This work extends the paradigm of evolutional deep neural networks (EDNNs) to solving parametric time-dependent partial differential equations (PDEs) on domains with geometric structure. By introducing positional embeddings based on eigenfunctions of the Laplace-Beltrami operator, geometric properties are encoded intrinsically and Dirichlet, Neumann and periodic boundary conditions of the PDE solution are enforced directly through the neural network architecture. The proposed embeddings lead to improved error convergence for static PDEs and extend EDNNs towards computational domains of realistic complexity. Several steps are taken to improve performance of EDNNs: Solving the EDNN update equation with a Krylov solver avoids the explicit assembly of Jacobians and enables scaling to larger neural networks. Computational efficiency is further improved by an ad-hoc active sampling scheme that uses the PDE dynamics to effectively sample collocation points. A modified linearly implicit Rosenbrock method is proposed to alleviate the time step requirements of stiff PDEs. Lastly, a completely training-free approach, which automatically enforces initial conditions and only requires time integration, is compared against EDNNs that are trained on the initial conditions. We report results for the Korteweg-de Vries equation, a nonlinear heat equation and (nonlinear) advection-diffusion problems on domains with and without holes and various boundary conditions, to demonstrate the effectiveness of the method. The numerical results highlight EDNNs as a promising surrogate model for parametrized PDEs with slow decaying Kolmogorov n-width.

翻译：本研究将演化深度神经网络（EDNN）框架扩展至求解具有几何结构的域上的参数化时间相关偏微分方程（PDE）。通过引入基于拉普拉斯-贝尔特拉米算子本征函数的位置嵌入，以固有方式编码几何特征，并直接通过神经网络架构施加PDE解的狄利克雷、诺伊曼及周期边界条件。所提出的嵌入方法可改善静态PDE的误差收敛性，并将EDNN推广至具有现实复杂度的计算域。本研究通过多项举措提升EDNN性能：采用Krylov求解器解算EDNN更新方程，避免显式组装雅可比矩阵，从而支持更大规模神经网络；提出基于PDE动力学的自适应主动采样方案以高效选取配置点，进一步优化计算效率；针对刚性PDE的时间步长限制，提出改进型线性隐式Rosenbrock方法。最后，将一种完全免训练的方法（自动满足初始条件且仅需时间积分）与基于初始条件训练的EDNN进行对比。我们报告了Korteweg-de Vries方程、非线性热方程及（非线性）对流扩散问题在含/不含孔洞域及多种边界条件下的计算结果，验证了该方法的有效性。数值结果凸显EDNN作为参数化PDE（具有慢衰减Kolmogorov n宽度）有前景的替代模型。