Related papers: Automatic Differentiation using Operator Overloadi…
We derive and analyze the alternating direction explicit (ADE) method for time evolution equations with the time-dependent Dirichlet boundary condition and with the zero Neumann boundary condition. The original ADE method is an additive…
The composition of pretraining data is a key determinant of foundation models' performance, but there is no standard guideline for allocating a limited computational budget across different data sources. Most current approaches either rely…
In this study, a novel semi-implicit second-order temporal scheme combined with the finite element method for space discretization is proposed to solve the coupled system of infiltration and solute transport in unsaturated porous media. The…
In this work, high-order discrete well-balanced methods for one-dimensional hyperbolic systems of balance laws are proposed. We aim to construct a method whose discrete steady states correspond to solutions of arbitrary high-order ODE…
Efficiently solving large-scale optimal power flow (OPF) problems is challenging due to the high dimensionality and interconnectivity of modern power systems. Decomposition methods offer a promising solution via partitioning large problems…
In this paper, we propose an adaptive high-order method for hyperbolic systems of conservation laws. The proposed method is based on a dual formulation approach: Two numerical solutions, corresponding to conservative and nonconservative…
Neural Operators (NOs) are a leading method for surrogate modeling of partial differential equations. Unlike traditional neural networks, which approximate individual functions, NOs learn the mappings between function spaces. While NOs have…
Neural operator surrogates for time-dependent partial differential equations (PDEs) conventionally employ autoregressive prediction schemes, which accumulate error over long rollouts and require uniform temporal discretization. We introduce…
Many applied time-dependent problems are characterized by an additive representation of the problem operator. Additive schemes are constructed using such a splitting and associated with the transition to a new time level on the basis of the…
An optimization based state and parameter estimation method is presented where the required Jacobian matrix of the cost function is computed via automatic differentiation. Automatic differentiation evaluates the programming code of the cost…
The splitting method is a powerful method for solving partial differential equations. Various splitting methods have been designed to separate different physics, nonlinearities, and so on. Recently, a new splitting approach has been…
Fully implicit timestepping methods have several potential advantages for atmosphere/ocean simulation. First, being unconditionally stable, they degrade more gracefully as the Courant number increases, typically requiring more solver…
Operator splitting methods allow to split the operator describing a complex dynamical system into a sequence of simpler subsystems and treat each part independently. In the modeling of dynamical problems, systems of (possibly coupled)…
Algorithmic Differentiation (AD) can be used to automate the generation of derivatives in arbitrary software projects. This will generate maintainable derivatives, that are always consistent with the computation of the software. If a domain…
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially…
This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order…
Automatic differentiation, also known as backpropagation, AD, autodiff, or algorithmic differentiation, is a popular technique for computing derivatives of computer programs accurately and efficiently. Sometimes, however, the derivatives…
We introduce a class of unconditionally energy stable, high order accurate schemes for gradient flows in a very general setting. The new schemes are a high order analogue of the minimizing movements approach for generating a time discrete…
In this article we develop an algorithm for the efficient simulation of electrolytes in the presence of physical boundaries. In previous work the Discrete Ion Stochastic Continuum Overdamped Solvent (DISCOS) algorithm was derived for triply…
In this letter, we address the task of adaptive sampling to model vector fields. When modeling environmental phenomena with a robot, gathering high resolution information can be resource intensive. Actively gathering data and modeling flows…