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 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 first perform temporal discretization on these variational systems. 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：我们提出了一种保持结构的欧拉算法用于求解$L^2$-梯度流，以及一种保持结构的拉格朗日算法用于求解广义扩散问题。两种算法均采用神经网络作为空间离散化工具。与大多数基于底层偏微分方程的强形式或弱形式构建数值离散格式的现有方法不同，本文提出的方案直接基于能量耗散定律构建。这保证了系统能量的单调衰减，从而避免了非物理状态的解，并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，我们首先对这些变分系统进行时间离散化。在执行基于神经网络的算法时，该方法在计算内存方面具有高效性。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数值格式的精度和能量稳定性。