We present a numerical framework for approximating unknown governing equations using
observation data and deep neural networks (DNN). In particular, we propose to use residual …

We present a framework for recovering/approximating unknown time-dependent partial
differential equation (PDE) using its solution data. Instead of identifying the terms in the …

Numerical schemes provably preserving the positivity of density and pressure are highly
desirable for ideal magnetohydrodynamics (MHD), but the rigorous positivity-preserving (PP) …

We present effective numerical algorithms for approximating unknown governing differential
equations from measurement data. We employ a set of standard basis functions, eg, …

The paper develops high-order accurate physical-constraints-preserving finite difference
WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax–…

We present a numerical approach for approximating unknown Hamiltonian systems using
observational data. A distinct feature of the proposed method is that it is structure-preserving, …

This paper proposes and analyzes arbitrarily high-order discontinuous Galerkin (DG) and
finite volume methods which provably preserve the positivity of density and pressure for the …

Invasive micropapillary carcinoma (IMPC) is a special histological subtype of breast cancer,
featured with extremely high rates of lymphovascular invasion and lymph node metastasis. …

We present a numerical framework for deep neural network (DNN) modeling of unknown
time-dependent partial differential equation (PDE) using their trajectory data. Unlike the recent …

Solutions to many partial differential equations satisfy certain bounds or constraints. For
example, the density and pressure are positive for equations of fluid dynamics, and in the …