利用演化深度神经网络求解偏微分方程的位置嵌入方法
Positional Embeddings for Solving PDEs with Evolutional Deep Neural Networks
AI总结：
本文提出基于Laplace-Beltrami算子特征函数的位置嵌入，将演化深度神经网络扩展到复杂几何区域上的参数化时变PDE求解，并通过Krylov求解、主动采样和改进Rosenbrock方法提升性能，数值实验验证了其有效性。
摘要：
本工作将演化深度神经网络（EDNNs）的范式扩展到在具有几何结构的区域上求解含参数的、依赖时间的偏微分方程（PDEs）。通过引入基于Laplace-Beltrami算子特征函数的位置嵌入，几何性质被内在地编码，并且PDE解的Dirichlet、Neumann和周期边界条件直接通过神经网络架构施加。所提出的嵌入改善了静态PDE的误差收敛性，并将EDNNs扩展到具有现实复杂性的计算区域。为提高EDNNs的性能采取了若干步骤：使用Krylov求解器求解EDNN更新方程，避免了显式组装Jacobian矩阵，并使得能够扩展到更大的神经网络。通过一种自适应的主动采样方案进一步提高计算效率，该方案利用PDE动力学有效地采样配点。提出了一种改进的线性隐式Rosenbrock方法，以缓解刚性PDE的时间步长要求。最后，将一种完全无需训练的方法（该方法自动施加初始条件且仅需要时间积分）与在初始条件上训练的EDNNs进行比较。我们报告了Korteweg-de Vries方程、非线性热方程以及在有孔和无孔区域上具有各种边界条件的（非线性）对流扩散问题的结果，以证明该方法的有效性。数值结果突出表明，EDNNs对于Kolmogorov n-宽度缓慢衰减的参数化PDE是一种有前景的替代模型。
英文摘要：
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.
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