ChatGPT-Enhanced ROC Analysis (CERA): A shiny web tool for finding optimal cutoff points in biomarker analysis.

Diagnostic tests play a crucial role in establishing the presence of a specific disease in an individual. Receiver Operating Characteristic (ROC) curve analyses are essential tools that provide performance metrics for diagnostic tests. Accurate determination of the cutoff point in ROC curve analyses is the most critical aspect of the process. A variety of methods have been developed to find the optimal cutoffs. Although the R programming language provides a variety of package programs for conducting ROC curve analysis and determining the appropriate cutoffs, it typically needs coding skills and a substantial investment of time. Specifically, the necessity for data preprocessing and analysis can present a significant challenge, especially for individuals without coding experience. We have developed the CERA (ChatGPT-Enhanced ROC Analysis) tool, a user-friendly ROC curve analysis web tool using the shiny interface for faster and more effective analyses to solve this problem. CERA is not only user-friendly, but it also interacts with ChatGPT, which interprets the outputs. This allows for an interpreted report generated by R-Markdown to be presented to the user, enhancing the accessibility and understanding of the analysis results.

AI-Aristotle: A physics-informed framework for systems biology gray-box identification.

Discovering mathematical equations that govern physical and biological systems from observed data is a fundamental challenge in scientific research. We present a new physics-informed framework for parameter estimation and missing physics identification (gray-box) in the field of Systems Biology. The proposed framework-named AI-Aristotle-combines the eXtreme Theory of Functional Connections (X-TFC) domain-decomposition and Physics-Informed Neural Networks (PINNs) with symbolic regression (SR) techniques for parameter discovery and gray-box identification. We test the accuracy, speed, flexibility, and robustness of AI-Aristotle based on two benchmark problems in Systems Biology: a pharmacokinetics drug absorption model and an ultradian endocrine model for glucose-insulin interactions. We compare the two machine learning methods (X-TFC and PINNs), and moreover, we employ two different symbolic regression techniques to cross-verify our results. To test the performance of AI-Aristotle, we use sparse synthetic data perturbed by uniformly distributed noise. More broadly, our work provides insights into the accuracy, cost, scalability, and robustness of integrating neural networks with symbolic regressors, offering a comprehensive guide for researchers tackling gray-box identification challenges in complex dynamical systems in biomedicine and beyond.

Two-component macrophage model for active phagocytosis with pseudopod formation.

Macrophage phagocytosis is critical for the immune response, homeostasis regulation, and tissue repair. This intricate process involves complex changes in cell morphology, cytoskeletal reorganization, and various receptor-ligand interactions controlled by mechanical constraints. However, there is a lack of comprehensive theoretical and computational models that investigate the mechanical process of phagocytosis in the context of cytoskeletal rearrangement. To address this issue, we propose a novel coarse-grained mesoscopic model that integrates a fluid-like cell membrane and a cytoskeletal network to study the dynamic phagocytosis process. The growth of actin filaments results in the formation of long and thin pseudopods, and the initial cytoskeleton can be disassembled upon target entry and reconstructed after phagocytosis. Through dynamic changes in the cytoskeleton, our macrophage model achieves active phagocytosis by forming a phagocytic cup utilizing pseudopods in two distinct ways. We have developed a new algorithm for modifying membrane area to prevent membrane rupture and ensure sufficient surface area during phagocytosis. In addition, the bending modulus, shear stiffness, and cortical tension of the macrophage model are investigated through computation of the axial force for the tubular structure and micropipette aspiration. With this model, we simulate active phagocytosis at the cytoskeletal level and investigate the mechanical process during the dynamic interplay between macrophage and target particles.

Rethinking skip connections in Spiking Neural Networks with Time-To-First-Spike coding.

Time-To-First-Spike (TTFS) coding in Spiking Neural Networks (SNNs) offers significant advantages in terms of energy efficiency, closely mimicking the behavior of biological neurons. In this work, we delve into the role of skip connections, a widely used concept in Artificial Neural Networks (ANNs), within the domain of SNNs with TTFS coding. Our focus is on two distinct types of skip connection architectures: (1) addition-based skip connections, and (2) concatenation-based skip connections. We find that addition-based skip connections introduce an additional delay in terms of spike timing. On the other hand, concatenation-based skip connections circumvent this delay but produce time gaps between after-convolution and skip connection paths, thereby restricting the effective mixing of information from these two paths. To mitigate these issues, we propose a novel approach involving a learnable delay for skip connections in the concatenation-based skip connection architecture. This approach successfully bridges the time gap between the convolutional and skip branches, facilitating improved information mixing. We conduct experiments on public datasets including MNIST and Fashion-MNIST, illustrating the advantage of the skip connection in TTFS coding architectures. Additionally, we demonstrate the applicability of TTFS coding on beyond image recognition tasks and extend it to scientific machine-learning tasks, broadening the potential uses of SNNs.

Sound propagation in realistic interactive 3D scenes with parameterized sources using deep neural operators.

We address the challenge of acoustic simulations in three-dimensional (3D) virtual rooms with parametric source positions, which have applications in virtual/augmented reality, game audio, and spatial computing. The wave equation can fully describe wave phenomena such as diffraction and interference. However, conventional numerical discretization methods are computationally expensive when simulating hundreds of source and receiver positions, making simulations with parametric source positions impractical. To overcome this limitation, we propose using deep operator networks to approximate linear wave-equation operators. This enables the rapid prediction of sound propagation in realistic 3D acoustic scenes with parametric source positions, achieving millisecond-scale computations. By learning a compact surrogate model, we avoid the offline calculation and storage of impulse responses for all relevant source/listener pairs. Our experiments, including various complex scene geometries, show good agreement with reference solutions, with root mean squared errors ranging from 0.02 to 0.10 Pa. Notably, our method signifies a paradigm shift as-to our knowledge-no prior machine learning approach has achieved precise predictions of complete wave fields within realistic domains.

TransformerG2G: Adaptive time-stepping for learning temporal graph embeddings using transformers.

Dynamic graph embedding has emerged as a very effective technique for addressing diverse temporal graph analytic tasks (i.e., link prediction, node classification, recommender systems, anomaly detection, and graph generation) in various applications. Such temporal graphs exhibit heterogeneous transient dynamics, varying time intervals, and highly evolving node features throughout their evolution. Hence, incorporating long-range dependencies from the historical graph context plays a crucial role in accurately learning their temporal dynamics. In this paper, we develop a graph embedding model with uncertainty quantification, TransformerG2G, by exploiting the advanced transformer encoder to first learn intermediate node representations from its current state (t) and previous context (over timestamps [t-1,t-l], l is the length of context). Moreover, we employ two projection layers to generate lower-dimensional multivariate Gaussian distributions as each node's latent embedding at timestamp t. We consider diverse benchmarks with varying levels of "novelty" as measured by the TEA (Temporal Edge Appearance) plots. Our experiments demonstrate that the proposed TransformerG2G model outperforms conventional multi-step methods and our prior work (DynG2G) in terms of both link prediction accuracy and computational efficiency, especially for high degree of novelty. Furthermore, the learned time-dependent attention weights across multiple graph snapshots reveal the development of an automatic adaptive time stepping enabled by the transformer. Importantly, by examining the attention weights, we can uncover temporal dependencies, identify influential elements, and gain insights into the complex interactions within the graph structure. For example, we identified a strong correlation between attention weights and node degree at the various stages of the graph topology evolution.

A combined computational and experimental investigation of the filtration function of splenic macrophages in sickle cell disease.

Being the largest lymphatic organ in the body, the spleen also constantly controls the quality of red blood cells (RBCs) in circulation through its two major filtration components, namely interendothelial slits (IES) and red pulp macrophages. In contrast to the extensive studies in understanding the filtration function of IES, fewer works investigate how the splenic macrophages retain the aged and diseased RBCs, i.e., RBCs in sickle cell disease (SCD). Herein, we perform a computational study informed by companion experiments to quantify the dynamics of RBCs captured and retained by the macrophages. We first calibrate the parameters in the computational model based on microfluidic experimental measurements for sickle RBCs under normoxia and hypoxia, as those parameters are not available in the literature. Next, we quantify the impact of key factors expected to dictate the RBC retention by the macrophages in the spleen, namely, blood flow conditions, RBC aggregation, hematocrit, RBC morphology, and oxygen levels. Our simulation results show that hypoxic conditions could enhance the adhesion between the sickle RBCs and macrophages. This, in turn, increases the retention of RBCs by as much as four-fold, which could be a possible cause of RBC congestion in the spleen of patients with SCD. Our study on the impact of RBC aggregation illustrates a 'clustering effect', where multiple RBCs in one aggregate can make contact and adhere to the macrophages, leading to a higher retention rate than that resulting from RBC-macrophage pair interactions. Our simulations of sickle RBCs flowing past macrophages for a range of blood flow velocities indicate that the increased blood velocity could quickly attenuate the function of the red pulp macrophages on detaining aged or diseased RBCs, thereby providing a possible rationale for the slow blood flow in the open circulation of the spleen. Furthermore, we quantify the impact of RBC morphology on their tendency to be retained by the macrophages. We find that the sickle and granular-shaped RBCs are more likely to be filtered by macrophages in the spleen. This finding is consistent with the observation of low percentages of these two forms of sickle RBCs in the blood smear of SCD patients. Taken together, our experimental and simulation results aid in our quantitative understanding of the function of splenic macrophages in retaining the diseased RBCs and provide an opportunity to combine such knowledge with the current knowledge of the interaction between IES and traversing RBCs to apprehend the complete filtration function of the spleen in SCD.

Physics-informed neural networks for predicting gas flow dynamics and unknown parameters in diesel engines.

This paper presents a physics-informed neural network (PINN) approach for monitoring the health of diesel engines. The aim is to evaluate the engine dynamics, identify unknown parameters in a "mean value" model, and anticipate maintenance requirements. The PINN model is applied to diesel engines with a variable-geometry turbocharger and exhaust gas recirculation, using measurement data of selected state variables. The results demonstrate the ability of the PINN model to predict simultaneously both unknown parameters and dynamics accurately with both clean and noisy data, and the importance of the self-adaptive weight in the loss function for faster convergence. The input data for these simulations are derived from actual engine running conditions, while the outputs are simulated data, making this a practical case study of PINN's ability to predict real-world dynamical systems. The mean value model of the diesel engine incorporates empirical formulae to represent certain states, but these formulae may not be generalizable to other engines. To address this, the study considers the use of deep neural networks (DNNs) in addition to the PINN model. The DNNs are trained using laboratory test data and are used to model the engine-specific empirical formulae in the mean value model, allowing for a more flexible and adaptive representation of the engine's states. In other words, the mean value model uses both the PINN model and the DNNs to represent the engine's states, with the PINN providing a physics-based understanding of the engine's overall dynamics and the DNNs offering a more engine-specific and adaptive representation of the empirical formulae. By combining these two approaches, the study aims to offer a comprehensive and versatile approach to monitoring the health and performance of diesel engines.

A combined computational and experimental investigation of the filtration function of splenic macrophages in sickle cell disease.

Being the largest lymphatic organ in the body, the spleen also constantly controls the quality of red blood cells (RBCs) in circulation through its two major filtration components, namely interendothelial slits (IES) and red pulp macrophages. In contrast to the extensive studies in understanding the filtration function of IES, there are relatively fewer works on investigating how the splenic macrophages retain the aged and diseased RBCs, i.e., RBCs in sickle cell disease (SCD). Herein, we perform a computational study informed by companion experiments to quantify the dynamics of RBCs captured and retained by the macrophages. We first calibrate the parameters in the computational model based on microfluidic experimental measurements for sickle RBCs under normoxia and hypoxia, as those parameters are not available in the literature. Next, we quantify the impact of a set of key factors that are expected to dictate the RBC retention by the macrophages in the spleen, namely, blood flow conditions, RBC aggregation, hematocrit, RBC morphology, and oxygen levels. Our simulation results show that hypoxic conditions could enhance the adhesion between the sickle RBCs and macrophages. This, in turn, increases the retention of RBCs by as much as five-fold, which could be a possible cause of RBC congestion in the spleen of patients with SCD. Our study on the impact of RBC aggregation illustrates a 'clustering effect', where multiple RBCs in one aggregate can make contact and adhere to the macrophages, leading to a higher retention rate than that resulting from RBC-macrophage pair interactions. Our simulations of sickle RBCs flowing past macrophages for a range of blood flow velocities indicate that the increased blood velocity could quickly attenuate the function of the red pulp macrophages on detaining aged or diseased RBCs, thereby providing a possible rationale for the slow blood flow in the open circulation of the spleen. Furthermore, we quantify the impact of RBC morphology on their tendency to be retained by the macrophages. We find that the sickle and granular-shaped RBCs are more likely to be filtered by macrophages in the spleen. This finding is consistent with the observation of low percentages of these two forms of sickle RBCs in the blood smear of SCD patients. Taken together, our experimental and simulation results aid in our quantitative understanding of the function of splenic macrophages in retaining the diseased RBCs and provide an opportunity to combine such knowledge with the current knowledge of the interaction between IES and traversing RBCs to apprehend the complete filtration function of the spleen in SCD.

In silico and in vitro study of the adhesion dynamics of erythrophagocytosis in sickle cell disease.

Erythrophagocytosis occurring in the spleen is a critical process for removing senescent and diseased red blood cells (RBCs) from the microcirculation. Although some progress has been made in understanding how the biological signaling pathways mediate the phagocytic processes, the role of the biophysical interaction between RBCs and macrophages, particularly under pathological conditions such as sickle cell disease, has not been adequately studied. Here, we combine computational simulations with microfluidic experiments to quantify RBC-macrophage adhesion dynamics under flow conditions comparable to those in the red pulp of the spleen. We also investigate the RBC-macrophage interaction under normoxic and hypoxic conditions. First, we calibrate key model parameters in the adhesion model using microfluidic experiments for normal and sickle RBCs under normoxia and hypoxia. We then study the adhesion dynamics between the RBC and the macrophage. Our simulation illustrates three typical adhesion states, each characterized by a distinct dynamic motion of the RBCs, namely firm adhesion, flipping adhesion, and no adhesion (either due to no contact with macrophages or detachment from the macrophages). We also track the number of bonds formed when RBCs and macrophages are in contact, as well as the contact area between the two interacting cells, providing mechanistic explanations for the three adhesion states observed in the simulations and microfluidic experiments. Furthermore, we quantify, for the first time to our knowledge, the adhesive forces between RBCs (normal and sickle) and macrophages under different oxygenated conditions. Our results show that the adhesive forces between normal cells and macrophages under normoxia are in the range of 33-58 pN and 53-92 pN for sickle cells under normoxia and 155-170 pN for sickle cells under hypoxia. Taken together, our microfluidic and simulation results improve our understanding of the biophysical interaction between RBCs and macrophages in sickle cell disease and provide a solid foundation for investigating the filtration function of the splenic macrophages under physiological and pathological conditions.

Systems Biology: Identifiability Analysis and Parameter Identification via Systems-Biology-Informed Neural Networks.

The dynamics of systems biological processes are usually modeled by a system of ordinary differential equations (ODEs) with many unknown parameters that need to be inferred from noisy and sparse measurements. Here, we introduce systems-biology-informed neural networks for parameter estimation by incorporating the system of ODEs into the neural networks. To complete the workflow of system identification, we also describe structural and practical identifiability analysis to analyze the identifiability of parameters. We use the ultradian endocrine model for glucose-insulin interaction as the example to demonstrate all these methods and their implementation.

Artificial intelligence velocimetry reveals in vivo flow rates, pressure gradients, and shear stresses in murine perivascular flows.

Quantifying the flow of cerebrospinal fluid (CSF) is crucial for understanding brain waste clearance and nutrient delivery, as well as edema in pathological conditions such as stroke. However, existing in vivo techniques are limited to sparse velocity measurements in pial perivascular spaces (PVSs) or low-resolution measurements from brain-wide imaging. Additionally, volume flow rate, pressure, and shear stress variation in PVSs are essentially impossible to measure in vivo. Here, we show that artificial intelligence velocimetry (AIV) can integrate sparse velocity measurements with physics-informed neural networks to quantify CSF flow in PVSs. With AIV, we infer three-dimensional (3D), high-resolution velocity, pressure, and shear stress. Validation comes from training with 70% of PTV measurements and demonstrating close agreement with the remaining 30%. A sensitivity analysis on the AIV inputs shows that the uncertainty in AIV inferred quantities due to uncertainties in the PVS boundary locations inherent to in vivo imaging is less than 30%, and the uncertainty from the neural net initialization is less than 1%. In PVSs of N = 4 wild-type mice we find mean flow speed 16.33 ± 11.09 µm/s, volume flow rate 2.22 ± 1.983 × 103 µm3/s, axial pressure gradient ( - 2.75 ± 2.01)×10-4 Pa/µm (-2.07 ± 1.51 mmHg/m), and wall shear stress (3.00 ± 1.45)×10-3 Pa (all mean ± SE). Pressure gradients, flow rates, and resistances agree with prior predictions. AIV infers in vivo PVS flows in remarkable detail, which will improve fluid dynamic models and potentially clarify how CSF flow changes with aging, Alzheimer's disease, and small vessel disease.

Accelerating gradient descent and Adam via fractional gradients.

We propose a class of novel fractional-order optimization algorithms. We define a fractional-order gradient via the Caputo fractional derivatives that generalizes integer-order gradient. We refer it to as the Caputo fractional-based gradient, and develop an efficient implementation to compute it. A general class of fractional-order optimization methods is then obtained by replacing integer-order gradients with the Caputo fractional-based gradients. To give concrete algorithms, we consider gradient descent (GD) and Adam, and extend them to the Caputo fractional GD (CfGD) and the Caputo fractional Adam (CfAdam). We demonstrate the superiority of CfGD and CfAdam on several large scale optimization problems that arise from scientific machine learning applications, such as ill-conditioned least squares problem on real-world data and the training of neural networks involving non-convex objective functions. Numerical examples show that both CfGD and CfAdam result in acceleration over GD and Adam, respectively. We also derive error bounds of CfGD for quadratic functions, which further indicate that CfGD could mitigate the dependence on the condition number in the rate of convergence and results in significant acceleration over GD.

Learning Poisson Systems and Trajectories of Autonomous Systems via Poisson Neural Networks.

We propose the Poisson neural networks (PNNs) to learn Poisson systems and trajectories of autonomous systems from data. Based on the Darboux-Lie theorem, the phase flow of a Poisson system can be written as the composition of: 1) a coordinate transformation; 2) an extended symplectic map; and 3) the inverse of the transformation. In this work, we extend this result to the unknotted trajectories of autonomous systems. We employ structured neural networks with physical priors to approximate the three aforementioned maps. We demonstrate through several simulations that PNNs are capable of handling very accurately several challenging tasks, including the motion of a particle in the electromagnetic potential, the nonlinear Schrödinger equation, and pixel observations of the two-body problem.

G2Φnet: Relating genotype and biomechanical phenotype of tissues with deep learning.

Many genetic mutations adversely affect the structure and function of load-bearing soft tissues, with clinical sequelae often responsible for disability or death. Parallel advances in genetics and histomechanical characterization provide significant insight into these conditions, but there remains a pressing need to integrate such information. We present a novel genotype-to-biomechanical phenotype neural network (G2Φnet) for characterizing and classifying biomechanical properties of soft tissues, which serve as important functional readouts of tissue health or disease. We illustrate the utility of our approach by inferring the nonlinear, genotype-dependent constitutive behavior of the aorta for four mouse models involving defects or deficiencies in extracellular constituents. We show that G2Φnet can infer the biomechanical response while simultaneously ascribing the associated genotype by utilizing limited, noisy, and unstructured experimental data. More broadly, G2Φnet provides a powerful method and a paradigm shift for correlating genotype and biomechanical phenotype quantitatively, promising a better understanding of their interplay in biological tissues.

AOSLO-net: A Deep Learning-Based Method for Automatic Segmentation of Retinal Microaneurysms From Adaptive Optics Scanning Laser Ophthalmoscopy Images.

Purpose: Accurate segmentation of microaneurysms (MAs) from adaptive optics scanning laser ophthalmoscopy (AOSLO) images is crucial for identifying MA morphologies and assessing the hemodynamics inside the MAs. Herein, we introduce AOSLO-net to perform automatic MA segmentation from AOSLO images of diabetic retinas. Method: AOSLO-net is composed of a deep neural network based on UNet with a pretrained EfficientNet as the encoder. We have designed customized preprocessing and postprocessing policies for AOSLO images, including generation of multichannel images, de-noising, contrast enhancement, ensemble and union of model predictions, to optimize the MA segmentation. AOSLO-net is trained and tested using 87 MAs imaged from 28 eyes of 20 subjects with varying severity of diabetic retinopathy (DR), which is the largest available AOSLO dataset for MA detection. To avoid the overfitting in the model training process, we augment the training data by flipping, rotating, scaling the original image to increase the diversity of data available for model training. Results: The validity of the model is demonstrated by the good agreement between the predictions of AOSLO-net and the MA masks generated by ophthalmologists and skillful trainees on 87 patient-specific MA images. Our results show that AOSLO-net outperforms the state-of-the-art segmentation model (nnUNet) both in accuracy (e.g., intersection over union and Dice scores), as well as computational cost. Conclusions: We demonstrate that AOSLO-net provides high-quality of MA segmentation from AOSLO images that enables correct MA morphological classification. Translational Relevance: As the first attempt to automatically segment retinal MAs from AOSLO images, AOSLO-net could facilitate the pathological study of DR and help ophthalmologists make disease prognoses.

Circulating cell clusters aggravate the hemorheological abnormalities in COVID-19.

Microthrombi and circulating cell clusters are common microscopic findings in patients with coronavirus disease 2019 (COVID-19) at different stages in the disease course, implying that they may function as the primary drivers in disease progression. Inspired by a recent flow imaging cytometry study of the blood samples from patients with COVID-19, we perform computational simulations to investigate the dynamics of different types of circulating cell clusters, namely white blood cell (WBC) clusters, platelet clusters, and red blood cell clusters, over a range of shear flows and quantify their impact on the viscosity of the blood. Our simulation results indicate that the increased level of fibrinogen in patients with COVID-19 can promote the formation of red blood cell clusters at relatively low shear rates, thereby elevating the blood viscosity, a mechanism that also leads to an increase in viscosity in other blood diseases, such as sickle cell disease and type 2 diabetes mellitus. We further discover that the presence of WBC clusters could also aggravate the abnormalities of local blood rheology. In particular, the extent of elevation of the local blood viscosity is enlarged as the size of the WBC clusters grows. On the other hand, the impact of platelet clusters on the local rheology is found to be negligible, which is likely due to the smaller size of the platelets. The difference in the impact of WBC and platelet clusters on local hemorheology provides a compelling explanation for the clinical finding that the number of WBC clusters is significantly correlated with thrombotic events in COVID-19 whereas platelet clusters are not. Overall, our study demonstrates that our computational models based on dissipative particle dynamics can serve as a powerful tool to conduct quantitative investigation of the mechanism causing the pathological alterations of hemorheology and explore their connections to the clinical manifestations in COVID-19.

Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysms.

Thoracic aortic aneurysm (TAA) is a localized dilatation of the aorta that can lead to life-threatening dissection or rupture. In vivo assessments of TAA progression are largely limited to measurements of aneurysm size and growth rate. There is promise, however, that computational modelling of the evolving biomechanics of the aorta could predict future geometry and properties from initiating mechanobiological insults. We present an integrated framework to train a deep operator network (DeepONet)-based surrogate model to identify TAA contributing factors using synthetic finite-element-based datasets. For training, we employ a constrained mixture model of aortic growth and remodelling to generate maps of local aortic dilatation and distensibility for multiple TAA risk factors. We evaluate the performance of the surrogate model for insult distributions varying from fusiform (analytically defined) to complex (randomly generated). We propose two frameworks, one trained on sparse information and one on full-field greyscale images, to gain insight into a preferred neural operator-based approach. We show that this continuous learning approach can predict the patient-specific insult profile associated with any given dilatation and distensibility map with high accuracy, particularly when based on full-field images. Our findings demonstrate the feasibility of applying DeepONet to support transfer learning of patient-specific inputs to predict TAA progression.

Approximation rates of DeepONets for learning operators arising from advection-diffusion equations.

We present the analysis of approximation rates of operator learning in Chen and Chen (1995) and Lu et al. (2021), where continuous operators are approximated by a sum of products of branch and trunk networks. In this work, we consider the rates of learning solution operators from both linear and nonlinear advection-diffusion equations with or without reaction. We find that the approximation rates depend on the architecture of branch networks as well as the smoothness of inputs and outputs of solution operators.

DynG2G: An Efficient Stochastic Graph Embedding Method for Temporal Graphs.

Dynamic graph embedding has gained great attention recently due to its capability of learning low-dimensional and meaningful graph representations for complex temporal graphs with high accuracy. However, recent advances mostly focus on learning node embeddings as deterministic "vectors" for static graphs, hence disregarding the key graph temporal dynamics and the evolving uncertainties associated with node embedding in the latent space. In this work, we propose an efficient stochastic dynamic graph embedding method (DynG2G) that applies an inductive feedforward encoder trained with node triplet energy-based ranking loss. Every node per timestamp is encoded as a time-dependent probabilistic multivariate Gaussian distribution in the latent space, and, hence, we are able to quantify the node embedding uncertainty on-the-fly. We have considered eight different benchmarks that represent diversity in size (from 96 nodes to 87 626 and from 13 398 edges to 4 870 863) as well as diversity in dynamics, from slowly changing temporal evolution to rapidly varying multirate dynamics. We demonstrate through extensive experiments based on these eight dynamic graph benchmarks that DynG2G achieves new state-of-the-art performance in capturing the underlying temporal node embeddings. We also demonstrate that DynG2G can simultaneously predict the evolving node embedding uncertainty, which plays a crucial role in quantifying the intrinsic dimensionality of the dynamical system over time. In particular, we obtain a "universal" relation of the optimal embedding dimension, Lo , versus the effective dimensionality of uncertainty, Du , and infer that Lo=Du for all cases. This, in turn, implies that the uncertainty quantification approach we employ in the DynG2G algorithm correctly captures the intrinsic dimensionality of the dynamics of such evolving graphs despite the diverse nature and composition of the graphs at each timestamp. In addition, this L0 - Du correlation provides a clear path to selecting adaptively the optimum embedding size at each timestamp by setting L ≥ Du .