We present a dimension-incremental method for function approximation in bounded orthonormal product bases to learn the solutions of various differential equations. Therefore, we deconstruct the source function of the differential equation into parameters like Fourier or Spline coefficients and treat the solution of the differential equation as a high-dimensional function w.r.t. the spatial variables, these parameters and also further possible parameters from the differential equation itself. Finally, we learn this function in the sense of sparse approximation in a suitable function space by detecting coefficients of the basis expansion with largest absolute value. Investigating the corresponding indices of the basis coefficients yields further insights on the structure of the solution as well as its dependency on the parameters and their interactions and allows for a reasonable generalization to even higher dimensions and therefore better resolutions of the deconstructed source function.

翻译：本文提出一种基于有界正交乘积基的维度递增函数逼近方法，用于学习各类微分方程的解。为此，我们将微分方程的源函数解构为傅里叶系数或样条系数等参数，并将微分方程的解视为关于空间变量、这些参数以及微分方程本身可能存在的其他参数的高维函数。最终，我们通过检测基展开中绝对值最大的系数，在合适函数空间中以稀疏逼近的方式学习该函数。研究基系数对应的指标不仅能进一步揭示解的结构及其对参数的依赖关系与交互作用，还能实现向更高维度的合理推广，从而获得解构后源函数更精细的分辨率。