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Numerical solution of partial differential equations (PDEs) plays a vital role in various fields of science and engineering. In recent years, deep neural networks (DNNs) have emerged as a powerful tool for solving PDEs, leveraging their…

Neural ordinary differential equations (Neural ODEs) is a class of machine learning models that approximate the time derivative of hidden states using a neural network. They are powerful tools for modeling continuous-time dynamical systems,…

Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…

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…

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to…

To derive the hidden dynamics from observed data is one of the fundamental but also challenging problems in many different fields. In this study, we propose a new type of interpretable network called the ordinary differential equation…

The bulk kinematics and thermodynamics of hot supernovae-driven galactic winds is critically dependent on both the amount of swept up cool clouds and non-spherical collimated flow geometry. However, accurately parameterizing these physics…

We introduce the framework of continuous-depth graph neural networks (GNNs). Neural graph differential equations (Neural GDEs) are formalized as the counterpart to GNNs where the input-output relationship is determined by a continuum of GNN…

Hamiltonian neural networks (HNNs) represent a promising class of physics-informed deep learning methods that utilize Hamiltonian theory as foundational knowledge within neural networks. However, their direct application to engineering…

Symbolic regression with polynomial neural networks and polynomial neural ordinary differential equations (ODEs) are two recent and powerful approaches for equation recovery of many science and engineering problems. However, these methods…

The existing Neural ODE formulation relies on an explicit knowledge of the termination time. We extend Neural ODEs to implicitly defined termination criteria modeled by neural event functions, which can be chained together and…

Tensorial neural networks (TNNs) combine the successes of multilinear algebra with those of deep learning to enable extremely efficient reduced-order models of high-dimensional problems. Here, I describe a deep neural network architecture…

Data-driven modeling and machine learning are widely used to model the behavior of dynamic systems. One application is the residual evaluation of technical systems where model predictions are compared with measurement data to create…

Modeling biological dynamical systems is challenging due to the interdependence of different system components, some of which are not fully understood. To fill existing gaps in our ability to mechanistically model physiological systems, we…

Neural Ordinary Differential Equations model dynamical systems with ODEs learned by neural networks. However, ODEs are fundamentally inadequate to model systems with long-range dependencies or discontinuities, which are common in…

Traditionally, calcium dynamics in neurons are modeled using partial differential equations (PDEs) and ordinary differential equations (ODEs). The PDE component focuses on reaction-diffusion processes, while the ODE component addresses…

The notion of an Evolutional Deep Neural Network (EDNN) is introduced for the solution of partial differential equations (PDE). The parameters of the network are trained to represent the initial state of the system only, and are…

Probabilistic theory and differential equation are powerful tools for the interpretability and guidance of the design of machine learning models, especially for illuminating the mathematical motivation of learning latent variable from…

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…

In this work, we explore the ability of NN (Neural Networks) to serve as a tool for finding eigen-pairs of ordinary differential equations. The question we aime to address is whether, given a self-adjoint operator, we can learn what are the…