4.7 Article

Application of a physics-informed neural network to solve the steady-state Bratu equation arising from solid biofuel combustion theory

Related references

Note: Only part of the references are listed.
Article Computer Science, Interdisciplinary Applications

When and why PINNs fail to train: A neural tangent kernel perspective

Sifan Wang et al.

Summary: This work investigates the Neural Tangent Kernel (NTK) of Physics-informed neural networks (PINNs) and demonstrates that it can converge to a deterministic kernel that remains constant during training under appropriate conditions. A novel gradient descent algorithm is proposed to adaptively calibrate the convergence rate of total training error using the eigenvalues of NTK. A series of numerical experiments are conducted to validate the theory and practical effectiveness of the proposed algorithms.

JOURNAL OF COMPUTATIONAL PHYSICS (2022)

Article Multidisciplinary Sciences

Analyses of internal structures and defects in materials using physics-informed neural networks

Enrui Zhang et al.

Summary: In this study, a general framework based on physics-informed neural networks is proposed for identifying unknown geometric and material parameters in materials' internal structures and defects. By using a mesh-free method, the geometry of the material is parameterized, and the effectiveness of the method is validated using constitutive models. The framework can be applied to other inverse problems involving unknown material properties and deformable geometries.

SCIENCE ADVANCES (2022)

Article Engineering, Multidisciplinary

Simulation of electromagnetic wave propagations in negative index materials by the localized RBF-collocation method

Hui Zheng et al.

Summary: In this paper, a fast simulation method for electromagnetic wave propagation in negative index materials is proposed using the localized meshfree radial basis function (RBF) collocation method. The efficiency of the method is demonstrated through numerical examples.

ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS (2022)

Article Engineering, Multidisciplinary

3D elastic dental analysis by a local RBF collocation method

Hui Zheng et al.

Summary: This paper presents a local radial basis function collocation method (LRBFCM) for analyzing 3D elastic problems of complex geometries, with a focus on a real tooth domain reconstructed from cone beam computer tomography (CBCT) data. The effectiveness of the approach is validated through linear elastic analyses of five different examples, investigating the stress difference between teeth with dental caries and healthy teeth through 3D elastic deformation and stress analysis.

APPLIED MATHEMATICAL MODELLING (2021)

Article Mathematics, Applied

An advancement approach of Haar wavelet method and Bratu-type equations

Swati et al.

Summary: This study adopts an efficient technique to solve a classical one-dimensional nonlinear eigenvalue problem of the famous Gelfand elliptic BVP, known as the Bratu problem. The advancement of the Haar wavelet method is proposed to enhance the precision and convergence rate of the solution, with numerical illustrations demonstrating the performance and efficiency of the method. The numerical findings indicate the improvement of the proposed method over various existing techniques such as splines, wavelets, and decomposition methods.

APPLIED NUMERICAL MATHEMATICS (2021)

Article Engineering, Multidisciplinary

Variational phase-field fracture modeling with interfaces

Keita Yoshioka et al.

Summary: This paper proposes a diffused approach to approximate failure at interfaces with negligible space, deriving an effective interface fracture toughness and verifying its effectiveness in various scenarios.

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING (2021)

Article Mathematics, Applied

NUMERICAL INVESTIGATION OF SPACE FRACTIONAL ORDER DIFFUSION EQUATION BY THE CHEBYSHEV COLLOCATION METHOD OF THE FOURTH KIND AND COMPACT FINITE DIFFERENCE SCHEME

Yones Esmaeelzade Aghdam et al.

Summary: This paper introduces a numerical scheme for finding approximate solutions of space fractional order of the diffusion equation by discretizing time derivative with compact finite difference and approximating spatial fractional derivative by Chebyshev collocation method. The time-discrete stability and convergence are analyzed, and the proposed method is validated with numerical examples demonstrating its performance and accuracy compared to existing methods.

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S (2021)

Review Physics, Applied

Physics-informed machine learning

George Em Karniadakis et al.

Summary: Physics-informed learning seamlessly integrates data and mathematical models through neural networks or kernel-based regression networks for accurate inference of realistic and high-dimensional multiphysics problems. Challenges remain in incorporating noisy data seamlessly, complex mesh generation, and addressing high-dimensional problems.

NATURE REVIEWS PHYSICS (2021)

Article Multidisciplinary Sciences

Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations

Maziar Raissi et al.

SCIENCE (2020)

Article Computer Science, Interdisciplinary Applications

A meshless collocation method for band structure simulation of nanoscale phononic crystals based on nonlocal elasticity theory

Hui Zheng et al.

JOURNAL OF COMPUTATIONAL PHYSICS (2020)

Article Engineering, Multidisciplinary

Probabilistic approach for optimal portfolio selection using a hybrid Monte Carlo simulation and Markowitz model

Mahboubeh Shadabfar et al.

ALEXANDRIA ENGINEERING JOURNAL (2020)

Article Computer Science, Interdisciplinary Applications

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

M. Raissi et al.

JOURNAL OF COMPUTATIONAL PHYSICS (2019)

Article Mathematics, Applied

An iterative finite difference method for approximating the two-branched solution of Bratu's problem

Mohamed Ben-Romdhane et al.

APPLIED NUMERICAL MATHEMATICS (2019)

Review Radiology, Nuclear Medicine & Medical Imaging

An overview of deep learning in medical imaging focusing on MRI

Alexander Selvikvag Lundervold et al.

ZEITSCHRIFT FUR MEDIZINISCHE PHYSIK (2019)

Article Mathematics, Applied

The Taylor wavelets method for solving the initial and boundary value problems of Bratu-type equations

E. Keshavarz et al.

APPLIED NUMERICAL MATHEMATICS (2018)

Article Computer Science, Interdisciplinary Applications

Hidden physics models: Machine learning of nonlinear partial differential equations

Maziar Raissi et al.

JOURNAL OF COMPUTATIONAL PHYSICS (2018)

Article Mathematics, Applied

Stability of a finite volume element method for the time-fractional advection-diffusion equation

M. Badr et al.

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS (2018)

Article Computer Science, Artificial Intelligence

Design of Mexican Hat Wavelet neural networks for solving Bratu type nonlinear systems

Zaheer Masood et al.

NEUROCOMPUTING (2017)

Article Engineering, Multidisciplinary

Bio-inspired computing platform for reliable solution of Bratu-type equations arising in the modeling of electrically conducting solids

Muhammad Asif Zahoor Raja et al.

APPLIED MATHEMATICAL MODELLING (2016)

Article Computer Science, Interdisciplinary Applications

Bratu's problem: A novel approach using fixed-point iterations and Green's functions

H. Q. Kafri et al.

COMPUTER PHYSICS COMMUNICATIONS (2016)

Article Mathematics, Applied

The Legendre wavelet method for solving initial value problems of Bratu-type

S. G. Venkatesh et al.

COMPUTERS & MATHEMATICS WITH APPLICATIONS (2012)

Article Mathematics, Applied

The Lie-group shooting method for solving the Bratu equation

S. Abbasbandy et al.

COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION (2011)

Article Computer Science, Interdisciplinary Applications

Non-polynomial spline method for solving Bratu's problem

R. Jalilian

COMPUTER PHYSICS COMMUNICATIONS (2010)

Article Mathematics, Applied

A general approach to obtain series solutions of nonlinear differential equations

S. Liao et al.

STUDIES IN APPLIED MATHEMATICS (2007)

Article Mathematics, Applied

From the fitting techniques to accurate schemes for the Liouville-Bratu-Gelfand problem

A. Serghini Mounim et al.

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS (2006)

Article Mathematics

The Liouville-Bratu-Gelfand problem for radial operators

J Jacobsen et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2002)