Related papers: Approximating Numerical Fluxes Using Fourier Neura…

The purpose of the current work is the development of an approach to account for quasi-static mechanical equilibrium in empirical (i.e., data-based) models for the stress field employing neural approximations (NAs), which include neural…

Lack of conservation has been the biggest drawback in meshfree generalized finite difference methods (GFDMs). In this paper, we present a novel modification of classical meshfree GFDMs to include local balances which produce an approximate…

When solving partial differential equations using classical schemes such as finite difference or finite volume methods, sufficiently fine meshes and carefully designed schemes are required to achieve high-order accuracy of numerical…

Radiative heat transfer is a fundamental process in high energy density physics and inertial fusion. Accurately predicting the behavior of Marshak waves across a wide range of material properties and drive conditions is crucial for design…

Neural operators generalize classical neural networks to maps between infinite-dimensional spaces, e.g., function spaces. Prior works on neural operators proposed a series of novel methods to learn such maps and demonstrated unprecedented…

In this paper we present a new strategy to model the subgrid-scale scalar flux in a three-dimensional turbulent incompressible flow using physics-informed neural networks (NNs). When trained from direct numerical simulation (DNS) data,…

The development of machine learning techniques enables us to construct surrogate models from data of direct numerical simulations, which has important implications for modeling complex physical systems. In this paper, based on the output…

Physics-Informed Neural Networks (PINNs) have emerged as a promising machine learning approach for solving partial differential equations (PDEs). However, PINNs face significant challenges in balancing multi-objective losses, as multiple…

We consider solving complex spatiotemporal dynamical systems governed by partial differential equations (PDEs) using frequency domain-based discrete learning approaches, such as Fourier neural operators. Despite their widespread use for…

Fourier Neural Operators are deep learning models that learn mappings between function spaces and can be used to learn and solve partial differential equations (PDEs), in some cases significantly faster than traditional PDE solvers. Within…

This paper presents a spectral scheme for the numerical solution of nonlinear conservation laws in non-periodic domains under arbitrary boundary conditions. The approach relies on the use of the Fourier Continuation (FC) method for spectral…

Neural ordinary differential equations (NODEs), one of the most influential works of the differential equation-based deep learning, are to continuously generalize residual networks and opened a new field. They are currently utilized for…

In this chapter, we aim at presenting the basic techniques necessary to go beyond the widely accepted paradigm of second-order numerics. We specifically focus on finite-volume schemes for hyperbolic conservation laws occuring in fluid…

In this paper, a simple fifth-order finite difference Hermite WENO (HWENO) scheme combined with limiter is proposed for one- and two- dimensional hyperbolic conservation laws. The fluxes in the governing equation are approximated by the…

Neural operators (NOs) are designed to learn maps between infinite-dimensional function spaces. We propose a novel reframing of their use. By introducing an auxiliary base-space, any finite-dimensional function can be viewed as an operator…

In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first…

Computational Fluid Dynamics (CFD) simulations are used for many air flow simulations including road vehicle aerodynamics. Numerical diffusion occurs when local flow direction is not aligned with the mesh lines and when there is a non-zero…

Fourier Neural Operator (FNO) is a powerful and popular operator learning method. However, FNO is mainly used in forward prediction, yet a great many applications rely on solving inverse problems. In this paper, we propose an invertible…

Simulation of urban wind environments is crucial for urban planning, pollution control, and renewable energy utilization. However, the computational requirements of high-fidelity computational fluid dynamics (CFD) methods make them…

When using Neural Networks as trial functions to numerically solve PDEs, a key choice to be made is the loss function to be minimised, which should ideally correspond to a norm of the error. In multiple problems, this error norm coincides…