Statistical Spatially Inhomogeneous Diffusion Inference

Inferring a diffusion equation from discretely-observed measurements is a statistical challenge of significant importance in a variety of fields, from single-molecule tracking in biophysical systems to modeling financial instruments. Assuming that the underlying dynamical process obeys a d-dimensional stochastic differential equation of the form dboldsymbolxt=boldsymbolb(boldsymbolxt)d t+Sigma(boldsymbolxt)dboldsymbolwt, we propose neural network-based estimators of both the drift boldsymbolb and the spatially-inhomogeneous diffusion tensor D = SigmaSigmaT and provide statistical convergence guarantees when boldsymbolb and D are s-H"older continuous. Notably, our bound aligns with the minimax optimal rate N^-frac2s2s+d for nonparametric function estimation even in the presence of correlation within observational data, which necessitates careful handling when establishing fast-rate generalization bounds. Our theoretical results are bolstered by numerical experiments demonstrating accurate inference of spatially-inhomogeneous diffusion tensors.