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Variational Physics-Informed Neural Networks often suffer from poor convergence when using stochastic gradient-descent-based optimizers. By introducing a Least Squares solver for the weights of the last layer of the neural network, we…
We consider the three-dimensional incompressible free-boundary magnetohydrodynamics (MHD) equations in a bounded domain with surface tension on the boundary. We establish a priori estimate for solutions in the Lagrangian coordinates with…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must confront the challenge of controlling errors in the discrete divergence of the magnetic field. One approach that has been…
Finding the distribution of the velocities and pressures of a fluid by solving the Navier-Stokes equations is a principal task in the chemical, energy, and pharmaceutical industries, as well as in mechanical engineering and the design of…
We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the boundary (or initial) conditions…
We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions…
Finite element methods usually construct basis functions and quadrature rules for multidimensional domains via tensor products of one-dimensional counterparts. While straightforward, this approach results in integration spaces larger than…
We show that the error achievable using physics-informed neural networks for solving systems of differential equations can be substantially reduced when these networks are trained using meta-learned optimization methods rather than to using…
Accurate quantification of uncertainty in neural network predictions remains a central challenge for scientific applications involving high-dimensional, correlated data. While existing methods capture either aleatoric or epistemic…
Fully adaptive computations of the resistive magnetohydrodynamic (MHD) equations are presented in two and three space dimensions using a finite volume discretization on locally refined dyadic grids. Divergence cleaning is used to control…
Rapid development in numerical modelling of materials and the complexity of new models increases quickly together with their computational demands. Despite the growing performance of modern computers and clusters, calibration of such models…
We describe and analyze a hybrid finite element/neural network method for predicting solutions of partial differential equations. The methodology is designed for obtaining fine scale fluctuations from neural networks in a local manner. The…
In this work, we propose a model order reduction framework to deal with inverse problems in a non-intrusive setting. Inverse problems, especially in a partial differential equation context, require a huge computational load due to the…
We present an adaptive multiresolution method for the numerical simulation of ideal magnetohydrodynamics in two space dimensions. The discretization uses a finite volume scheme based on a Cartesian mesh and an explicit compact Rung-Kutta…
The numerical simulation and optimization of technical systems described by partial differential equations is expensive, especially in multi-query scenarios in which the underlying equations have to be solved for different parameters. A…
We propose a second-order accurate semi-implicit and well-balanced finite volume scheme for the equations of ideal magnetohydrodynamics (MHD) including gravitational source terms. The scheme treats all terms associated with the acoustic…
Mixed-precision Deep Neural Networks achieve the energy efficiency and throughput needed for hardware deployment, particularly when the resources are limited, without sacrificing accuracy. However, the optimal per-layer bit precision that…
Existing techniques for the cost optimization of water distribution networks either employ meta-heuristics, or try to develop problem-specific optimization techniques. Instead, we exploit recent advances in generic NLP solvers and explore a…
Inertial measurement units are commonly used to estimate the attitude of moving objects. Numerous nonlinear filter approaches have been proposed for solving the inherent sensor fusion problem. However, when a large range of different…