Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910018453897216 |
|---|---|
| author | Shahab, Muhammad Luthfi Mukhlash, Imam Susanto, Hadi |
| author_facet | Shahab, Muhammad Luthfi Mukhlash, Imam Susanto, Hadi |
| contents | This work investigates the use of shallow physics-informed neural networks (PINNs) for solving forward and inverse problems of nonlinear partial differential equations (PDEs). By reformulating PINNs as nonlinear systems, the Levenberg-Marquardt (LM) algorithm is employed to efficiently optimize the network parameters. Analytical expressions for the neural network derivatives with respect to the input variables are derived, enabling accurate and efficient computation of the Jacobian matrix required by LM. The proposed approach is tested on several benchmark problems, including the Burgers, Schrödinger, Allen-Cahn, and three-dimensional Bratu equations. Numerical results demonstrate that LM significantly outperforms BFGS in terms of convergence speed, accuracy, and final loss values, even when using shallow network architectures with only two hidden layers. These findings indicate that, for a wide class of PDEs, shallow PINNs combined with efficient second-order optimization methods can provide accurate and computationally efficient solutions for both forward and inverse problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08515 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm Shahab, Muhammad Luthfi Mukhlash, Imam Susanto, Hadi Numerical Analysis Machine Learning Neural and Evolutionary Computing Optimization and Control This work investigates the use of shallow physics-informed neural networks (PINNs) for solving forward and inverse problems of nonlinear partial differential equations (PDEs). By reformulating PINNs as nonlinear systems, the Levenberg-Marquardt (LM) algorithm is employed to efficiently optimize the network parameters. Analytical expressions for the neural network derivatives with respect to the input variables are derived, enabling accurate and efficient computation of the Jacobian matrix required by LM. The proposed approach is tested on several benchmark problems, including the Burgers, Schrödinger, Allen-Cahn, and three-dimensional Bratu equations. Numerical results demonstrate that LM significantly outperforms BFGS in terms of convergence speed, accuracy, and final loss values, even when using shallow network architectures with only two hidden layers. These findings indicate that, for a wide class of PDEs, shallow PINNs combined with efficient second-order optimization methods can provide accurate and computationally efficient solutions for both forward and inverse problems. |
| title | Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm |
| topic | Numerical Analysis Machine Learning Neural and Evolutionary Computing Optimization and Control |
| url | https://arxiv.org/abs/2602.08515 |