Solving Inverse Problems in Stochastic Self-Organizing Systems through Invariant Representations

TL;DR

This work tackles inverse problems in stochastic self-organizing systems by replacing pixel-based discrepancy with invariant, perceptual embeddings. It uses a pre-trained visual embedding model (e.g., CLIP) to map patterns into a stable embedding space and employs CMA-ES to optimize forward-model parameters so that the target and generated patterns have close embeddings, thereby handling stochastic observable space. The method is demonstrated across physics (Gray-Scott reaction-diffusion), biology (embryonic development via Cellular Potts models), and sociology (Schelling segregation ABM), with additional tests on natural patterns such as ocellated lizard and zebra skin patterns. The results show robust parameter recovery and offer a practical tool for theorists and experimentalists to investigate and refine mechanistic explanations of complex stochastic pattern formation, without relying on handcrafted loss functions. By revealing when a model lacks expressivity, the approach also guides iterative model development toward more faithful representations of underlying dynamics.

Abstract

Self-organizing systems demonstrate how simple local rules can generate complex stochastic patterns. Many natural systems rely on such dynamics, making self-organization central to understanding natural complexity. A fundamental challenge in modeling such systems is solving the inverse problem: finding the unknown causal parameters from macroscopic observations. This task becomes particularly difficult when observations have a strong stochastic component, yielding diverse yet equivalent patterns. Traditional inverse methods fail in this setting, as pixel-wise metrics cannot capture feature similarities between variable outcomes. In this work, we introduce a novel inverse modeling method specifically designed to handle stochasticity in the observable space, leveraging the capacity of visual embeddings to produce robust representations that capture perceptual invariances. By mapping the pattern representations onto an invariant embedding space, we can effectively recover unknown causal parameters without the need for handcrafted objective functions or heuristics. We evaluate the method on three self-organizing systems: a physical, a biological, and a social one; namely, a reaction-diffusion system, a model of embryonic development, and an agent-based model of social segregation. We show that the method reliably recovers parameters despite stochasticity in the pattern outcomes. We further apply the method to real biological patterns, highlighting its potential as a tool for both theorists and experimentalists to investigate the dynamics underlying complex stochastic pattern formation.

Figures (16)

• Figure 1: Inverse problem diagram. Inverse modeling consists in finding the mapping from observations to their causal space (also referred to as parameter space, solution space or domain space in the literature). It presents several challenges: (i) solutions may not exist for all observations; (ii) the problem of uniqueness: multiple causes may produce identical observations; and (iii) observations may be stochastic, resulting in different observed patterns for the same model parameters. Our method seeks to address the challenge of stochasticity in the observable space.
• Figure 2: Stochasticity in self-organizing systems. Three simulation instances of two self-organizing models run with the same parameters. (Top): Reaction-diffusion model; the stochasticity results from varying initial conditions $s_0\sim\mathcal{S}$. (Bottom): Schelling's model segregation; the stochasticity is present in the model $F$ itself in the form of asynchronous updates. Both systems are examples of stochasticity in the observable space.
• Figure 3: Invariant representations. (a): Diagram representing models parameters $\theta_i$, generating observable patterns $y_i$, and the visual patterns being mapped to an embedding $z_i$. (b): Distance matrix between embedding representations $z$ of four patterns (two reaction-diffusion and two ABM segregation patterns). Notice that each of the patterns is a unique instantiation of the model. Similarity clusters appear, suggesting that the embedding model represents patterns with similar features as similar vectors. The values displayed are pairwise cosine distances $1 - \cos(z_i, z_j)$.
• Figure 4: Method overview. A target pattern is mapped onto an embedding space $z_{\text{target}} \in \mathbb{R}^\mathcal{D}$ using the embedding model $E$, where $D$ is the embedding space dimension. A forward model, parametrised by $\theta \in \mathbb{R}^\mathcal{M}$, where $\mathcal{M}$ is the degrees of freedom of the model, generates a pattern which is mapped to the same embedding space resulting in $z_{\text{solution}}$. The loss $\mathcal{L}$ between the target and the candidate solution embedding guides the optimizer, iteratively updating $\theta$ until convergence. By operating in embedding space, the method offers robustness against the inherent stochasticity of emergent patterns.
• Figure 5: Results for Gray-Scott reaction-diffusion system. (a): A selection of three target patterns $y_i$ (left column), and three reconstructions $y_i', y_i", y_i"'$ of the causal parameters found by the inverse method on three independent training runs. The method reconstructs patterns with features that qualitatively match the targets. (b): Loss curves showing the average loss and variance of 10 training runs for 12 different patterns. (c): Histogram showing the mean-square errors between the causal $\theta$ and optimized parameters $\theta'$; error bars indicate one standard deviation of the 10 training runs averaged for all the 12 target patterns.