# Statistical Spatially Inhomogeneous Diffusion Inference

arXiv: https://arxiv.org/abs/2312.05793 · HTML: https://mlsanity.com/paper/2312.05793/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 80 references that are in the map.
- First posted on arXiv 10 Dec 2023.

## Builds on (most cited first, 16 of 16)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networks](https://mlsanity.com/paper/1903.10047/) (2019)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Parametric Complexity Bounds for Approximating PDEs with Neural Networks](https://mlsanity.com/paper/2103.02138/) (2021)
- [Some observations on high-dimensional partial differential equations with Barron data](https://mlsanity.com/paper/2012.01484/) (2020)
- [Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective](https://mlsanity.com/paper/2210.12101/) (2022)
- [Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent](https://mlsanity.com/paper/2205.07331/) (2022)
- [Learning theory estimates with observations from general stationary stochastic processes](https://mlsanity.com/paper/1605.02887/) (2016)
- [Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions](https://mlsanity.com/paper/1510.05526/) (2015)

## Built on by (most cited first, 2 of 2)

- [Approximation Bounds for Recurrent Neural Networks with Application to Regression](https://mlsanity.com/paper/2409.05577/) (2024)
- [Approximation Bounds for Transformer Networks with Application to Regression](https://mlsanity.com/paper/2504.12175/) (2025)

## Abstract (arXiv)

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 dboldsymbolxt=boldsymbolb(boldsymbolxt)d t+Sigma(boldsymbolxt)dboldsymbolwt, we propose neural network-based estimators of both the drift boldsymbolb and the spatially-inhomogeneous diffusion tensor D = SigmaSigmaT and provide statistical convergence guarantees when boldsymbolb and D are s-H"older continuous. Notably, our bound aligns with the minimax optimal rate N^-frac2s2s+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.