Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations
This incremental improvement addresses function approximation for sparse data in applications like parametric differential equations, benefiting researchers in computational science and engineering.
The paper tackles the problem of approximating functions from sparse data by proposing a modified Nonlinear Level set Learning (NLL) method that reduces input dimensions to a theoretical lower bound with minimal accuracy loss, achieving faster training and higher accuracy than Active Subspaces and original NLL on high-dimensional and parametric differential equation examples.
A dimension reduction method based on the "Nonlinear Level set Learning" (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and sensitivity analysis. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method. While accommodating sparse input data, the proposed algorithm is shown to train quickly and provide a much more accurate and informative reduction than either AS or the original NLL on two example functions with high-dimensional domains, as well as two state-dependent quantities depending on the solutions to parametric differential equations.