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.

翻译：自组织系统展示了简单的局部规则如何生成复杂的随机模式。许多自然系统依赖于此类动力学机制，使得自组织成为理解自然复杂性的核心。对此类系统进行建模的一个根本挑战在于求解逆问题：从宏观观测中推断未知的因果参数。当观测结果具有强烈随机性、产生多样却等效的模式时，这一任务变得尤为困难。传统逆问题求解方法在此场景下失效，因为逐像素度量无法捕捉可变结果之间的特征相似性。本研究提出一种专门设计用于处理可观测空间随机性的新型逆建模方法，该方法利用视觉嵌入技术生成能够捕捉感知不变性的鲁棒表示。通过将模式表示映射到不变嵌入空间，我们能够有效恢复未知的因果参数，而无需依赖人工设计的损失函数或启发式规则。我们在三个自组织系统上评估了该方法：物理系统、生物系统和社会系统，具体包括反应-扩散系统、胚胎发育模型以及基于智能体的社会隔离模型。实验表明，即使在模式结果存在随机性的情况下，该方法仍能可靠地恢复参数。我们进一步将该方法应用于真实生物模式，凸显其作为理论研究者与实验工作者探究复杂随机模式形成背后动力学的潜在工具价值。