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 dx_t = b(x_t)dt + \Sigma(x_t)dw_t, we propose neural network-based estimators of both the drift b and the spatially-inhomogeneous diffusion tensor D = \Sigma\Sigma^T/2 and provide statistical convergence guarantees when b and D are s-Hölder continuous. Notably, our bound aligns with the minimax optimal rate N^{-\frac{2s}{2s+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.

Authors

• Yinuo Ren Stanford University
• Yiping Lu NYU
• Lexing Ying Stanford University
• Grant M. Rotskoff Stanford University

Keywords

• ML: Applications
• ML: Classification and Regression
• ML: Evaluation and Analysis
• ML: Learning Theory
• ML: Other Foundations of Machine Learning
• ML: Time-Series/Data Streams