Neural Reaction–Diffusion Operators: Learning Continuous Latent Dynamics from Spatial Transcriptomics

Graph-based spatial-transcriptomics models often universally tie predictions over measured spots, which leaves expression undefined between any two nodes. In this paper, we introduce \textbf{Neural Reaction--Diffusion Operators} (NRDO), which encodes irregular samples into a continuous latent field, evolves it under a learned anisotropic reaction--diffusion equation, and decodes expression at arbitrary coordinates. Via Strang splitting, the solver integrates diffusion exactly in the Fourier domain to provide stable, second-order schemes. Across \MSSections{} sections from four technologies, NRDO improves masked-region reconstruction over graph, Gaussian-process, neural-field, and imputation baselines. These improvements also survive Holm correction against \MSSpecNSig{} of \MSNBaselines{} baselines, one of which also includes a graph model. Parameter-matched control attributes the gains to operator structure rather than the increased model capacity, and improvements have been shown to increase with gene-level spatial autocorrelation and vanish for genes that lack spatial structure. We prove that a single-stationary snapshot constrains the reaction network to only a Lebesgue-null subset of latent space, which bounds any method of this class that could identify from static tissue. Two limitations still need to be addressed: a comparison against the strongest graph baseline to the available number of specimens, and the predicted fields that are smoother than the measurements.