DeepBrownianMotion

A deep learning classification tool for anomalous diffusion trajectories.

Introduction

MSD against time for different fractional Brownian motion realizations ($\alpha$ = 0.5, 1.0, and 1.5).

Let's consider the motion a particle immersed in a medium. Since its dynamics is very complicated and depends on a large number of parameters, it is not straightforward to identify useful quantities to characterize it. Of fundamental importance is the mean squared displacement (MSD), which connects time to the average of the squared distance travelled. Given a continuous random variable X(t) describing the position of a particle at time t, MSD is defined as:

$\mathrm{MSD}\left(\tau\right)$ $=$ $\langle\left|\boldsymbol{X}\left(t+\tau\right)-\boldsymbol{X}\left(t\right)\right|^2\rangle$

where $\left|\right|^2$ refers to the squared norm and $\langle\rangle$ to the mean value over time. When ordinary diffusion takes place, $MSD(\tau)$ is proportional to $\tau$, but there exist cases where $\mathrm{MSD}\left(\tau\right)$ $\sim$ $\tau^\alpha$. This naturally leads to a classication of diffusion dynamics into three main classes:

• Subdiffusion when $\alpha < 1$
• Ordinary diffusion when $\alpha = 1$
• Superdiffusion when $\alpha > 1$

Whenever $\alpha \neq 1$ the process is called anomalous diffusion.

There have been attemps to use deep learning techniques to overperform traditional methods to infer the $\alpha$ parameter from a trajectory¹. This project is meant to extend the previously used state of the art approach -an LSTM neural network- with a Transformer Encoder²

Training

Methods based on fractional Brownian motion are a simple, though mathematically rigorous, way to simulate diffusion trajectories starting from the $\alpha$ parameter. A simulator³ was used to generate 60000, 100 step long, trajectories with $\alpha$ sampled from a uniform distribution 0-1.8. A Transformer Encoder was then trained to predict $\alpha$ using a pytorch implementation. Preliminary analysis suggests that an embedding size close to 30, together with 2 attention heads, is enough to fully take advantage of the information contained in the trajectories. After 3 epochs with a learning rate of 0.001, MAE loss, Adam optimizer and batch size of 10, the model achieved an average loss of 0.22 on the validation set.

Mean absolute error (MAE) as a function of the number of iterations. Vertical red dashed lines represent epochs.

Requirements

pip install numpy, stochastic, torch

Usage

This script takes as input a vector containing the positions of the particle and returns the $\alpha$ parameter.

from DeepBrownianMotion import DeepBrownianMotion
from stochastic.processes import FractionalBrownianMotion
import numpy as np

# Choose trajectory parameters
alpha = 0.7
trajectory_length = 100

# Instantiate simulator object
f = FractionalBrownianMotion(hurst=0.5*alpha, t=1)
trajectory = f.sample(trajectory_length-1).reshape(1, trajectory_length)

# Instantiate deep learning model
dpm = DeepBrownianMotion(device='cpu')
alpha_pred = np.around(dpm.inference(trajectory).item(), 3)

print('Real alpha {0}, predicted alpha {1}'.format(alpha, alpha_pred) )

Output:

Real alpha 0.7, predicted alpha 0.746

Credits

I want to thank Marco Gherardi from the University of Milan for his support on a previous closely related project. This work was inspired by this paper from A. Argun, G. Volpe and S. Bo, and would not have been possible without the stochastic simulation tool. As always, I also want to thank the pytorch team for their amazing library.

Footnotes

1. Aykut Argun et al 2021 J. Phys. A: Math. Theor. 54 294003 ↩

2. Attention Is All You Need ↩

3. stochastic by crflynn ↩