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Dynamic astrophysical phenomena are predominantly described by differential equations, yet our understanding of these systems is constrained by our incomplete grasp of non-linear physics and scarcity of comprehensive datasets. As such,…

Modeling and controlling complex spatiotemporal dynamical systems driven by partial differential equations (PDEs) often necessitate dimensionality reduction techniques to construct lower-order models for computational efficiency. This paper…

Optimal control problems naturally arise in many scientific applications where one wishes to steer a dynamical system from a certain initial state $\mathbf{x}_0$ to a desired target state $\mathbf{x}^*$ in finite time $T$. Recent advances…

We show that Neural Ordinary Differential Equations (ODEs) learn representations that preserve the topology of the input space and prove that this implies the existence of functions Neural ODEs cannot represent. To address these…

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…

The high computational cost associated with solving for detailed chemistry poses a significant challenge for predictive computational fluid dynamics (CFD) simulations of turbulent reacting flows. These models often require solving a system…

Neural networks have the ability to serve as universal function approximators, but they are not interpretable and don't generalize well outside of their training region. Both of these issues are problematic when trying to apply standard…

The unprecedented availability of large-scale datasets in neuroscience has spurred the exploration of artificial deep neural networks (DNNs) both as empirical tools and as models of natural neural systems. Their appeal lies in their ability…

Neural Ordinary Differential Equations (NODEs) have proven to be a powerful modeling tool for approximating (interpolation) and forecasting (extrapolation) irregularly sampled time series data. However, their performance degrades…

Stochastic differential equations (SDEs) are used to describe a wide variety of complex stochastic dynamical systems. Learning the hidden physics within SDEs is crucial for unraveling fundamental understanding of these systems' stochastic…

We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the…

The identification of a mathematical dynamics model is a crucial step in the designing process of a controller. However, it is often very difficult to identify the system's governing equations, especially in complex environments that…

Normalizing flows are a powerful technique for obtaining reparameterizable samples from complex multimodal distributions. Unfortunately current approaches fall short when the underlying space has a non trivial topology, and are only…

Neural Ordinary Differential Equations (NODEs) use a neural network to model the instantaneous rate of change in the state of a system. However, despite their apparent suitability for dynamics-governed time-series, NODEs present a few…

Modeling continuous dynamical systems from discretely sampled observations is a fundamental problem in data science. Often, such dynamics are the result of non-local processes that present an integral over time. As such, these systems are…

We develop compositional learning algorithms for coupled dynamical systems, with a particular focus on electrical networks. While deep learning has proven effective at modeling complex relationships from data, compositional couplings…

Forecasting system behaviour near and across bifurcations is crucial for identifying potential shifts in dynamical systems. While machine learning has recently been used to learn critical transitions and bifurcation structures from data,…

In this work, we investigate a method for simulation-free training of Neural Ordinary Differential Equations (NODEs) for learning deterministic mappings between paired data. Despite the analogy of NODEs as continuous-depth residual…

Modern power systems require fast and accurate dynamic simulations for stability assessment, digital twins, and real-time control, but classical ODE solvers are often too slow for large-scale or online applications. We propose a…

Owing to the remarkable development of deep learning technology, there have been a series of efforts to build deep learning-based climate models. Whereas most of them utilize recurrent neural networks and/or graph neural networks, we design…