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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2510.09693 |
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| _version_ | 1866909836754550784 |
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| author | Chen, Jiakang |
| author_facet | Chen, Jiakang |
| contents | Partial differential equations (PDEs) underpin models across science and engineering, yet analytical solutions are atypical and classical mesh-based solvers can be costly in high dimensions. This dissertation presents a unified comparison of three mesh-free neural PDE solvers, physics-informed neural networks (PINNs), the deep Ritz method (DRM), and weak adversarial networks (WANs), on Poisson problems (up to 5D) and the time-independent Schrödinger equation in 1D/2D (infinite well and harmonic oscillator), and extends the study to a laser-driven case of Schrödinger's equation via the Kramers-Henneberger (KH) transformation.
Under a common protocol, all methods achieve low $L_2$ errors ($10^{-6}$-$10^{-9}$) when paired with forced boundary conditions (FBCs), forced nodes (FNs), and orthogonality regularization (OG). Across tasks, PINNs are the most reliable for accuracy and recovery of excited spectra; DRM offers the best accuracy-runtime trade-off on stationary problems; WAN is more sensitive but competitive when weak-form constraints and FN/OG are used effectively. Sensitivity analyses show that FBC removes boundary-loss tuning, network width matters more than depth for single-network solvers, and most gains occur within 5000-10,000 epochs. The same toolkit solves the KH case, indicating transfer beyond canonical benchmarks.
We provide practical guidelines for method selection and outline the following extensions: time-dependent formulations for DRM and WAN, adaptive residual-driven sampling, parallel multi-state training, and neural domain decomposition. These results support physics-guided neural solvers as credible, scalable tools for solving complex PDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09693 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Neural PDE Solvers with Physics Constraints: A Comparative Study of PINNs, DRM, and WANs Chen, Jiakang Machine Learning Quantum Physics Partial differential equations (PDEs) underpin models across science and engineering, yet analytical solutions are atypical and classical mesh-based solvers can be costly in high dimensions. This dissertation presents a unified comparison of three mesh-free neural PDE solvers, physics-informed neural networks (PINNs), the deep Ritz method (DRM), and weak adversarial networks (WANs), on Poisson problems (up to 5D) and the time-independent Schrödinger equation in 1D/2D (infinite well and harmonic oscillator), and extends the study to a laser-driven case of Schrödinger's equation via the Kramers-Henneberger (KH) transformation. Under a common protocol, all methods achieve low $L_2$ errors ($10^{-6}$-$10^{-9}$) when paired with forced boundary conditions (FBCs), forced nodes (FNs), and orthogonality regularization (OG). Across tasks, PINNs are the most reliable for accuracy and recovery of excited spectra; DRM offers the best accuracy-runtime trade-off on stationary problems; WAN is more sensitive but competitive when weak-form constraints and FN/OG are used effectively. Sensitivity analyses show that FBC removes boundary-loss tuning, network width matters more than depth for single-network solvers, and most gains occur within 5000-10,000 epochs. The same toolkit solves the KH case, indicating transfer beyond canonical benchmarks. We provide practical guidelines for method selection and outline the following extensions: time-dependent formulations for DRM and WAN, adaptive residual-driven sampling, parallel multi-state training, and neural domain decomposition. These results support physics-guided neural solvers as credible, scalable tools for solving complex PDEs. |
| title | Neural PDE Solvers with Physics Constraints: A Comparative Study of PINNs, DRM, and WANs |
| topic | Machine Learning Quantum Physics |
| url | https://arxiv.org/abs/2510.09693 |