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.

翻译：我们提出了一种用于求解$L^2$-梯度流的结构保持欧拉算法，以及一种用于求解广义扩散方程的结构保持拉格朗日算法。这两种算法均采用神经网络作为空间离散化的工具。与大多数基于偏微分方程强形式或弱形式构建数值离散化的现有方法不同，所提出的方案直接基于能量耗散律构建。这保证了系统自由能的单调衰减，避免了非物理解的出现，对数值计算的长期稳定性至关重要。为应对非线性神经网络离散化带来的挑战，我们在空间离散化之前对这些变分系统进行时间离散化。这种方法在实现基于神经网络的算法时，具有计算内存效率高的优势。所提出的基于神经网络的方案是无网格的，使我们能够求解高维梯度流问题。通过多种数值实验验证了所提出数值格式的精确性和能量稳定性。