对比三种神经微分方程求解器，给出实际应用建议。
Neural PDE Solvers with Physics Constraints: A Comparative Study of PINNs, DRM, and WANs
- 统一框架下比较PINNs、DRM、WAN在不同问题上的表现。
- 在5D泊松方程和1-2维薛定谔方程上误差低至10^-6~10^-9。
- 提供选型指南，适合做物理建模与高维偏微分方程求解的研究者。
偏微分方程（PDEs）广泛存在于科学与工程领域，但解析解罕见，传统网格方法在高维场景下计算成本高昂。本文对三种无网格神经微分方程求解器——物理信息神经网络（PINNs）、深度里茨法（DRM）与弱对抗网络（WANs）——进行了统一比较，涵盖5D泊松问题、1D/2D定态薛定谔方程（无限势阱与简谐振子），并扩展至通过克雷默斯-亨内伯格（KH）变换处理的激光驱动薛定谔方程。在统一实验协议下，所有方法结合强制边界条件（FBCs）、强制节点（FNs）与正交性正则化（OG）后，均实现 $L_2$ 误差 $10^{-6}$ 到 $10^{-9}$ 的高精度。其中，PINNs 在准确性和激发谱恢复上最可靠；DRM 在静态问题中表现最佳的精度-速度平衡；WAN 在有效使用弱形式约束与 FN/OG 时敏感但具竞争力。敏感性分析表明：FBC 可消除边界损失调参需求，网络宽度比深度更重要，大部分性能提升发生在5000–10000轮训练内。相同方法框架成功求解KH案例，表明其超越经典基准的迁移能力。本文提出实用选型指南，并展望未来方向：DRM与WAN的时间依赖形式、残差驱动自适应采样、多态并行训练与神经域分解。结果支持物理引导的神经求解器作为解决复杂PDE的可信、可扩展工具。
原文摘要 · Abstract (English)
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.
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