Machine Learning Sparse Reaction-Diffusion Models from Stochastic Dynamics and Spatiotemporal Patterns

Pattern-forming stochastic systems arise throughout biology, with dynamic molecular waves observed in biochemical networks regulating critical cellular processes. Modeling these reaction-diffusion systems using handcrafted stochastic partial differential equations (PDEs) requires extensive trial-and-error tuning. Data-driven approaches for improved modeling are needed but have been hindered by data scarcity and noise. Here, we present a solution to the inverse problem of learning stochastic reaction-diffusion models from limited data by optimizing two spatiotemporal features: (1) stochastic dynamics and (2) spatiotemporal patterns. Combined with sparsity enforcement, this method identifies novel activator-inhibitor models with interpretable structure. We demonstrate robust learning from simulations of excitable systems with varying data scarcity, as well as noisy live-cell imaging data with low temporal resolution and a single observed biomolecule. This generalizable approach to learning governing stochastic PDEs enhances our ability to model and understand complex spatiotemporal systems from limited, real-world data.