Numerical methods for approximately solving partial differential equations (PDE) are at the core of scientific computing. Often, this requires high-resolution or adaptive discretization grids to capture relevant spatio-temporal features in the PDE solution, e.g., in applications like turbulence, combustion, and shock propagation. Numerical approximation also requires knowing the PDE in order to construct problem-specific discretizations. Systematically deriving such solution-adaptive discrete operators, however, is a current challenge. Here we present STENCIL-NET, an artificial neural network architecture for data-driven learning of problem- and resolution-specific local discretizations of nonlinear PDEs. STENCIL-NET achieves numerically stable discretization of the operators in an unknown nonlinear PDE by spatially and temporally adaptive parametric pooling on regular Cartesian grids, and by incorporating knowledge about discrete time integration. Knowing the actual PDE is not necessary, as solution data is sufficient to train the network to learn the discrete operators. A once-trained STENCIL-NET model can be used to predict solutions of the PDE on larger spatial domains and for longer times than it was trained for, hence addressing the problem of PDE-constrained extrapolation from data. To support this claim, we present numerical experiments on long-term forecasting of chaotic PDE solutions on coarse spatio-temporal grids. We also quantify the speed-up achieved by substituting base-line numerical methods with equation-free STENCIL-NET predictions on coarser grids with little compromise on accuracy.
翻译:大约解决部分差异方程式(PDE)的数值方法(PDE)是科学计算的核心。这通常需要高分辨率或适应性离散格网格结构,以捕捉PDE解决方案中与问题和解决方案特定地方的离散性数据。STENCIL-NET网络通过空间和时间适应性参数集合,在正常的Cartes电网中以未知的非线性PDE实现操作者数字离散化,并纳入离散时间整合知识。但是,系统生成这种解决方案适应性离散操作器是一个当前的挑战。在这里,我们提供了StenCIL-NET,这是一个由数据驱动的人工神经网络结构,用于学习非线性PDE的当地数据驱动和解决方案。SteningCIL-NET模型经过一次的Stencial-ele-nal-nalal-loral-loral-lational-loral-lational-lal-lal-lational-lational-lational-lational-lational-lational-lational-mo-lational-lational-lational-lational-lal-lal-lational-lational-lational-lal-lal-lal-lal-lal-lal-lal-lal-lational-l-l-lal-lal-lational-l-l-l-lational-lational-lal-l)模型,也用来用来用来用来用来用来用来用来用来用来用来用来预测法,用来预测,用来预测,用来预测-lation-l-l-l-l-l-al-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-