Learning Biological Patterns with Differentiable Reaction-Diffusion Systems

Theoretical frameworks of morphogenesis have largely focused on reaction–diffusion systems, where partial differential equations capture the temporal dynamics of a few interacting species. These models, despite their simplicity, can generate strikingly complex patterns. Yet classical formulations remain hand-tuned and narrow in scope, whereas biological problems demand scalable and trainable approaches. To address this gap, we integrate dynamical systems theory with modern optimization to uncover general principles of biological pattern generation. We introduce a differentiable reaction–diffusion network optimized via auto-differentiation to reproduce target shapes across high-dimensional parameter spaces. An auxiliary energy functional enables us to reverse-engineer and categorize the strategies acquired by the networks, revealing that multiple mechanisms can approximate the same pattern. Our findings show that even simple morphogenetic tasks admit diverse solutions, indicating that the solution space of morphogenesis is both rich and degenerate.