Neural Ordinary Differential Equations Research Articles

Continuous-discrete extended Kalman filtering based on the neural ordinary differential equations method

In bearings-only tracking (BOT), the accuracy of state estimation significantly diminishes when confronted with an unknown target motion model. Therefore, this study proposed a novel continuous-discrete filtering algorithm, the neural ordinary differential equations-based continuous-discrete extended Kalman filtering (NODE-CD-EKF). The approach utilizes the neural ordinary differential equation (NODE) to build the target's system model, which is then used in the time update to precisely predict the target state at the next time point within the continuous-discrete Kalman framework. The NODE concisely and accurately describes the motion model in the form of ordinary differential equations in a data-driven way. This approach addresses the limitations of traditional Kalman filters, which suffer from degraded performance due to a lack of a precise mathematical model of target states. Our method is evaluated and compared to existing multiple-model and other data-driven algorithms in simulations of BOT and another scenario using the Accumulated Root Mean Square Error (ARMSE) of position. The ARMSE of our method is only 16.30 % of the multiple-model methods at least. Overall, this approach outperforms other multiple-model and data-driven algorithms and possesses the potential for solving other nonlinear tasks in real-world applications.

Biologically informed NeuralODEs for genome-wide regulatory dynamics

BackgroundGene regulatory network (GRN) models that are formulated as ordinary differential equations (ODEs) can accurately explain temporal gene expression patterns and promise to yield new insights into important cellular processes, disease progression, and intervention design. Learning such gene regulatory ODEs is challenging, since we want to predict the evolution of gene expression in a way that accurately encodes the underlying GRN governing the dynamics and the nonlinear functional relationships between genes. Most widely used ODE estimation methods either impose too many parametric restrictions or are not guided by meaningful biological insights, both of which impede either scalability, explainability, or both.ResultsWe developed PHOENIX, a modeling framework based on neural ordinary differential equations (NeuralODEs) and Hill-Langmuir kinetics, that overcomes limitations of other methods by flexibly incorporating prior domain knowledge and biological constraints to promote sparse, biologically interpretable representations of GRN ODEs. We tested the accuracy of PHOENIX in a series of in silico experiments, benchmarking it against several currently used tools. We demonstrated PHOENIX’s flexibility by modeling regulation of oscillating expression profiles obtained from synchronized yeast cells. We also assessed the scalability of PHOENIX by modeling genome-scale GRNs for breast cancer samples ordered in pseudotime and for B cells treated with Rituximab.ConclusionsPHOENIX uses a combination of user-defined prior knowledge and functional forms from systems biology to encode biological “first principles” as soft constraints on the GRN allowing us to predict subsequent gene expression patterns in a biologically explainable manner.

Applying Neural ODEs to Derive a Mechanism-Based Model for Characterizing Maturation-Related Serum Creatinine Dynamics in Preterm Newborns.

Serum creatinine in neonates follows complex dynamics due to maturation processes, most pronounced in the first few weeks of life. The development of a mechanism-based model describing complex dynamics requires high expertise in pharmacometric (PMX) modeling and substantial model development time. A recently published machine learning (ML) approach of low-dimensional neural ordinary differential equations (NODEs) is capable of modeling such data from newborns automatically. However, this efficient data-driven approach in itself does not result in a clinically interpretable model. In this work, an approach to deriving an interpretable model with reasonable PMX-type functions is presented. This "translation" was applied to derive a PMX model for serum creatinine in neonates considering maturation processes and covariates. The developed model was compared to a previously published mechanism-based PMX model whereas both models had similar mechanistic structures. The developed model was then utilized to simulate serum creatinine concentrations in the first few weeks of life considering different covariate values for gestational age and birth weight. The reference serum creatinine values derived from these simulations are consistent with observed serum creatinine values and previously published reference values. Thus, the presented NODE-based ML approach to model complex serum creatinine dynamics in newborns and derive interpretable, mathematical-statistical components similar to those in a conventional PMX model demonstrates a novel, viable approach to facilitate the modeling of complex dynamics in clinical settings and pediatric drug development.

An embedding model for temporal knowledge graphs with long and irregular intervals

As the basis of various downstream applications such as automated question answering and knowledge reasoning, Temporal Knowledge Graph (TKG) embedding learning has attracted many interests. Recent works have explored several methods to learn dynamic embedding for TKGs and led to success in many applications, however, they are difficult to be optimized. In order to accurately catch semantic information propagation of entity-to-entity and entity-to-relation, as well as track dynamics of entities and relations in continuous time, we proposed a novel TKG embedding model ODETKGE. Instead of using recurrent neural network to model dynamics of semantic representations in continuous time, we leverage neural ordinary differential equations to model the dynamic evolution, which enable us to capture long-range dependencies between embeddings at different timestamps, track the dynamics of TKGs with irregular intervals and remain computationally efficient. Furthermore, to effectively model semantics of knowledge graph structure, we combine entity-relation collaboratively updated graph convolutional networks and neural ordinary differential equations. Additionally, to acquire important semantics such as creation and disappearance of relations, we utilize a graph transition layer. Comparing with RNN based models, the proposed ODETKGE can use fewer parameters to achieve deeper layers, and make the transformation of representations between two timestamps smoother. Extensive experiments have been executed on real-world datasets and the results verify that ODETKGE is superior than the state-of-the-art models in link forecasting, long-interval link forecasting, irregular-interval link forecasting, ablation study and training time.

Continuous patient state attention model for addressing irregularity in electronic health records

BackgroundIrregular time series (ITS) are common in healthcare as patient data is recorded in an electronic health record (EHR) system as per clinical guidelines/requirements but not for research and depends on a patient’s health status. Due to irregularity, it is challenging to develop machine learning techniques to uncover vast intelligence hidden in EHR big data, without losing performance on downstream patient outcome prediction tasks.MethodsIn this paper, we propose Perceiver, a cross-attention-based transformer variant that is computationally efficient and can handle long sequences of time series in healthcare. We further develop continuous patient state attention models, using Perceiver and transformer to deal with ITS in EHR. The continuous patient state models utilise neural ordinary differential equations to learn patient health dynamics, i.e., patient health trajectory from observed irregular time steps, which enables them to sample patient state at any time.ResultsThe proposed models’ performance on in-hospital mortality prediction task on PhysioNet-2012 challenge and MIMIC-III datasets is examined. Perceiver model either outperforms or performs at par with baselines, and reduces computations by about nine times when compared to the transformer model, with no significant loss of performance. Experiments to examine irregularity in healthcare reveal that continuous patient state models outperform baselines. Moreover, the predictive uncertainty of the model is used to refer extremely uncertain cases to clinicians, which enhances the model’s performance. Code is publicly available and verified at https://codeocean.com/capsule/4587224.ConclusionsPerceiver presents a computationally efficient potential alternative for processing long sequences of time series in healthcare, and the continuous patient state attention models outperform the traditional and advanced techniques to handle irregularity in the time series. Moreover, the predictive uncertainty of the model helps in the development of transparent and trustworthy systems, which can be utilised as per the availability of clinicians.

Using neural ordinary differential equations to predict complex ecological dynamics from population density data.

Simple models have been used to describe ecological processes for over a century. However, the complexity of ecological systems makes simple models subject to modelling bias due to simplifying assumptions or unaccounted factors, limiting their predictive power. Neural ordinary differential equations (NODEs) have surged as a machine-learning algorithm that preserves the dynamic nature of the data (Chen et al. 2018 Adv. Neural Inf. Process. Syst.). Although preserving the dynamics in the data is an advantage, the question of how NODEs perform as a forecasting tool of ecological communities is unanswered. Here, we explore this question using simulated time series of competing species in a time-varying environment. We find that NODEs provide more precise forecasts than autoregressive integrated moving average (ARIMA) models. We also find that untuned NODEs have a similar forecasting accuracy to untuned long-short term memory neural networks and both are outperformed in accuracy and precision by empirical dynamical modelling . However, we also find NODEs generally outperform all other methods when evaluating with the interval score, which evaluates precision and accuracy in terms of prediction intervals rather than pointwise accuracy. We also discuss ways to improve the forecasting performance of NODEs. The power of a forecasting tool such as NODEs is that it can provide insights into population dynamics and should thus broaden the approaches to studying time series of ecological communities.

LFT: Neural Ordinary Differential Equations With Learnable Final-Time.

Since the last decade, deep neural networks have shown remarkable capability in learning representations. The recently proposed neural ordinary differential equations (NODEs) can be viewed as the continuous-time equivalence of residual neural networks. It has been shown that NODEs have a tremendous advantage over the conventional counterparts in terms of spatial complexity for modeling continuous-time processes. However, existing NODEs methods entail their final time to be specified in advance, precluding the models from choosing a desirable final time and limiting their expressive capabilities. In this article, we propose learnable final-time (LFT) NODEs to overcome this limitation. LFT rebuilds the NODEs learning process as a final-time-free optimal control problem and employs the calculus of variations to derive the learning algorithm of NODEs. In contrast to existing NODEs methods, the new approach empowers the NODEs models to choose their suitable final time, thus being more flexible in adjusting the model depth for given tasks. Additionally, we analyze the gradient estimation errors caused by numerical ordinary differential equations (ODEs) solvers and employ checkpoint-based methods to obtain accurate gradients. We demonstrate the effectiveness of the proposed method with experimental results on continuous normalizing flows (CNFs) and feedforward models.

Mutli-modal straight flow matching for accelerated MR imaging

Diffusion models have garnered great interest lately in Magnetic Resonance (MR) image reconstruction. A key component of generating high-quality samples from noise is iterative denoising for thousands of steps. However, the complexity of inference steps has limited its applications. To solve the challenge in obtaining high-quality reconstructed images with fewer inference steps and computational complexity, we introduce a novel straight flow matching, based on a neural ordinary differential equation (ODE) generative model. Our model creates a linear path between undersampled images and reconstructed images, which can be accurately simulated with a few Euler steps. Furthermore, we propose a multi-modal straight flow matching model, which uses relatively easily available modalities as supplementary information to guide the reconstruction of target modalities. We introduce the low frequency fusion layer and the high frequency fusion layer into our multi-modal model, which has been proved to produce promising results in fusion tasks. The proposed multi-modal straight flow matching (MMSflow) achieves state-of-the-art performances in task of reconstruction in fastMRI and Brats-2020 and improves the sampling rate by an order of magnitude than other methods based on stochastic differential equations (SDE).

Towards complex dynamic physics system simulation with graph neural ordinary equations

The great learning ability of deep learning facilitates us to comprehend the real physical world, making learning to simulate complicated particle systems a promising endeavour both in academia and industry. However, the complex laws of the physical world pose significant challenges to the learning based simulations, such as the varying spatial dependencies between interacting particles and varying temporal dependencies between particle system states in different time stamps, which dominate particles’ interacting behavior and the physical systems’ evolution patterns. Existing learning based methods fail to fully account for the complexities, making them unable to yield satisfactory simulations. To better comprehend the complex physical laws, we propose a novel model – Graph Networks with Spatial–Temporal neural Ordinary Differential Equations (GNSTODE) – that characterizes the varying spatial and temporal dependencies in particle systems using a united end-to-end framework. Through training with real-world particle–particle interaction observations, GNSTODE can simulate any possible particle systems with high precisions. We empirically evaluate GNSTODE’s simulation performance on two real-world particle systems, Gravity and Coulomb, with varying levels of spatial and temporal dependencies. The results show that GNSTODE yields better simulations than state-of-the-art methods, showing that GNSTODE can serve as an effective tool for particle simulation in real-world applications. Our code is made available at https://github.com/Guangsi-Shi/AI-for-physics-GNSTODE.

ASV station keeping under wind disturbances using neural network simulation error minimization model predictive control

AbstractStation keeping is an essential maneuver for autonomous surface vehicles (ASVs), mainly when used in confined spaces, to carry out surveys that require the ASV to keep its position or in collaboration with other vehicles where the relative position has an impact over the mission. However, this maneuver can become challenging for classic feedback controllers due to the need for an accurate model of the ASV dynamics and the environmental disturbances. This work proposes a model predictive controller using neural network simulation error minimization (NNSEM–MPC) to accurately predict the dynamics of the ASV under wind disturbances. The performance of the proposed scheme under wind disturbances is tested and compared against other controllers in simulation, using the robotics operating system and the multipurpose simulation environment Gazebo. A set of six tests was conducted by combining two varying wind speeds that are modeled as the Harris spectrum and three wind directions (, , and ). The simulation results clearly show the advantage of the NNSEM–MPC over the following methods: backstepping controller, sliding mode controller, simplified dynamics MPC (SD‐MPC), neural ordinary differential equation MPC (NODE‐MPC), and knowledge‐based NODE MPC. The proposed NNSEM–MPC approach performs better than the rest in five out of the six test conditions, and it is the second best in the remaining test case, reducing the mean position and heading error by at least % ( m) and % (), respectively, across all the test cases. In terms of execution speed, the proposed NNSEM–MPC is at least 36% faster than the rest of the MPC controllers. The field experiments on two different ASV platforms showed that ASVs can effectively keep the station utilizing the proposed method, with a position error as low as m and a heading error as low as within time windows of at least s. This would increase the potential applications of ASVs for launch, recovery, and replenishment in long‐term surveys in collaboration with other autonomous systems.

Verifying safety of neural networks from topological perspectives

Neural networks (NNs) are increasingly applied in safety-critical systems such as autonomous vehicles. However, they are fragile and are often ill-behaved. Consequently, their behaviors should undergo rigorous guarantees before deployment in practice. In this paper, we propose a set-boundary reachability method to investigate the safety verification problem of NNs from topological perspectives. Given an NN with an input set and a safe set, the safety verification problem is to determine whether all outputs of the NN resulting from the input set fall within the safe set. In our method, the homeomorphism property of NNs is first exploited, which establishes rigorous guarantees between the boundaries of the input set and the boundaries of the output set. A homeomorphism is a special case of open maps, and consequently our set-boundary method is considered to be generalized to situations with open map property then2. The exploitation of these two properties facilitates reachability computations via extracting subsets of the input set rather than the entire input set, thus controlling the wrapping effect in reachability analysis and facilitating the reduction of computation burdens for safety verification. The homeomorphism property exists in some widely used NNs such as invertible residual networks (i-ResNets) and Neural ordinary differential equations (Neural ODEs), and the open map is a less strict topological property and is easier to satisfy compared with homeomorphisms. For NNs establishing either of these two properties, our set-boundary reachability method only needs to perform reachability analysis on the boundary of the input set. Moreover, for NNs that do not feature these properties with respect to the input set, we also explore subsets of the input set for establishing the local homeomorphism property and then abandon these subsets for reachability computations. Finally, some examples demonstrate the performance of our proposed method.

Frontiers in Operations: Equitable Data-Driven Facility Location and Resource Allocation to Fight the Opioid Epidemic

Problem definition: The opioid epidemic is a crisis that has plagued the United States for decades. One central issue of the epidemic is inequitable access to treatment for opioid use disorder (OUD), which puts certain populations at a higher risk of opioid overdose. Methodology/results: We integrate a predictive dynamical model and a prescriptive optimization problem to compute high-quality opioid treatment facility and treatment budget allocations for each U.S. state. Our predictive model is a differential equation-based epidemiological model that captures the dynamics of the opioid epidemic. We use a process inspired by neural ordinary differential equations to fit this model to opioid epidemic data for each state and obtain estimates for unknown parameters in the model. We then incorporate this epidemiological model into a corresponding mixed-integer optimization problem (MIP) that aims to minimize the number of opioid overdose deaths and the number of people with OUD. We develop strong relaxations based on McCormick envelopes to efficiently compute approximate solutions to our MIPs that have a mean optimality gap of 3.99%. Our method provides socioeconomically equitable solutions, as it incentivizes investments in areas with higher social vulnerability (from the U.S. Centers for Disease Control’s Social Vulnerability Index) and opioid prescribing rates. On average, when allowing for overbudget solutions, our approach decreases the number of people with OUD by [Formula: see text], increases the number of people in treatment by [Formula: see text], and decreases the number of opioid-related deaths by [Formula: see text] after 2 years compared with the baseline epidemiological model’s predictions. Managerial implications: Our solutions show that policymakers should target adding treatment facilities to counties that have significantly fewer facilities than their population share and are more socially vulnerable. Furthermore, we demonstrate that our optimization approach, guided by epidemiological and socioeconomic factors, should help inform these strategic decisions, as it yields population health benefits in comparison with benchmarks based solely on population and social vulnerability. History: This paper has been accepted in the Manufacturing & Service Operations Management Frontiers in Operations Initiative. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0042 .

Enhancing continuous time series modelling with a latent ODE-LSTM approach

Due to their dynamic properties such as irregular sampling rate and high-frequency sampling, Continuous Time Series (CTS) are found in many applications. Since CTS with irregular sampling rate are difficult to model with standard Recurrent Neural Networks (RNNs), RNNs have been generalised to have continuous-time hidden dynamics defined by a Neural Ordinary Differential Equation (Neural ODE), leading to the ODE-RNN model. Another approach that provides a better modelling is that of the Latent ODE model, which constructs a continuous-time model where a latent state is defined at all times. The Latent ODE model uses a standard RNN as the encoder and a Neural ODE as the decoder. However, since the RNN encoder leads to difficulties with missing data and ill-defined latent variables, a Latent ODE-RNN model has recently been proposed that uses a ODE-RNN model as the encoder instead.Both the Latent ODE and Latent ODE-RNN models are difficult to train due to the vanishing and exploding gradients problem. To overcome this problem, the main contribution of this paper is to propose and illustrate a new model based on a new Latent ODE using an ODE-LSTM (Long Short-Term Memory) network as an encoder - the Latent ODE-LSTM model. To limit the growth of the gradients, the Norm Gradient Clipping strategy was embedded on the Latent ODE-LSTM model.The performance evaluation of the new Latent ODE-LSTM (with and without Norm Gradient Clipping) for modelling CTS with regular and irregular sampling rates is then demonstrated. Numerical experiments show that the new Latent ODE-LSTM performs better than Latent ODE-RNNs and can avoid the vanishing and exploding gradients during training.Code implementations developed in this work are available at github.com/CeciliaCoelho/LatentODELSTM.

Whole-heart electromechanical simulations using Latent Neural Ordinary Differential Equations

Cardiac digital twins provide a physics and physiology informed framework to deliver personalized medicine. However, high-fidelity multi-scale cardiac models remain a barrier to adoption due to their extensive computational costs. Artificial Intelligence-based methods can make the creation of fast and accurate whole-heart digital twins feasible. We use Latent Neural Ordinary Differential Equations (LNODEs) to learn the pressure-volume dynamics of a heart failure patient. Our surrogate model is trained from 400 simulations while accounting for 43 parameters describing cell-to-organ cardiac electromechanics and cardiovascular hemodynamics. LNODEs provide a compact representation of the 3D-0D model in a latent space by means of an Artificial Neural Network that retains only 3 hidden layers with 13 neurons per layer and allows for numerical simulations of cardiac function on a single processor. We employ LNODEs to perform global sensitivity analysis and parameter estimation with uncertainty quantification in 3 hours of computations, still on a single processor.

Designing a set of criteria for evaluating artificial neural networks trained with physics-based data to replicate molecular dynamics and other particle method trajectories

This article presents an in-depth analysis and evaluation of artificial neural networks (ANNs) when applied to replicate trajectories in molecular dynamics (MD) simulations or other particle methods. This study focuses on several architectures—feedforward neural networks (FNNs), convolutional neural networks (CNNs), recurrent neural networks (RNNs), time convolutions (TCs), self-attention (SA), graph neural networks (GNNs), neural ordinary differential equation (ODENets), and an example of physics-informed machine learning (PIML) model—assessing their effectiveness and limitations in understanding and replicating the underlying physics of particle systems. Through this analysis, this paper introduces a comprehensive set of criteria designed to evaluate the capability of ANNs in this context. These criteria include the minimization of losses, the permutability of particle indices, the ability to predict trajectories recursively, the conservation of particles, the model’s handling of boundary conditions, and its scalability. Each network type is systematically examined to determine its strengths and weaknesses in adhering to these criteria. While, predictably, none of the networks fully meets all criteria, this study extends beyond the simple conclusion that only by integrating physics-based models into ANNs is it possible to fully replicate complex particle trajectories. Instead, it probes and delineates the extent to which various neural networks can “understand” and interpret aspects of the underlying physics, with each criterion targeting a distinct aspect of this understanding.

Forecasting with an N-dimensional Langevin equation and a neural-ordinary differential equation.

Accurate prediction of electricity day-ahead prices is essential in competitive electricity markets. Although stationary electricity-price forecasting techniques have received considerable attention, research on non-stationary methods is comparatively scarce, despite the common prevalence of non-stationary features in electricity markets. Specifically, existing non-stationary techniques will often aim to address individual non-stationary features in isolation, leaving aside the exploration of concurrent multiple non-stationary effects. Our overarching objective here is the formulation of a framework to systematically model and forecast non-stationary electricity-price time series, encompassing the broader scope of non-stationary behavior. For this purpose, we develop a data-driven model that combines an N-dimensional Langevin equation (LE) with a neural-ordinary differential equation (NODE). The LE captures fine-grained details of the electricity-price behavior in stationary regimes but is inadequate for non-stationary conditions. To overcome this inherent limitation, we adopt a NODE approach to learn, and at the same time predict, the difference between the actual electricity-price time series and the simulated price trajectories generated by the LE. By learning this difference, the NODE reconstructs the non-stationary components of the time series that the LE is not able to capture. We exemplify the effectiveness of our framework using the Spanish electricity day-ahead market as a prototypical case study. Our findings reveal that the NODE nicely complements the LE, providing a comprehensive strategy to tackle both stationary and non-stationary electricity-price behavior. The framework's dependability and robustness is demonstrated through different non-stationary scenarios by comparing it against a range of basic naïve methods.

Enhancing predictive capabilities in data-driven dynamical modeling with automatic differentiation: Koopman and neural ODE approaches.

Data-driven approximations of the Koopman operator are promising for predicting the time evolution of systems characterized by complex dynamics. Among these methods, the approach known as extended dynamic mode decomposition with dictionary learning (EDMD-DL) has garnered significant attention. Here, we present a modification of EDMD-DL that concurrently determines both the dictionary of observables and the corresponding approximation of the Koopman operator. This innovation leverages automatic differentiation to facilitate gradient descent computations through the pseudoinverse. We also address the performance of several alternative methodologies. We assess a "pure" Koopman approach, which involves the direct time-integration of a linear, high-dimensional system governing the dynamics within the space of observables. Additionally, we explore a modified approach where the system alternates between spaces of states and observables at each time step-this approach no longer satisfies the linearity of the true Koopman operator representation. For further comparisons, we also apply a state-space approach (neural ordinary differential equations). We consider systems encompassing two- and three-dimensional ordinary differential equation systems featuring steady, oscillatory, and chaotic attractors, as well as partial differential equations exhibiting increasingly complex and intricate behaviors. Our framework significantly outperforms EDMD-DL. Furthermore, the state-space approach offers superior performance compared to the "pure" Koopman approach where the entire time evolution occurs in the space of observables. When the temporal evolution of the Koopman approach alternates between states and observables at each time step, however, its predictions become comparable to those of the state-space approach.

Spatial-temporal graph neural ODE networks for skeleton-based action recognition.

In the field of skeleton-based action recognition, accurately recognizing human actions is crucial for applications such as virtual reality and motion analysis. However, this task faces challenges such intraindividual action differences and long-term temporal dependencies. To address these challenges, we propose an innovative model called spatial-temporal graph neural ordinary differential equations (STG-NODE). First, in the data preprocessing stage, the dynamic time warping (DTW) algorithm is used to normalize and calculate 3D skeleton data to facilitate the derivation of customized adjacency matrices for improving the influence of intraindividual action differences. Secondly, a custom ordinary differential equation (ODE) integrator is applied based on the initial conditions of the temporal features, producing a solution function that simulates the dynamic evolution trend of the events of interest. Finally, the outstanding ODE solver is used to numerically solve the time features based on the solution function to increase the influence of long-term dependencies on the recognition accuracy of the model and provide it with a more powerful temporal modeling ability. Through extensive experiments conducted on the NTU RGB+D 60 and Kinetics Skeleton 400 benchmark datasets, we demonstrate the superior performance of STG-NODE in the action recognition domain. The success of the STG-NODE model also provides new ideas and methods for the future development of the action recognition field.

Physics-informed data-driven model of dehydration reaction stage in the sintering process of ternary cathode materials

The sintering process is the main process that affects the performance of ternary cathode materials. Due to the closed, continuous, and long-term characteristics of the sintering process carried out in a roller kiln, the reaction state including reaction degree, particle size, and etc., of the raw materials cannot be directly obtained through sampling detection, resulting in ineffective temperature control. This article mainly focuses on modeling the dehydration reaction stage of the sintering process to obtain the reaction state of the raw materials. Firstly, a two-sphere model is established based on Newton’s second law and the particle volume conservation equation to describe the relationship between the particle radius and sintering temperature of lithium hydroxide and ternary precursor; Secondly, the chemical kinetic parameters of the dehydration reaction in the two-sphere model are obtained by combining the single heating rate scanning method with the multiple heating rate scanning method; Then, in order to improve the accuracy of the two-sphere model and reduce the impact of external environmental temperature changes, a physics-informed neural network model is constructed by combining the two-sphere model with the neural ordinary differential equation model. Finally, the effectiveness of the modeling method is verified through numerical simulation and empirical data.

RNA velocity prediction via neural ordinary differential equation

RNA velocity is a crucial tool for unraveling the trajectory of cellular responses. Several approaches, including ordinary differential equations and machine learning models, have been proposed to interpret velocity. However, the practicality of these methods is constrained by underlying assumptions. In this study, we introduce SymVelo, a dual-path framework that effectively integrates high- and low-dimensional information. Rigorous benchmarking and extensive studies demonstrate that SymVelo is capable of inferring differentiation trajectories in developing organs, analyzing gene responses to stimulation, and uncovering transcription dynamics. Moreover, the adaptable architecture of SymVelo enables customization to accommodate intricate data and diverse modalities in forthcoming research, thereby providing a promising avenue for advancing our understanding of cellular behavior.