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The paper is devoted to the study of the backward behavior of solutions of the initial boundary value problem for the chevron pattern equations under homogeneous Dirichlet’s boundary conditions. We prove that, as t → ∞, the asymptotic behavior of ...

Solving the three-dimensional (3D) Bratu equation is highly challenging due to the presence of multiple and sharp solutions. Research on this equation began in the late 1990s, but there are no satisfactory results to date. To address this issue, ...

• We develop a symmetric finite difference method (SFDM) to address the challenges of solving the 3D Bratu equation.
• We modify the Bratu equation by incorporating a new constraint that facilitates the construction of bifurcation ...

Conventionally, piecewise polynomials have been used in the boundary element method (BEM) to approximate unknown boundary values. However, since infinitely smooth radial basis functions (RBFs) are more stable and accurate than the polynomials for ...

• This paper introduces the radial BEM, using radial basis functions for improved stability and accuracy.
• A new distribution of boundary source points eliminates singularities in boundary integrals.
• The radial BEM allows precise ...

Utilizing physics-informed neural networks (PINN) to solve partial differential equations (PDEs) has become a hot issue and also shown its great powers, but still suffers from the dilemmas of limited predicted accuracy in the sampling domain and ...

• Designing a symmetry-enhanced deep neural network (sDNN) which makes the architecture invariant under the finite group.
• Proving rigorously the sDNN has the universal approximation ability.
• Improving the accuracies in and beyond the ...

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for solving partial differential equation (PDE) systems in computer mechanics. However, PINNs still struggle in simulating systems whose solution functions ...

This paper addresses the input-to-state stability (ISS) of infinite-dimensional systems by introducing a novel notion named generalized ISS-Lyapunov functional (GISS-LF) and the corresponding ISS Lyapunov theorem. Unlike the classical ISS-...

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving partial differential equations (PDEs) in various scientific and engineering applications. PINNs integrate the PDEs into the loss functions of neural networks ...$$

As a grid-independent approach for solving partial differential equations (PDEs), Physics-Informed Neural Networks (PINNs) have garnered significant attention due to their unique capability to simultaneously learn from both data and the governing ...

• FPINNs is proposed to incorporate fuzzy learning for reducing uncertainty in PINNs.
• Fuzzy rules and Gaussian membership functions is used to improve solution accuracy.
• FPINNs outperform existing methods in solving both forward and ...

The aim of this note is to construct a neural network for which the linear finite element approximation of a simple one dimensional boundary value problem is a minimum of the cost function to find out if the neural network is able to reproduce ...

We develop an optimal control framework that enables to determine the most-beneficial ways of investing in technology and directing capital within an economy. Our developed framework features three main novelties: the optimization of a cross–...${}_{}\mathrm{}$${}^{}\mathrm{}$${}_{}{}^{}$${}_{}{}^{}$

Physics-informed neural networks (PINNs), rooted in deep learning, have emerged as a promising approach for solving partial differential equations (PDEs). By embedding the physical information described by PDEs into feedforward neural networks, ...

• We propose BsPINNs, a binary structured neural network framework addressing PINNs' limits with rapidly changing solutions.
• BsPINNs partition the network into channels with partially independent parameters, enabling decomposition into ...

The application of Artificial Neural Networks (ANNs) has been extensively investigated for fluid dynamic problems. A specific form of ANNs are Physics-Informed Neural Networks (PINNs). They incorporate physical laws in the training and have ...

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• PINNs improved the ANN prediction accuracy for the potential flow cases in this work.
• Random distribution of training data lead to a higher prediction accuracy of ANNs.
• A Sequence-to-sequence method enabled temporal interpolation ...

Control of mixed-autonomy traffic where Human-driven Vehicles (HVs) and Autonomous Vehicles (AVs) coexist on the road has gained increasing attention over the recent decades. This paper addresses the boundary stabilization problem for mixed ...

A novel mesh-free kernel method is presented for solving function interpolation problems and partial differential equations (PDEs). Despite the simplicity and accuracy of the radial basis function (RBF) method, two main difficulties have been ...

In this study, we investigate how the updating of weights during forward operation and the computation of gradients during backpropagation impact the optimization process, training procedure, and overall performance of the neural network, ...

• SQR-SkipResNet, inspired by highway networks, ensures stability and convergence.
• Plain NN's instability due to weight norm fluctuations hampers convergence.
• Deeper networks' impact on accuracy varies with problem complexity.
• ...

The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics-Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their lack ...

• Limitations of auto-differentiation hinder its accuracy to compute physical gradients.
• Foreign numerical operators can be used to compute the physical gradients.
• A model enriched with physical invariances knowledge can manage ...

We present an efficient dimension-by-dimension finite-volume method which solves the adiabatic magnetohydrodynamics equations at high discretization order, using the constrained-transport approach on Cartesian grids. Results are presented up to ...

• A novel physics-informed convolutional neural network f-PICNN for PDEs without any labelled data.
• Nonlinear convolutional units (NCUs).
• Memory mechanism (numerical results show it can considerably speed up the convergence).
• ...

The physics and interdisciplinary problems in science and engineering are mainly described as partial differential equations (PDEs). Recently, a novel method using physics-informed neural networks (PINNs) to solve PDEs by employing deep neural ...

A novel framework is proposed that utilizes symbolic regression via genetic programming to identify free-form partial differential equations from scarce and noisy data. The framework successfully identified ground truth models for four synthetic ...

• Symbolic regression identifies partial differential equations from scarce and noisy data.
• Framework successfully identified ground truth models of four synthetic systems.
• Framework identifies partial differential equation models ...

We consider the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic problems can be decoupled via the introduction of a new unknown, corresponding to the ...