Recurrent neural network analysis of single trajectories switching between anomalous diffusion states

TL;DR

This work tackles inferring time-varying anomalous diffusion parameters from short single trajectories that switch between diffusive states. It presents sequandi, an LSTM-based method that outputs a time-resolved $\hat{\alpha}(t)$ and $\hat{K}(t)$ along with a changepoint detector $\hat{t}^{\textrm{CP}}$, enabling simultaneous parameter estimation and state-change localization. The approach is trained on a large, mixed dataset inspired by AnDi2024 and validated on benchmark and test sets, achieving competitive point-wise accuracy (e.g., $\text{MAE}_{\alpha(t)}\approx0.19$ and $\text{MSLE}_{K(t)}\approx0.02$) and CP-detection performance (e.g., $\text{JSC}_{\textrm{CP}}\approx0.58$). Compared against the classical CPDA algorithm, sequandi often yields superior changepoint detection with fewer false positives, while still preserving high fidelity in time-resolved diffusion parameters. The work advances practical analysis of complex diffusion in biology and materials, and points to future improvements via transformers and 3D extensions.

Abstract

Diffusive dynamics abound in nature and have been especially studied in physical, biological, and financial systems. These dynamics are characterised by a linear growth of the mean squared displacement (MSD) with time. Often, the conditions that give rise to simple diffusion are violated, and many systems, such as biomolecules inside cells, microswimmers, or particles in turbulent flows, undergo anomalous diffusion, featuring an MSD that grows following a power law with an exponent $α$. Precisely determining this exponent and the generalised diffusion coefficient provides valuable information on the systems under consideration, but it is a very challenging task when only a few short trajectories are available, which is common in non-equilibrium and living systems. Estimating the exponent becomes overwhelmingly difficult when the diffusive dynamics switches between different behaviours, characterised by different exponents $α$ or diffusion coefficients $K$. We develop a method based on recurrent neural networks that successfully estimates the anomalous diffusion exponents and generalised diffusion coefficients of individual trajectories that switch between multiple diffusive states. Our method returns the $α$ and $K$ as a function of time and identifies the times at which the dynamics switches between different behaviours. We showcase the method's capabilities on the dataset of the 2024 Anomalous Diffusion Challenge.

Figures (18)

• Figure 1: Neural Network Architecture of sequandi. 2D trajectory data is normalised and separated into $M$ blocks $b_\tau$ of 3 coordinate increments each. The blocks are then fed into two adjacent LSTM layers of dimensions 250 and 50. Next, outputs from the second LSTM layer are passed through a dense TimeDistributed layer, which is adapted depending on the task (inference of $\alpha$, $K$ or $t^{\textrm{CP}}$) and returns a sequence of length $M$, ready for post-processing.
• Figure 2: Block assignment and point-wise inference. The input trajectory is split into blocks $b_\tau$ containing three increments per spatial dimension as stated in Eq. \ref{['eq:block']}. The network returns block-wise predictions $\hat{\alpha} (b_\tau), \hat{K}(b_\tau)$. To obtain a point-wise prediction, for each increment, we shift the trajectory by one and two increments, respectively, changing which increments are assigned to each block $b'_\tau=[\Delta x_{3\tau+1},\Delta y_{3\tau+1},\sigma ,\Delta x_{3\tau+2},\Delta y_{3\tau+2},\sigma, \Delta x_{3\tau+3},\Delta y_{3\tau+3},\sigma ]$, and $b"_\tau=[\Delta x_{3\tau+2},\Delta y_{3\tau+2},\sigma ,\Delta x_{3\tau+3},\Delta y_{3\tau+3},\sigma, \Delta x_{3\tau+4},\Delta y_{3\tau+4},\sigma ]$. This returns staggered predictions $\hat{\alpha} (b_\tau'),\, \hat{K}(b_\tau'), \, \hat{\alpha} (b_\tau"),\, \hat{K}(b_\tau")$. We obtain the point-wise $\hat{\alpha}(t)$ and $\hat{K}(t)$ by averaging the block predictions of the blocks that contain the time $t$, illustrated by the shifting red square. For instance, consider the vertical slice corresponding to $t=3$. The point-wise dynamical prediction is obtained as $\hat{\alpha}(t=3) = [(\hat{\alpha}(b_1)+\hat{\alpha}(b'_0)+\hat{\alpha}(b"_0)]/3$. For $t=0\textrm{ and }1$, we only use $\hat{\alpha}(b_\tau)$ predictions from blocks $\tau=0$ and $\tau=0,1$, respectively.
• Figure 3: Example of $\hat{\alpha}(t)$ and $\hat{K}(t)$ and ground truth. Results from example trajectory in Fig. \ref{['fig:fig1']}. (a) The orange line is sequandi's point-wise prediction of the anomalous exponent $\hat{\alpha}(t)$, and the blue piece-wise constant line is the ground truth $\alpha(t)$. The grey line is the prediction of the network developed in Ref. Bo2019 and the brown line is from the TA-MSD. Since these methods require longer trajectories to work, the original trajectory was repeated (see \ref{['app:compare']} for details). (b) The orange line is based on sequandi's point-wise prediction of the generalised diffusion coefficient $\hat{K}(t)$, and is $\log(\hat{K}(t)+1)$. The blue piece-wise constant line is the ground truth $\log(K(t)+1)$, and the brown line is obtained via a TA-MSD estimate. sequandi's prediction tracks the ground truth with an $\textrm{MAE}_{\alpha(t)}=0.134$ and $\textrm{MSLE}_{K(t)}=0.016$. In this window of 200 frames, the error from the network of Ref. Bo2019 is $\textrm{MAE}_{\alpha(t)}=0.268$, while for the TA-MSD they are $\textrm{MAE}_{\alpha(t)}=0.295$ and $\textrm{MSLE}_{K(t)}=0.194$
• Figure 4: Changepoint detection comparison: sequandi vs. CPDA Using the trajectory illustrated in Fig. \ref{['fig:fig1']}, the two vertical solid black lines are sequandi's prediction $\hat{t}^{\textrm{CP}}$. (a) As in Fig. \ref{['fig:fig3']}, the orange line is $\hat{\alpha}(t)$, and the blue piece-wise constant line is the ground truth $\alpha(t)$. The three vertical dashed gray lines are obtained by the CPDA algorithm described in Section \ref{['sec:CPalg']} applied to $\hat{\alpha}(t)$ with a confidence level of $99.99\%$. (b) As in Fig. \ref{['fig:fig3']}, the orange line is $\log(\hat{K}(t)+1)$ and the blue piece-wise constant line is the ground truth $\log(K(t)+1)$. The six vertical dashed gray lines are obtained by the CPDA algorithm described in Section \ref{['sec:CPalg']} applied to $\log(\hat{K}(t)+1)$ with a confidence level of $99.99\%$.
• Figure 5: Segmentation of $\hat{\alpha}(t)$ and $\hat{K}(t)$. We segment the trajectory shown in Fig. \ref{['fig:fig1']} based on the CPs predicted by sequandi$\hat{t}^{\textrm{CP}}$, which were indicated by the solid black lines shown in Fig. \ref{['fig:fig4']}. (a) As in Fig. \ref{['fig:fig3']}, the orange line is $\hat{\alpha}(t)$, and the blue piece-wise constant line is the ground truth $\alpha(t)$. The dashed brown line is the time averaged exponent for each segment $\hat{\alpha}_{i}$, following Eq. \ref{['eq:alphaK_seg']}. (b) As in Fig. \ref{['fig:fig3']}, the orange line is $\log(\hat{K}(t)+1)$ and the blue piece-wise constant line is the ground truth $\log(K(t)+1)$. The dashed brown line is based on the time averaged generalised diffusion for each segment $\hat{K}_i$, following Eq. \ref{['eq:alphaK_seg']}, and is $\log(\hat{K}_i+1)$.