Statistical Spatially Inhomogeneous Diffusion Inference

2023/12/10 by Ren, Yinuo, Lu, Yiping, Ying, Lexing +1 · 1 citation
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2312.05793

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 d\boldsymbolx_t=\boldsymbolb(\boldsymbolx_t)d t+Σ(\boldsymbolx_t)d\boldsymbolw_t, we propose neural network-based estimators of both the drift \boldsymbolb and the spatially-inhomogeneous diffusion tensor D = ΣΣ^T and provide statistical convergence guarantees when \boldsymbolb and D are s-Hölder continuous. Notably, our bound aligns with the minimax optimal rate N^-(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.

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