Statistical Spatially Inhomogeneous Diffusion Inference
Yinuo Ren, Yiping Lu, Lexing Ying, Grant M. Rotskoff
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 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.
BibTeX
@article{Ren_Lu_Ying_Rotskoff_2024, title={Statistical Spatially Inhomogeneous Diffusion Inference}, volume={38}, url={https://ojs.aaai.org/index.php/AAAI/article/view/29401}, DOI={10.1609/aaai.v38i13.29401}, abstractNote={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.}, number={13}, journal={Proceedings of the AAAI Conference on Artificial Intelligence}, author={Ren, Yinuo and Lu, Yiping and Ying, Lexing and Rotskoff, Grant M.}, year={2024}, month={Mar.}, pages={14820-14828} }