Function-valued RKHS-based Operator Learning for Differential Equations

Recently, a steam of works seek for solving a family of partial differential equations, which consider solving partial differential equations as computing the inverse operator map between the input and solution space. Toward this end, we incorporate function-valued reproducing kernel Hilbert spaces into our operator learning model, which shows that the approximate solution of target operator has a special form. With an appropriate kernel and growth of the data, the approximation solution will converge to the exact one. Then we propose a neural network architecture based on the special form. We perform various experiments and show that the proposed architecture has a desirable accuracy on linear and non-linear partial differential equations even in a small amount of data. By learning the mappings between function spaces, the proposed method has the ability to find the solution of a high-resolution input after learning from lower-resolution data.