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Long after Turing's seminal Reaction-Diffusion (RD) model, the elegance ofhis fundamental equations alleviated much of the skepticism surrounding patternformation. Though Turing model is a simplification and an idealization, it isone of the best-known theoretical models to explain patterns as a reminiscentof those observed in nature. Over the years, concerted efforts have been madeto align theoretical models to explain patterns in real systems. The apparentdifficulty in identifying the specific dynamics of the RD system makes theproblem particularly challenging. Interestingly, we observe Turing-likepatterns in a system of neurons with adversarial interaction. In this study, weestablish the involvement of Turing instability to create such patterns. Bytheoretical and empirical studies, we present a pseudo-reaction-diffusion modelto explain the mechanism that may underlie these phenomena. While supervisedlearning attains homogeneous equilibrium, this paper suggests that theintroduction of an adversary helps break this homogeneity to createnon-homogeneous patterns at equilibrium. Further, we prove that randomlyinitialized gradient descent with over-parameterization can convergeexponentially fast to an $\epsilon$-stationary point even under adversarialinteraction. In addition, different from sole supervision, we show that thesolutions obtained under adversarial interaction are not limited to a tinysubspace around initialization.

Why Adversarial Interaction Creates Non-Homogeneous Patterns: A Pseudo-Reaction-Diffusion Model for Turing Instability