Statistical Spatially Inhomogeneous Diffusion Inference

TL;DR

This paper introduces neural network estimators for inferring spatially-inhomogeneous diffusion processes from data, providing theoretical guarantees and demonstrating accurate numerical results in complex stochastic systems.

Contribution

It develops neural network-based methods for estimating drift and diffusion tensors in inhomogeneous diffusion models, with proven convergence rates and handling data correlations.

Findings

Achieves minimax optimal convergence rates for diffusion tensor estimation.

Provides theoretical guarantees for neural network estimators in stochastic differential equations.

Demonstrates accurate numerical inference of spatially-inhomogeneous diffusion tensors.

Abstract

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 $$\mathrm{d}\boldsymbol{x}_t=\boldsymbol{b}(\boldsymbol{x}_t)\mathrm{d} t+\Sigma(\boldsymbol{x}_t)\mathrm{d}\boldsymbol{w}_t,$$ we propose neural network-based estimators of both the drift $\boldsymbol{b}$ and the spatially-inhomogeneous diffusion tensor $D = \Sigma\Sigma^{T}$ and provide statistical convergence guarantees when $\boldsymbol{b}$ and $D$ are $s$-H\"older 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.