Many successful methods to learn dynamical systems from data have recently been introduced. Such methods often rely on the availability of the system's full state. However, this underlying hypothesis is rather restrictive as it is typically not confirmed in practice, leaving us with partially observed systems. Utilizing the Mori-Zwanzig (MZ) formalism from statistical physics, we demonstrate that Constant Lag Neural Delay Differential Equations (ND-DEs) naturally serve as suitable models for partially observed states. In empirical evaluation, we show that such models outperform existing methods on both synthetic and experimental data.

We propose a unified framework for delay differential equations (DDEs) based on deep neural networks (DNNs) - the neural delay differential equations (NDDEs), aimed at solving the forward and inverse problems of delay differential equations. This framework could embed delay differential equations into neural networks to accommodate the diverse requirements of DDEs in terms of initial conditions, control equations, and known data. NDDEs adjust the network parameters through automatic differentiation and optimization algorithms to minimize the loss function, thereby obtaining numerical solutions to the delay differential equations without the grid dependence and polynomial interpolation typical of traditional numerical methods. In addressing inverse problems, the NDDE framework can utilize observational data to perform precise estimation of single or multiple delay parameters, which is very important in practical mathematical modeling. The results of multiple numerical experiments have shown that NDDEs demonstrate high precision in both forward and inverse problems, proving their effectiveness and promising potential in dealing with delayed differential equation issues.

Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with representative datasets. Recently, an augmented framework has been developed to overcome some limitations that emerged in the application of the original framework. In this paper, we propose a new class of continuous-depth neural networks with delay, named Neural Delay Differential Equations (NDDEs). To compute the corresponding gradients, we use the adjoint sensitivity method to obtain the delayed dynamics of the adjoint. Differential equations with delays are typically seen as dynamical systems of infinite dimension that possess more fruitful dynamics. Compared to NODEs, NDDEs have a stronger capacity of nonlinear representations. We use several illustrative examples to demonstrate this outstanding capacity. Firstly, we successfully model the delayed dynamics where the trajectories in the lower-dimensional phase space could be mutually intersected and even chaotic in a model-free or model-based manner. Traditional NODEs, without any argumentation, are not directly applicable for such modeling. Secondly, we achieve lower loss and higher accuracy not only for the data produced synthetically by complex models but also for the CIFAR10, a well-known image dataset. Our results on the NDDEs demonstrate that appropriately articulating the elements of dynamical systems into the network design is truly beneficial in promoting network performance.

Continuous-time dynamics models, such as neural ordinary differential equations, have enabled the modeling of underlying dynamics in time-series data and accurate forecasting. However, parameterization of dynamics using a neural network makes it difficult for humans to identify causal structures in the data. In consequence, this opaqueness hinders the use of these models in the domains where capturing causal relationships carries the same importance as accurate predictions, e.g., tsunami forecasting. In this paper, we address this challenge by proposing a mechanism for mining causal structures from continuous-time models. We train models to capture the causal structure by enforcing sparsity in the weights of the input layers of the dynamics models. We first verify the effectiveness of our method in the scenario where the exact causal-structures of time-series are known as a priori. We next apply our method to a real-world problem, namely tsunami forecasting, where the exact causal-structures are difficult to characterize. Experimental results show that the proposed method is effective in learning physically-consistent causal relationships while achieving high forecasting accuracy.

A novel way of using neural networks to learn the dynamics of time delay systems from sequential data is proposed. A neural network with trainable delays is used to approximate the right hand side of a delay differential equation. We relate the delay differential equation to an ordinary differential equation by discretizing the time history and train the corresponding neural ordinary differential equation (NODE) to learn the dynamics. An example on learning the dynamics of the Mackey-Glass equation using data from chaotic behavior is given. After learning both the nonlinearity and the time delay, we demonstrate that the bifurcation diagram of the neural network matches that of the original system.

Continuous-depth neural networks, such as the Neural Ordinary Differential Equations (ODEs), have aroused a great deal of interest from the communities of machine learning and data science in recent years, which bridge the connection between deep neural networks and dynamical systems. In this article, we introduce a new sort of continuous-depth neural network, called the Neural Piecewise-Constant Delay Differential Equations (PCDDEs). Here, unlike the recently proposed framework of the Neural Delay Differential Equations (DDEs), we transform the single delay into the piecewise-constant delay(s). The Neural PCDDEs with such a transformation, on one hand, inherit the strength of universal approximating capability in Neural DDEs. On the other hand, the Neural PCDDEs, leveraging the contributions of the information from the multiple previous time steps, further promote the modeling capability without augmenting the network dimension. With such a promotion, we show that the Neural PCDDEs do outperform the several existing continuous-depth neural frameworks on the one-dimensional piecewise-constant delay population dynamics and real-world datasets, including MNIST, CIFAR10, and SVHN.

Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with some representative datasets. Recently, an augmented framework has been successfully developed for conquering some limitations emergent in application of the original framework. Here we propose a new class of continuous-depth neural networks with delay, named as Neural Delay Differential Equations (NDDEs), and, for computing the corresponding gradients, we use the adjoint sensitivity method to obtain the delayed dynamics of the adjoint. Since the differential equations with delays are usually seen as dynamical systems of infinite dimension possessing more fruitful dynamics, the NDDEs, compared to the NODEs, own a stronger capacity of nonlinear representations. Indeed, we analytically validate that the NDDEs are of universal approximators, and further articulate an extension of the NDDEs, where the initial function of the NDDEs is supposed to satisfy ODEs. More importantly, we use several illustrative examples to demonstrate the outstanding capacities of the NDDEs and the NDDEs with ODEs' initial value. Specifically, (1) we successfully model the delayed dynamics where the trajectories in the lower-dimensional phase space could be mutually intersected, while the traditional NODEs without any argumentation are not directly applicable for such modeling, and (2) we achieve lower loss and higher accuracy not only for the data produced synthetically by complex models but also for the real-world image datasets, i.e., CIFAR10, MNIST, and SVHN. Our results on the NDDEs reveal that appropriately articulating the elements of dynamical systems into the network design is truly beneficial to promoting the network performance.