Reconstructing Dynamics from Steady Spatial Patterns with Partial Observations

Self-organized spatial patterns, ubiquitous in biological and chemical systems, are often modeled via reaction-diffusion equations. However, real-world scenarios frequently provide only partial observations—such as a single component’s steady-state snapshot—challenging the discovery of underlying dynamics. In this work, we address the inverse problem of identifying reaction-diffusion systems from partial observations. We establish the theoretical feasibility of identifying reaction terms and their corresponding coefficients, and introduce a constructive two-stage approach that combines hidden component inference with reaction coefficient identification. Numerical experiments validate the approach’s effectiveness. This work provides a novel framework with theoretical guarantees, advancing the study of pattern dynamics with limited data and offering new perspectives for uncovering unknown reaction-diffusion dynamics in real-world scenarios.