用神经网络解偏微分方程，能精准处理复杂边界条件。
Solving PDEs With Deep Neural Nets under General Boundary Conditions
- 结合自然梯度与时间推进法，稳定求解带狄利克雷边界的PDE
- 海恩法比欧拉法更准（二阶修正），欧拉法更快更简单
- 适合需要高精度边值控制的物理模拟场景
偏微分方程在物理、生物和工程领域建模中至关重要，但传统数值方法在高维或复杂问题上表现受限。物理信息神经网络（PINNs）通过将物理约束嵌入深度学习框架成为高效替代方案，但在精度和复杂边界条件处理上仍有挑战。本文将时间演化自然梯度（TENG）框架拓展至狄利克雷边界条件，结合自然梯度优化与数值时间推进法（包括欧拉法和海恩法），确保求解稳定性和准确性。通过在损失函数中引入边界惩罚项，实现对狄利克雷约束的精确施加。热方程实验表明，海恩法因具有二阶修正而精度更高，欧拉法在简单场景下计算效率更优。本工作为拓展至诺伊曼及混合边界条件和更广泛的PDE类奠定了基础，推动了神经网络求解器在真实问题中的应用。
原文摘要 · Abstract (English)
Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed Neural Networks (PINNs) have emerged as an efficient alternative by embedding physics-based constraints into deep learning frameworks, but they face challenges in achieving high accuracy and handling complex boundary conditions. In this work, we extend the Time-Evolving Natural Gradient (TENG) framework to address Dirichlet boundary conditions, integrating natural gradient optimization with numerical time-stepping schemes, including Euler and Heun methods, to ensure both stability and accuracy. By incorporating boundary condition penalty terms into the loss function, the proposed approach enables precise enforcement of Dirichlet constraints. Experiments on the heat equation demonstrate the superior accuracy of the Heun method due to its second-order corrections and the computational efficiency of the Euler method for simpler scenarios. This work establishes a foundation for extending the framework to Neumann and mixed boundary conditions, as well as broader classes of PDEs, advancing the applicability of neural network-based solvers for real-world problems.
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