We present a framework for solving time-dependent partial differential equations (PDEs) in the spirit of the random feature method. The numerical solution is constructed using a space-time partition of unity and random feature functions. Two different ways of constructing the random feature functions are investigated: feature functions that treat the spatial and temporal variables (STC) on the same footing, or functions that are the product of two random feature functions depending on spatial and temporal variables separately (SoV). Boundary and initial conditions are enforced by penalty terms. We also study two ways of solving the resulting least-squares problem: the problem is solved as a whole or solved using the block time-marching strategy. The former is termed ``the space-time random feature method'' (ST-RFM). Numerical results for a series of problems show that the proposed method, i.e. ST-RFM with STC and ST-RFM with SoV, have spectral accuracy in both space and time. In addition, ST-RFM only requires collocation points, not a mesh. This is important for solving problems with complex geometry. We demonstrate this by using ST-RFM to solve a two-dimensional wave equation over a complex domain. The two strategies differ significantly in terms of the behavior in time. In the case when block time-marching is used, we prove a lower error bound that shows an exponentially growing factor with respect to the number of blocks in time. For ST-RFM, we prove an upper bound with a sublinearly growing factor with respect to the number of subdomains in time. These estimates are also confirmed by numerical results.

翻译：我们提出了一种基于随机特征方法框架求解时间依赖偏微分方程(PDEs)的方法。数值解采用时空单位分解与随机特征函数构造。研究了两种不同的随机特征函数构造方式：将空间和时间变量同等对待的特征函数(STC)，以及分别依赖空间和时间变量的两个随机特征函数乘积形式的函数(SoV)。边界条件和初始条件通过惩罚项施加。我们还研究了解算所得最小二乘问题的两种方式：整体求解或采用块时间推进策略求解。前者称为"时空随机特征方法"(ST-RFM)。一系列问题的数值结果表明，所提出的方法（即STC型ST-RFM与SoV型ST-RFM）在空间和时间上均具有谱精度。此外，ST-RFM仅需配置点，无需网格。这对于求解具有复杂几何形状的问题至关重要。我们通过使用ST-RFM求解复杂域上的二维波动方程来证明这一点。两种策略在时间行为上存在显著差异。当使用块时间推进时，我们证明了一个下界误差估计，该估计显示存在随时间块数量呈指数增长的因子。对于ST-RFM，我们证明了一个上界估计，其时域子域数量呈次线性增长因子。这些估计也得到了数值结果的验证。