The neural network collocation method for solving partial differential equations

Brink, Adam Ray; Najera-Flores, David; Martinez, C.

doi:10.1007/s00521-020-05340-5

Cited by 25 publications

(9 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…a range [0.5, 3], [0.5, 4.5], [1, 2.5], [1,4], [2.5, 6.5] for first, second, third, fourth and fifth layer respectively. The minimum RMSE for S6 and S7 tuples (S6 * and S7 * ) are 0.0017 and 0.0009 respectively.…”

Section: Declarationsmentioning

confidence: 99%

“…Recent development in the field is done by Jafarian and Baleanu [17], Weinan et al [11], Raissi et. al [21], Brink et al [4], Moseley and Markham [1], Dwivedi and Srinivasan [10], Samaniego et al [31]. Moseley and Markham [1] defined the extended version of NN called physics informed neural networks (PINNs) and Dwivedi and Srinivasan [10] discussed Physics informed extreme learning machine (PIELM).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Time and Error Analysis of Neural Network Model of Rectangular Vibrating Membrane

Pandey

Agarwal

Aggarwal

2022

Preprint

View full text Add to dashboard Cite

Series solution of rectangular vibrating membrane (SSRVM) of the motion has been represented by Neural Network (NN) model. Experiments have been performed to find a combination of NN structure and size of sample data such that error and time cost of SSRVM are the minimum. Error and time cost of NN models have been studied w.r.t structure, size of sample, random choice of initial weights and selection of data. Peak signal to noise ratio (PSNR) between series solution and solution by NN model is approximately 50 dB. This ensures that solution obtained by NN model is acceptable at human visual system. The average training time of constructed NN models is better than state-of-the-art related to solution of differential equation by NN. NN models are 26 times faster than SSRVM for solution prediction.

show abstract

Section: Declarationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Time and Error Analysis of Neural Network Model of Rectangular Vibrating Membrane

Pandey

Agarwal

Aggarwal

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In general, the activation function selects nonlinear functions, such as sigmoid and tanh. Here, we choose tanh as the activation function 22,66 , i.e.,…”

Section: Neural Network and Data Sampling Pinn With Residual Unitsmentioning

confidence: 99%

A hybrid neural network and data sampling solver for forward and backward modified diffusion equations

Mou

Fang

2022

Preprint

View full text Add to dashboard Cite

We developed a hybrid solver based on physics-informed neural networks and a mixture of grid and Latin hypercube sampling to solve forward and backward modified diffusion equations. We optimized the parameters in the neural networks by considering the squeeze boundary condition and the parameter in the mixed data sampling by adjusting the mixture coefficient. Then,we used a given modified diffusion equation as an example to demonstrate the efficiency of the hybrid solver for forward and backward problems. The neural network results were compared with the numerical solutions, and good agreement with high precision was observed. This hybrid neural network solver can be generalized to other partial differential equations.

show abstract

“…These neurons, which are simple processing units, each have weights that return weighted signals and an output signal, which is achieved using an activation function. The MLP reduces error by optimisation algorithms or functions, such as backpropagation [10,25,47].…”

Section: Multi-layer Perceptron (Mlp)mentioning

confidence: 99%

The stochastic aeroelastic response analysis of helicopter rotors using deep and shallow machine learning

Chatterjee

Essien

Ganguli

et al. 2021

Neural Comput & Applic

View full text Add to dashboard Cite

This paper addresses the influence of manufacturing variability of a helicopter rotor blade on its aeroelastic responses. An aeroelastic analysis using finite elements in spatial and temporal domains is used to compute the helicopter rotor frequencies, vibratory hub loads, power required and stability in forward flight. The novelty of the work lies in the application of advanced data-driven machine learning (ML) techniques, such as convolution neural networks (CNN), multi-layer perceptron (MLP), random forests, support vector machines and adaptive Gaussian process (GP) for capturing the nonlinear responses of these complex spatio-temporal models to develop an efficient physics-informed ML framework for stochastic rotor analysis. Thus, the work is of practical significance as (i) it accounts for manufacturing uncertainties, (ii) accurately quantifies their effects on nonlinear response of rotor blade and (iii) makes the computationally expensive simulations viable by the use of ML. A rigorous performance assessment of the aforementioned approaches is presented by demonstrating validation on the training dataset and prediction on the test dataset. The contribution of the study lies in the following findings: (i) The uncertainty in composite material and geometric properties can lead to significant variations in the rotor aeroelastic responses and thereby highlighting that the consideration of manufacturing variability in analyzing helicopter rotors is crucial for assessing their behaviour in real-life scenarios. (ii) Precisely, the substantial effect of uncertainty has been observed on the six vibratory hub loads and the damping with the highest impact on the yawing hub moment. Therefore, sufficient factor of safety should be considered in the design to alleviate the effects of perturbation in the simulation results. (iii) Although advanced ML techniques are harder to train, the optimal model configuration is capable of approximating the nonlinear response trends accurately. GP and CNN followed by MLP achieved satisfactory performance. Excellent accuracy achieved by the above ML techniques demonstrates their potential for application in the optimization of rotors under uncertainty.

show abstract

The neural network collocation method for solving partial differential equations

Cited by 25 publications

References 24 publications

Time and Error Analysis of Neural Network Model of Rectangular Vibrating Membrane

Time and Error Analysis of Neural Network Model of Rectangular Vibrating Membrane

A hybrid neural network and data sampling solver for forward and backward modified diffusion equations

The stochastic aeroelastic response analysis of helicopter rotors using deep and shallow machine learning

Contact Info

Product

Resources

About