Neural PDE Solvers with Physics Constraints: A Comparative Study of PINNs, DRM, and WANs

• URL: http://arxiv.org/abs/2510.09693v1
• Date: Thu, 09 Oct 2025 13:41:51 GMT
• Title: Neural PDE Solvers with Physics Constraints: A Comparative Study of PINNs, DRM, and WANs
• Authors: Jiakang Chen,
• Abstract summary: Partial equations (PDEs) underpin models across science and engineering, yet analytical solutions are atypical and classical mesh-based solvers can be costly in high dimensions.<n>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 Schr"odinger equation in 1D/2D.
• Score: 1.131316248570352
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 Schr\"odinger equation in 1D/2D (infinite well and harmonic oscillator), and extends the study to a laser-driven case of Schr\"odinger'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.

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