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Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDE) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations…
Several forms for constructing novel physics-informed neural-networks (PINN) for the solution of partial-differential-algebraic equations based on derivative operator splitting are proposed, using the nonlinear Kirchhoff rod as a prototype…
Recent studies shows that the majority of existing deep steganalysis models have a large amount of redundancy, which leads to a huge waste of storage and computing resources. The existing model compression method cannot flexibly compress…
Whilst the partial differential equations that govern the dynamics of our world have been studied in great depth for centuries, solving them for complex, high-dimensional conditions and domains still presents an incredibly large…
Traditional physics-informed neural networks (PINNs) do not always satisfy physics based constraints, especially when the constraints include differential operators. Rather, they minimize the constraint violations in a soft way. Strict…
The automatic differentiation (AD) in the vanilla physics-informed neural networks (PINNs) is the computational bottleneck for the high-efficiency analysis. The concept of derivative discretization in smoothed particle hydrodynamics (SPH)…
Time-dependent Partial Differential Equations with given initial conditions are considered in this paper. New differentiation techniques of the unknown solution with respect to time variable are proposed. It is shown that the proposed…
We present strongly convergent explicit and semi-implicit adaptive numerical schemes for systems of stiff stochastic differential equations (SDEs) where both the drift and diffusion are non-globally Lipschitz continuous. This stiffness may…
Multiscale stochastic dynamical systems have been widely adopted to a variety of scientific and engineering problems due to their capability of depicting complex phenomena in many real world applications. This work is devoted to…
Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every…
Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and…
Accurately learning solution operators for time-dependent partial differential equations (PDEs) from sparse and irregular data remains a challenging task. Recurrent DeepONet extensions inherit the discrete-time limitations of…
In this work, we present a machine learning approach for reducing the error when numerically solving time-dependent partial differential equations (PDE). We use a fully convolutional LSTM network to exploit the spatiotemporal dynamics of…
Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…
Numerical discretisations of partial differential equations (PDEs) can be written as discrete convolutions, which, themselves, are a key tool in AI libraries and used in convolutional neural networks (CNNs). We therefore propose to…
Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…
The measured spatiotemporal response of various physical processes is utilized to infer the governing partial differential equations (PDEs). We propose SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE), a technique…
Physics-informed neural networks (PINNs) have recently emerged as an alternative way of solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In…
Traffic forecasting is one of the most popular spatio-temporal tasks in the field of machine learning. A prevalent approach in the field is to combine graph convolutional networks and recurrent neural networks for the spatio-temporal…
In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the solution data can reveal the PDE operator depending on the…