Statistical Spatially Inhomogeneous Diffusion Inference
DOI:
https://doi.org/10.1609/aaai.v38i13.29401Keywords:
ML: Learning Theory, ML: Classification and Regression, ML: Applications, ML: Time-Series/Data Streams, ML: Evaluation and Analysis, ML: Other Foundations of Machine LearningAbstract
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.Downloads
Published
2024-03-24
How to Cite
Ren, Y., Lu, Y., Ying, L., & Rotskoff, G. M. (2024). Statistical Spatially Inhomogeneous Diffusion Inference. Proceedings of the AAAI Conference on Artificial Intelligence, 38(13), 14820-14828. https://doi.org/10.1609/aaai.v38i13.29401
Issue
Section
AAAI Technical Track on Machine Learning IV