Combining machine learning with physics: A framework for tracking and sorting multiple dark solitons

In ultracold-atom experiments, data often comes in the form of images which suffer information loss inherent in the techniques used to prepare and measure the system. This is particularly problematic when the processes of interest are complicated, such as interactions among excitations in Bose-Einstein condensates (BECs). In this paper, we describe a framework combining machine learning (ML) models with physics-based traditional analyses to identify and track multiple solitonic excitations in images of BECs. We use an ML-based object detector to locate the solitonic excitations and develop a physics-informed classifier to sort solitonic excitations into physically motivated subcategories. Lastly, we introduce a quality metric quantifying the likelihood that a specific feature is a longitudinal soliton. Our trained implementation of this framework, soldet, is publicly available as an open-source python package. soldet is broadly applicable to feature identification in cold-atom images when trained on a suitable user-provided dataset.

Figure 1

Overview of the framework. The colored arrows link the preparation (Secs. 2a, 2b, and 2c), validation (Secs. 3a and 3b), and application (Sec. 3d) phases of the framework. The red path represents the preparation and implementation of the physics-based-approximation module of the framework. The blue path represents the ML modules.

Figure 2

Representative data. The top panels plot preprocessed images from our dataset and the bottom panels plot profiles: profile of full image (green), TF fits (black), and density fluctuations (blue). The red lines mark the location of the deepest depletion in the density fluctuations, while the orange lines mark the soliton locations found from our OD. (a) An element of the no-excitation class. (b) Three elements of the single-excitation class: (i) a single longitudinal soliton, (ii) an off-center longitudinal soliton, and (iii) a solitonic vortex. (c) Two representative elements of the other excitations class.

Figure 3

Correlations between parameters implemented in PIE classifier. The top panel shows the distribution of center position difference versus the amplitude ratio (on a logarithmic scale). The bottom panel shows the correlation between the center position difference and the asymmetrical shoulder height difference for the gray points from the top panel. Both panels are colored based on the cut-off points discussed in Sec. 3b

Figure 4

OD performance compared to ground truth (top), and the CNN classifier prediction (bottom). For ground truth and the CNN classifier, the ticks '0,' '1,' and 'other' represent no, single, and other excitation classes. For OD, ticks represent the total number of positive excitations within an image.

Figure 5

The flow of the PIE classifier with example images for classification categories. Flow pathways and nodes are square-root scaled.

Figure 6

Quality estimate performance on no excitation and single excitation classes. In all cases we use the following color scheme: longitudinal solitons (orange), all other solitonic excitations (green), and all nonsolitonic local minima (blue). (a) Power transformed fit coefficient distributions, with untransformed variables labeled on the top axis. (b) Distribution of quality estimate. (c) Performance of quality estimate quantified by F1 score; the stars indicate the optimal F1 value and the circles mark the threshold we use for classification. The inset shows precision/recall curves.

Figure 7

Performance of quality estimate on other excitation (orange) and mislabeled (red) classes. (a) Power-transformed fit coefficient distributions, with untransformed variables labeled on the top axis. (b) Distribution of quality estimate of all longitudinal solitons. (c) Representative images from the other excitation [(i),(ii)] and mislabeled [(iii),(iv)] classes, with $\mathrm{OD}+\mathrm{PIE}$ identified longitudinal soliton marked with arrows. The quality estimate for these excitations is as follows: $M_{\left(\text{i}\right)}=[0.74,0.86]$, $M_{\left(\text{ii}\right)}=[0.00,0.01]$, $M_{\left(\text{iii}\right)}=[0.92,0.02]$ (all from left to right), and $M_{\left(\text{iv}\right)}=0.82$.

Figure 8

The soldet flow chart. The black line follows the soldet dataflow and contains the labels added by each module (rectangles). Blue blocks represent ML modules; red blocks represent physics-based modules.

Figure 9

Illustration of (a) OD and (b) CNN classifier neural network architectures. Yellow-orange boxes show convolutional layers while orange-red boxes show max-pooling layers. The horizontal lengths of boxes represent the number of filters and the other two dimensions represent the image sizes. The horizontal blue and purple rectangles in (a) denote output vectors. Each cell of the blue vector describes the probability that it contains a soliton and and the purple vector contains the position of a soliton within the cell. And the vertical blue-green rectangles in (b) are three fully connected layers and the output layer. The lengths of edges are logarithmically scaled.