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.

翻译：偏微分方程（PDEs）是物理、生物和工程领域中复杂系统建模的核心工具，然而传统数值方法在处理高维或复杂问题时常常面临困难。物理信息神经网络（PINNs）通过将基于物理的约束嵌入深度学习框架，已成为一种高效的替代方案，但其在实现高精度和处理复杂边界条件方面仍存在挑战。本研究将时间演化自然梯度（TENG）框架扩展至处理狄利克雷边界条件，通过将自然梯度优化与数值时间步进格式（包括欧拉法和Heun法）相结合，确保了方法的稳定性与精度。通过在损失函数中加入边界条件惩罚项，所提方法能够精确实施狄利克雷约束。在热传导方程上的实验表明，Heun方法因其二阶修正而具有更高的精度，而欧拉方法在简单场景中展现出更高的计算效率。本研究为将该框架扩展至诺伊曼边界条件、混合边界条件以及更广泛的PDE类型奠定了基础，从而推进了基于神经网络的求解器在实际问题中的应用。