Neural ordinary differential equations (ODEs) have attracted much attention as continuous-time counterparts of deep residual neural networks (NNs), and numerous extensions for recurrent NNs have been proposed. Since the 1980s, ODEs have also been used to derive theoretical results for NN learning rules, e.g., the famous connection between Oja's rule and principal component analysis. Such rules are typically expressed as additive iterative update processes which have straightforward ODE counterparts. Here we introduce a novel combination of learning rules and Neural ODEs to build continuous-time sequence processing nets that learn to manipulate short-term memory in rapidly changing synaptic connections of other nets. This yields continuous-time counterparts of Fast Weight Programmers and linear Transformers. Our novel models outperform the best existing Neural Controlled Differential Equation based models on various time series classification tasks, while also addressing their fundamental scalability limitations.

Many successful methods to learn dynamical systems from data have recently been introduced.

Despite the promise of scientific machine learning (SciML) in combining data-driven techniques with mechanistic modeling, existing approaches for incorporating hard constraints in neural differential equations (NDEs) face significant limitations. Scalability issues and poor numerical properties prevent these neural models from being used for modeling physical systems with complicated conservation laws. We propose Manifold-Projected Neural ODEs (PNODEs), a method that explicitly enforces algebraic constraints by projecting each ODE step onto the constraint manifold. This framework arises naturally from semi-explicit differential-algebraic equations (DAEs), and includes both a robust iterative variant and a fast approximation requiring a single Jacobian factorization. We further demonstrate that prior works on relaxation methods are special cases of our approach. PNODEs consistently outperform baselines across six benchmark problems achieving a mean constraint violation error below $10^{-10}$. Additionally, PNODEs consistently achieve lower runtime compared to other methods for a given level of error tolerance. These results show that constraint projection offers a simple strategy for learning physically consistent long-horizon dynamics.

Time series modeling and analysis has become critical in various domains. Conventional methods such as RNNs and Transformers, while effective for discrete-time and regularly sampled data, face significant challenges in capturing the continuous dynamics and irregular sampling patterns inherent in real-world scenarios. Neural Differential Equations (NDEs) represent a paradigm shift by combining the flexibility of neural networks with the mathematical rigor of differential equations. This paper presents a comprehensive review of NDE-based methods for time series analysis, including neural ordinary differential equations, neural controlled differential equations, and neural stochastic differential equations. We provide a detailed discussion of their mathematical formulations, numerical methods, and applications, highlighting their ability to model continuous-time dynamics. Furthermore, we address key challenges and future research directions. This survey serves as a foundation for researchers and practitioners seeking to leverage NDEs for advanced time series analysis.

In transportation systems and autonomous vehicles, intelligent agents must understand the future motion of traffic participants to effectively plan motion trajectories. At the same time, the motion of traffic participants is inherently uncertain. In this paper, we propose TrajFlow, a generative framework for estimating the occupancy density of traffic participants. Our framework utilizes a causal encoder to extract semantically meaningful embeddings of the observed trajectory, as well as a normalizing flow to decode these embeddings and determine the most likely future location of traffic participants at some time point in the future. Our formulation differs from existing approaches because we model the marginal distribution of spatial locations instead of the joint distribution of unobserved trajectories. The advantages of a marginal formulation are numerous. First, we demonstrate that the marginal formulation produces higher accuracy on challenging trajectory forecasting benchmarks. Second, the marginal formulation allows for a fully continuous sampling of future locations. Finally, marginal densities are better suited for downstream tasks as they allow for the computation of per-agent motion trajectories and occupancy grids, the two most commonly used representations for motion forecasting. We present a novel architecture based entirely on neural differential equations as an implementation of this framework and provide ablations to demonstrate the advantages of a continuous implementation over a more traditional discrete neural network based approach. The code is available at https://github.com/kosieram21/TrajFlow .

Neural ordinary differential equations (ODEs) have attracted much attention as continuous-time counterparts of deep residual neural networks (NNs), and numerous extensions for recurrent NNs have been proposed. Since the 1980s, ODEs have also been used to derive theoretical results for NN learning rules, e.g., the famous connection between Oja's rule and principal component analysis. Such rules are typically expressed as additive iterative update processes which have straightforward ODE counterparts. Here we introduce a novel combination of learning rules and Neural ODEs to build continuous-time sequence processing nets that learn to manipulate short-term memory in rapidly changing synaptic connections of other nets. This yields continuous-time counterparts of Fast Weight Programmers and linear Transformers.

We present a novel dataset of simulations of the photodissociation region (PDR) in the Orion Bar and provide benchmarks of emulators for the dataset. Numerical models of PDRs are computationally expensive since the modeling of these changing regions requires resolving the thermal balance and chemical composition along a line-of-sight into an interstellar cloud. This often makes it a bottleneck for 3D simulations of these regions. In this work, we provide a dataset of 8192 models with different initial conditions simulated with 3D-PDR.

Neural differential equations offer a powerful approach for learning dynamics from data. However, they do not impose known constraints that should be obeyed by the learned model. It is well-known that enforcing constraints in surrogate models can enhance their generalizability and numerical stability. In this paper, we introduce projected neural differential equations (PNDEs), a new method for constraining neural differential equations based on projection of the learned vector field to the tangent space of the constraint manifold. In tests on several challenging examples, including chaotic dynamical systems and state-of-the-art power grid models, PNDEs outperform existing methods while requiring fewer hyperparameters. The proposed approach demonstrates significant potential for enhancing the modeling of constrained dynamical systems, particularly in complex domains where accuracy and reliability are essential.

Long-term traffic flow forecasting plays a crucial role in intelligent transportation as it allows traffic managers to adjust their decisions in advance. However, the problem is challenging due to spatio-temporal correlations and complex dynamic patterns in continuous-time stream data. Neural Differential Equations (NDEs) are among the state-of-the-art methods for learning continuous-time traffic dynamics. However, the traditional NDE models face issues in long-term traffic forecasting due to failures in capturing delayed traffic patterns, dynamic edge (location-to-location correlation) patterns, and abrupt trend patterns. To fill this gap, we propose a new NDE architecture called Multi-View Neural Differential Equations. Our model captures current states, delayed states, and trends in different state variables (views) by learning latent multiple representations within Neural Differential Equations. Extensive experiments conducted on several real-world traffic datasets demonstrate that our proposed method outperforms the state-of-the-art and achieves superior prediction accuracy for long-term forecasting and robustness with noisy or missing inputs.