论文标题
梯度流的能量变分神经网络离散
Energetic Variational Neural Network Discretizations of Gradient Flows
论文作者
论文摘要
我们提出了一种具有结构的Eulerian算法，用于求解$ l^2 $ - 级别流量和一种具有结构的Lagrangian算法，用于求解广义扩散。两种算法都采用神经网络作为空间离散化的工具。与大多数基于基础PDE的强或弱形式构建数值离散化的现有方法不同，拟议方案是基于能量驱散法直接构建的。这保证了系统自由能的单调衰减，从而避免了溶液的非物理状态，并且对于数值计算的长期稳定性至关重要。为了解决非线性神经网络离散化引起的挑战，我们在空间离散之前对这些变异系统进行时间离散。在实现基于神经网络的算法时，这种方法在计算内存效率。提出的基于神经网络的方案无网状，使我们能够在高维度上解决梯度流。提出了各种数值实验，以证明所提出的数值方案的准确性和能量稳定性。
We present a structure-preserving Eulerian algorithm for solving $L^2$-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial discretization. Unlike most existing methods that construct numerical discretizations based on the strong or weak form of the underlying PDE, the proposed schemes are constructed based on the energy-dissipation law directly. This guarantees the monotonic decay of the system's free energy, which avoids unphysical states of solutions and is crucial for the long-term stability of numerical computations. To address challenges arising from nonlinear neural network discretization, we perform temporal discretizations on these variational systems before spatial discretizations. This approach is computationally memory-efficient when implementing neural network-based algorithms. The proposed neural network-based schemes are mesh-free, allowing us to solve gradient flows in high dimensions. Various numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed numerical schemes.
