scholarly journals Comparison and Analysis of Neural Solver Methods for Differential Equations in Physical Systems

ELKHA ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 134
Author(s):  
Fabio M Sim ◽  
Eka Budiarto ◽  
Rusman Rusyadi
Keyword(s):  
Neural Networks ◽  
Numerical Methods ◽  
Differential Equations ◽  
Electric Fields ◽  
Test Cases ◽  
Mass System ◽  
Time Stepping ◽  
Computational Speed ◽  
Nonlinear Solutions ◽  
Near Future

Differential equations are ubiquitous in many fields of study, yet not all equations, whether ordinary or partial, can be solved analytically. Traditional numerical methods such as time-stepping schemes have been devised to approximate these solutions. With the advent of modern deep learning, neural networks have become a viable alternative to traditional numerical methods. By reformulating the problem as an optimisation task, neural networks can be trained in a semi-supervised learning fashion to approximate nonlinear solutions. In this paper, neural solvers are implemented in TensorFlow for a variety of differential equations, namely: linear and nonlinear ordinary differential equations of the first and second order; Poisson’s equation, the heat equation, and the inviscid Burgers’ equation. Different methods, such as the naive and ansatz formulations, are contrasted, and their overall performance is analysed. Experimental data is also used to validate the neural solutions on test cases, specifically: the spring-mass system and Gauss’s law for electric fields. The errors of the neural solvers against exact solutions are investigated and found to surpass traditional schemes in certain cases. Although neural solvers will not replace the computational speed offered by traditional schemes in the near future, they remain a feasible, easy-to-implement substitute when all else fails.

Rattlebacks for Undergraduate Engineers: Modelling Complex Behavior Using Introductory Dynamics and Numerical Methods

2018 ◽  
Author(s):  
Simon Jones ◽  
Kirby Kern
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Reference Frame ◽  
Engineering Students ◽  
Rotary Motion ◽  
Complex Dynamic ◽  
Time Stepping ◽  
Undergraduate Engineering ◽  
Clockwise Direction

Rattlebacks are semi-ellipsoidal tops that have a preferred direction of spin. If spun in, say, the clockwise direction, the rattleback will exhibit stable rotary motion. If spun in the counter-clockwise direction, the rattleback’s rotary motion will transition to a rattling motion, and then reverse its spin resulting in clockwise rotation. This counter-intuitive dynamic behavior has long been a favored subject of study in graduate-level dynamics classes. Previous literature on rattleback dynamics offer insight into a myriad of advanced topics, including three-dimensional motion, sliding and rolling friction models, stability regions, nondimensionalization, etc. However, it is the current authors’ view that focusing on these advanced topics clouds the students’ understanding of the fundamental kinetics of the body. The goal of this paper is to demonstrate that accurately simulating rattleback behavior need not be complicated; undergraduate engineering students can accurately model the behavior using concepts from introductory dynamics and numerical methods. The current paper develops an accurate dynamic model of a rattleback from first principles. All necessary steps are discussed in detail, including computing the mass moment of inertia, choice of reference frame, conservation of momenta equations, and application of kinematic constraints. Basic numerical techniques like Gaussian quadrature, Newton-Raphson root-finding, and Runge-Kutta time-stepping are employed to solve the necessary integrals, nonlinear algebraic equations, and ordinary differential equations. Since not all undergraduate engineering students are familiar with 3D dynamics, a simpler 2D rocking semi-ellipse example is first introduced to develop the transformation matrix between an inertial reference frame and a body-fixed reference frame. This provides the framework to transition seamlessly into 3D dynamics using roll, pitch, and yaw angles, concepts that are widely understood by engineering students. In fact, when written in vector notation, the governing equations for the rocking ellipse and the spin-biased rattleback are shown to be the same, enforcing the concept that 3D dynamics need not be intimidating. The purpose of this paper is to guide a typical undergraduate engineering student through a complex dynamic simulation, and to demonstrate that he or she already has the tools necessary to simulate complex dynamic behavior. Conservation of momenta will account for the dynamics, intimidating integrals and differentials can be tackled numerically, and classic time-stepping approaches make light work of nonlinear differential equations.

scholarly journals Solving Partial Differential Equations Using Deep Learning and Physical Constraints

2020 ◽  
Vol 10 (17) ◽  
pp. 5917
Author(s):  
Yanan Guo ◽  
Xiaoqun Cao ◽  
Bainian Liu ◽  
Mei Gao
Keyword(s):  
Neural Networks ◽  
Deep Learning ◽  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Language Processing ◽  
Kdv Equation ◽  
Great Success ◽  
Partial Differential ◽  
Physical Constraints

The various studies of partial differential equations (PDEs) are hot topics of mathematical research. Among them, solving PDEs is a very important and difficult task. Since many partial differential equations do not have analytical solutions, numerical methods are widely used to solve PDEs. Although numerical methods have been widely used with good performance, researchers are still searching for new methods for solving partial differential equations. In recent years, deep learning has achieved great success in many fields, such as image classification and natural language processing. Studies have shown that deep neural networks have powerful function-fitting capabilities and have great potential in the study of partial differential equations. In this paper, we introduce an improved Physics Informed Neural Network (PINN) for solving partial differential equations. PINN takes the physical information that is contained in partial differential equations as a regularization term, which improves the performance of neural networks. In this study, we use the method to study the wave equation, the KdV–Burgers equation, and the KdV equation. The experimental results show that PINN is effective in solving partial differential equations and deserves further research.

Glycemia monitoring

1998 ◽  
Vol 2 ◽  
pp. 23-30
Author(s):  
Igor Basov ◽  
Donatas Švitra
Keyword(s):  
Diabetes Mellitus ◽  
Mathematical Model ◽  
Numerical Methods ◽  
Differential Equations ◽  
Blood Sugar ◽  
Self Regulation ◽  
Physiological System ◽  
Non Linear ◽  
The Mathematical Model

Here a system of two non-linear difference-differential equations, which is mathematical model of self-regulation of the sugar level in blood, is investigated. The analysis carried out by qualitative and numerical methods allows us to conclude that the mathematical model explains the functioning of the physiological system "insulin-blood sugar" in both normal and pathological cases, i.e. diabetes mellitus and hyperinsulinism.

scholarly journals Discretization and machine learning approximation of BSDEs with a constraint on the Gains-process

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Idris Kharroubi ◽  
Thomas Lim ◽  
Xavier Warin
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Neural Networks ◽  
Differential Equations ◽  
Numerical Experiments ◽  
Optimization Problem ◽  
Learning Approach ◽  
The Neural Network ◽  
Machine Learning Approach ◽  
Mesh Grid

AbstractWe study the approximation of backward stochastic differential equations (BSDEs for short) with a constraint on the gains process. We first discretize the constraint by applying a so-called facelift operator at times of a grid. We show that this discretely constrained BSDE converges to the continuously constrained one as the mesh grid converges to zero. We then focus on the approximation of the discretely constrained BSDE. For that we adopt a machine learning approach. We show that the facelift can be approximated by an optimization problem over a class of neural networks under constraints on the neural network and its derivative. We then derive an algorithm converging to the discretely constrained BSDE as the number of neurons goes to infinity. We end by numerical experiments.

Numerical solution of fractal-fractional Mittag–Leffler differential equations with variable-order using artificial neural networks

2021 ◽  
Author(s):  
C. J. Zúñiga-Aguilar ◽  
J. F. Gómez-Aguilar ◽  
H. M. Romero-Ugalde ◽  
R. F. Escobar-Jiménez ◽  
G. Fernández-Anaya ◽  
...  
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Differential Equations ◽  
Numerical Solution ◽  
Variable Order ◽  
Artificial Neural

scholarly journals A New Model for the Stochastic Point Reactor: Development and Comparison with Available Models

Energies ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 955
Author(s):  
Alamir Elsayed ◽  
Mohamed El-Beltagy ◽  
Amnah Al-Juhani ◽  
Shorooq Al-Qahtani
Keyword(s):  
Kinetic Model ◽  
Differential Equations ◽  
Test Cases ◽  
System Of Differential Equations ◽  
Reactor Model ◽  
Time System ◽  
New Model ◽  
Reactor Dynamics ◽  
Mean And Variance ◽  
The Mean

The point kinetic model is a system of differential equations that enables analysis of reactor dynamics without the need to solve coupled space-time system of partial differential equations (PDEs). The random variations, especially during the startup and shutdown, may become severe and hence should be accounted for in the reactor model. There are two well-known stochastic models for the point reactor that can be used to estimate the mean and variance of the neutron and precursor populations. In this paper, we reintroduce a new stochastic model for the point reactor, which we named the Langevin point kinetic model (LPK). The new LPK model combines the advantages, accuracy, and efficiency of the available models. The derivation of the LPK model is outlined in detail, and many test cases are analyzed to investigate the new model compared with the results in the literature.

scholarly journals New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1159
Author(s):  
Shyam Sundar Santra ◽  
Omar Bazighifan ◽  
Mihai Postolache
Keyword(s):  
Fluid Dynamics ◽  
Neural Networks ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Delay Differential

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

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