Novel Adaptive Bayesian Regularization Networks for Peristaltic Motion of a Third-Grade Fluid in a Planar Channel
Abstract
:1. Introduction
2. Problem Formulation
3. Methodology with Performance Metrics
3.1. Adams Numerical Scheme
3.2. Bayesian Regularization Neural Networks
- The execution of maximum epochs.
- The maximum set time is exceeded.
- Performance goal is attained.
- The performance gradient is less than the minimum gradient value, i.e., min_grad.
- Adaptive parameter mu is greater than its maximum value, i.e., mu_max.
3.3. Performance Measuring Metrics
4. Results and Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
References
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Scenario | Case | β | θ | ϕ | F |
---|---|---|---|---|---|
1 | 1 | 0.08 | π/2 | 0.4 | −0.1 |
2 | 0.07 | π/2 | 0.4 | −0.1 | |
3 | 0.04 | π/2 | 0.4 | −0.1 | |
4 | 0.02 | π/2 | 0.4 | −0.1 | |
2 | 2 | 0.08 | π/5 | 0.4 | −0.1 |
3 | 0.08 | π/4 | 0.4 | −0.1 | |
4 | 0.08 | π | 0.4 | −0.1 | |
3 | 2 | 0.08 | π/2 | 0.1 | −0.1 |
3 | 0.08 | π/2 | 0.6 | −0.1 | |
4 | 0.08 | π/2 | 0.7 | −0.1 | |
4 | 2 | 0.08 | π/2 | 0.4 | 1.2 |
3 | 0.08 | π/2 | 0.4 | 1.4 | |
4 | 0.08 | π/2 | 0.4 | 2.0 |
Index | Settings |
---|---|
Maximum epochs for training | 1000 |
Performance/fitness goal | 0 |
Marquardt adjustment parameter, i.e., mu | 0.005 |
Decreeing factor of mu, i.e., mu_dec | 0.1 |
Increasing factor of mu i.e., mu_inc | 10 |
Maximum of mu | 1010 |
Maximum number of validation failures | Inf |
Minimum value of performance gradient | 10−7 |
Number of hidden neurons | 35 |
Training and testing samples | 80 and 20 percentage |
Sample selection | Randomize |
Input, output, and hidden layers | Single for all three |
Reference data set generation | Adams method |
Scenario | Case | C | Performance Attained | Gradient Value | Mu Step Size | Epoch Executed | Consumed Time | |
---|---|---|---|---|---|---|---|---|
Training | Testing | |||||||
1 | 1 | 2.00 × 10−13 | 1.03 × 10−9 | 2.20 × 10−13 | 9.28 × 10−8 | 5.00 | 148 | 0.00.06 |
2 | 2.55 × 10−13 | 2.07 × 10−10 | 2.55 × 10−13 | 9.77 × 10−8 | 5.00 | 66 | 0.00.02 | |
3 | 4.34 × 10−13 | 5.37 × 10−12 | 4.34 × 10−13 | 1.40 × 10−8 | 5.00 | 13 | 0.00.01 | |
4 | 2.10 × 10−12 | 2.77 × 10−9 | 2.10 × 10−12 | 1.89 × 10−8 | 500 | 12 | 0.00.01 | |
2 | 2 | 1.33 × 10−12 | 3.98 × 10−12 | 1.33 × 10−12 | 1.40 × 10−8 | 500 | 11 | 0.00.01 |
3 | 1.59 × 10−12 | 2.50 × 10−10 | 1.59 × 10−12 | 3.82 × 10−8 | 500 | 12 | 0.00.01 | |
4 | 1.91 × 10−13 | 2.19 × 10−9 | 1.91 × 10−13 | 9.95 × 10−8 | 500 | 48 | 0.00.02 | |
3 | 2 | 7.14 × 10−11 | 3.32 × 10−6 | 7.14 × 10−11 | 5.23 × 10−8 | 500 | 64 | 0.00.02 |
3 | 9.90 × 10−13 | 1.65 × 10−11 | 9.89 × 10−13 | 9.80 × 10−8 | 500 | 22 | 0.00.01 | |
4 | 1.32 × 10−13 | 8.14 × 10−13 | 1.32 × 10−13 | 8.19 × 10−8 | 500 | 14 | 0.00.01 | |
4 | 2 | 3.26 × 10−12 | 3.82 × 10−11 | 3.26 × 10−12 | 9.94 × 10−8 | 50 | 30 | 0.00.01 |
3 | 1.06 × 10−11 | 1.05 × 10−10 | 1.06 × 10−11 | 9.84 × 10−8 | 50 | 32 | 0.00.01 | |
4 | 7.33 × 10−12 | 2.18 × 10−11 | 7.33 × 10−12 | 9.81 × 10−8 | 50 | 172 | 0.00.04 |
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Mahmood, T.; Ali, N.; Chaudhary, N.I.; Cheema, K.M.; Milyani, A.H.; Raja, M.A.Z. Novel Adaptive Bayesian Regularization Networks for Peristaltic Motion of a Third-Grade Fluid in a Planar Channel. Mathematics 2022, 10, 358. https://doi.org/10.3390/math10030358
Mahmood T, Ali N, Chaudhary NI, Cheema KM, Milyani AH, Raja MAZ. Novel Adaptive Bayesian Regularization Networks for Peristaltic Motion of a Third-Grade Fluid in a Planar Channel. Mathematics. 2022; 10(3):358. https://doi.org/10.3390/math10030358
Chicago/Turabian StyleMahmood, Tariq, Nasir Ali, Naveed Ishtiaq Chaudhary, Khalid Mehmood Cheema, Ahmad H. Milyani, and Muhammad Asif Zahoor Raja. 2022. "Novel Adaptive Bayesian Regularization Networks for Peristaltic Motion of a Third-Grade Fluid in a Planar Channel" Mathematics 10, no. 3: 358. https://doi.org/10.3390/math10030358
APA StyleMahmood, T., Ali, N., Chaudhary, N. I., Cheema, K. M., Milyani, A. H., & Raja, M. A. Z. (2022). Novel Adaptive Bayesian Regularization Networks for Peristaltic Motion of a Third-Grade Fluid in a Planar Channel. Mathematics, 10(3), 358. https://doi.org/10.3390/math10030358