A comparative investigation of a time‐dependent mesh method and physics‐informed neural networks to analyze the generalized Kolmogorov–Petrovsky–Piskunov equation

Sultan, Saad; Zhang, Zhengce

doi:10.1002/fld.5259

Cited by 6 publications

(1 citation statement)

References 26 publications

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“…At the very outset, we note that variants of Burgers equation continue to be explored via ever more sophisticated methods not using deep-learning [51,52]. But in such studies, neither are the setups chosen corresponding to the challenging edge-case of finite-time blow-up as we consider here nor is there a study of the accuracy of any theoretical error bound in implementations.…”

Section: Methodsmentioning

confidence: 99%

Investigating the ability of PINNs to solve Burgers’ PDE near finite-time blowup

Kumar,

Mukherjee

2024

Mach. Learn.: Sci. Technol.

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Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated PDEs numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive generalization bounds for PINNs for Burgers' PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the l2-distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.

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Section: Methodsmentioning

confidence: 99%

Investigating the ability of PINNs to solve Burgers’ PDE near finite-time blowup

Kumar,

Mukherjee

2024

Mach. Learn.: Sci. Technol.

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Adaptive temporal mesh equidistribution for studying the singularly perturbed delay-in-time parabolic equations

Sultan,

Zhang

2024

Z. Angew. Math. Phys.

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Various optimized artificial neural network simulations of advection-diffusion processes

Sari,

Gulen,

Celenk

2024

Phys. Scr.

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The aim of this research is to describe an artificial neural network (ANN) based method to approximate the solutions of the natural advection-diffusion equations. Although the solutions of these equations can be obtained by various effective numerical methods, feed forward neural network (FFNN) techniques combined with different optimization techniques offer a more practicable and flexible alternative than the traditional approaches to solve those equations. However, the ability of FFNN techniques to solve partial differential equations is a questionable issue and has not yet been fully concluded in the existing literature. The reliability and accuracy of computational results can be advanced by the choice of optimization techniques. Therefore, this study aims to take an effective step towards presenting the ability to solve the advection-diffusion equations by leveraging the inherent benefits of ANN methods while avoiding some of the limitations of traditional approaches. In this technique, the solution process requires minimizing the error generated by using a differential equation whose solution is considered as a trial solution. More specifically, this study uses a FFNN and backpropagation technique, one of the variants of the ANN method, to minimize the error and the adjustment of parameters. In the solution process, the loss function (error) needs to be minimized; this is accomplished by fitting the trial function into the differential equation using appropriate optimization techniques and obtaining the network output. Therefore, in this study, the commonly used techniques in the literature, namely gradient descent (GD), particle swarm optimization (PSO) and artificial bee colony (ABC), are selected to compare the effectiveness of gradient and gradient-free optimization techniques in solving the advection-diffusion equation. The calculations with all three optimization techniques for linear and nonlinear advection-diffusion equations have been run several times to obtain the optimum accuracy of the results. The computed results are seen to be very promising and in good agreement with the effective numerical methods and the physics-informed neural network (PINN) method in the literature. It is also concluded that the PSO-based algorithm outperforms other methods in terms of accuracy.

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A comparative investigation of a time‐dependent mesh method and physics‐informed neural networks to analyze the generalized Kolmogorov–Petrovsky–Piskunov equation

Cited by 6 publications

References 26 publications

Investigating the ability of PINNs to solve Burgers’ PDE near finite-time blowup

Investigating the ability of PINNs to solve Burgers’ PDE near finite-time blowup

Adaptive temporal mesh equidistribution for studying the singularly perturbed delay-in-time parabolic equations

Various optimized artificial neural network simulations of advection-diffusion processes

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