Physics-Informed Neural Network for Nonlinear Bending Analysis of Nano-Beams: A Systematic Hyperparameter Optimization
Abstract
1. Introduction
2. Materials and Methods
2.1. Mathematical Formulation
2.1.1. Nonlocal Strain Gradient Theory
2.1.2. Governing Equations
2.2. Solution Methodology
2.2.1. Physics-Informed Neural Network
2.2.2. Training Procedure
2.2.3. HPO via Gaussian Process-Based Bayesian Optimization
3. Results and Discussion
3.1. HPO via GP for Linear Bending
3.2. HPO via GP for S-S for Nonlinear Bending
3.3. HPO via GP for C-C for Nonlinear Bending
3.4. PINN Results for HPO Hyperparameters
4. Conclusions
- The results indicate that when the dimensionless strain gradient parameter is elevated, the beam displays decreased deflection. Physically, a higher nonlocal parameter implies that each material point is influenced by a larger neighborhood, effectively increasing the material’s resistance to deformation. As a result, the beam exhibits higher stiffness and consequently lower deflection under the same loading conditions.
- Due to the geometric nonlinearity which causes a stiffening effect, the nonlinear dimensionless bending deflection is smaller than the linear deflection.
- HPO via GP can decrease the loss function in order to tune the hyperparameters, and the results are more reliable than manually tuning mostly for complicated problems.
- Although the HPO–GP method demonstrates great performance in finding the optimal HPs, decreasing the loss function, and improving accuracy, it comes with a high computational cost limitation, especially when applied to a large number of HPs or complex models.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
PINN | Physics-Informed Neural Network |
NN | Neural Network |
HP | Hyperparameter |
HPO | Hyperparameter Optimization |
GP | Gaussian processes |
BO | Bayesian Optimization |
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Type of Boundary Condition | Constrained Items |
---|---|
Free edge | |
Clamped edge | |
Simply supported edge |
HP | LR | NH | σ | Epochs | ND | ||
---|---|---|---|---|---|---|---|
Case of Study | |||||||
C-C | [0.0001, 0.05] | [1, 5] | 5 | ] | [20,000, 300,000] | 50 | |
S-S |
Case | Optimum Loss Value | ) |
---|---|---|
C-C | 10−7 | |
S-S | 10−9 |
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Mirsadeghi Esfahani, S.S.; Fallah, A.; Mohammadi Aghdam, M. Physics-Informed Neural Network for Nonlinear Bending Analysis of Nano-Beams: A Systematic Hyperparameter Optimization. Math. Comput. Appl. 2025, 30, 72. https://doi.org/10.3390/mca30040072
Mirsadeghi Esfahani SS, Fallah A, Mohammadi Aghdam M. Physics-Informed Neural Network for Nonlinear Bending Analysis of Nano-Beams: A Systematic Hyperparameter Optimization. Mathematical and Computational Applications. 2025; 30(4):72. https://doi.org/10.3390/mca30040072
Chicago/Turabian StyleMirsadeghi Esfahani, Saba Sadat, Ali Fallah, and Mohammad Mohammadi Aghdam. 2025. "Physics-Informed Neural Network for Nonlinear Bending Analysis of Nano-Beams: A Systematic Hyperparameter Optimization" Mathematical and Computational Applications 30, no. 4: 72. https://doi.org/10.3390/mca30040072
APA StyleMirsadeghi Esfahani, S. S., Fallah, A., & Mohammadi Aghdam, M. (2025). Physics-Informed Neural Network for Nonlinear Bending Analysis of Nano-Beams: A Systematic Hyperparameter Optimization. Mathematical and Computational Applications, 30(4), 72. https://doi.org/10.3390/mca30040072