels are constructed to ensure divergence-free flows.

Figure 6: Score function estimation via diffusion models and construction of stochastic dynamics combining learned flows and score functions.

Long-Term Distribution Preservation

The method's ability to preserve distributions over long time horizons is validated using Maximum Mean Discrepancy (MMD) metrics. Both learned and training-free dynamics maintain significantly lower MMD values compared to random or score-only approaches, demonstrating superior long-term invariance.

Figure 7: MMD evolution shows that learned and training-free dynamics preserve distributions over time, outperforming random and score-only baselines.

Theoretical Implications and Future Directions

The empirical findings suggest that the uniqueness of recovered dynamics is governed by both data sparsity (manifold constraints) and model inductive bias. The framework provides a tractable approach to the abstract problem of linking Kolmogorov complexity to dynamical inference. Future work should focus on quantifying the influence of data topology and sparsity on solution uniqueness, developing more efficient one-stage models, and exploring alternative divergence estimation techniques. The inverse design paradigm opens new avenues for controlled synthesis of complex systems in artificial life and beyond.

Figure 8: Example of a skew-symmetric convolutional kernel, ensuring divergence-free transformation in high-dimensional pattern dynamics.

Conclusion

The Equilibrium flow method offers a principled approach for inferring distribution-preserving dynamics from static data, validated across low- and high-dimensional systems, including chaotic and pattern-forming regimes. The framework leverages score-based generative modeling, neural parameterization, and training-free constructions to recover plausible dynamics, analyze solution uniqueness, and enable inverse design of emergent behaviors. The dual contribution lies in both advancing computational tools for process inference and providing practical methodologies for temporal reconstruction from snapshot data, with broad implications for scientific discovery and synthetic system design.