# Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations

arXiv: https://arxiv.org/abs/2104.14072 · HTML: https://mlsanity.com/paper/2104.14072/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 42 references that are in the map.
- First posted on arXiv 29 Apr 2021.

## Builds on (most cited first, 6 of 6)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Reversible Architectures for Arbitrarily Deep Residual Neural Networks](https://mlsanity.com/paper/1709.03698/) (2017)
- [Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs](https://mlsanity.com/paper/2011.15110/) (2020)
- [Parallel Transport Unfolding: A Connection-based Manifold Learning Approach](https://mlsanity.com/paper/1806.09039/) (2018)
- [Active Manifolds: A non-linear analogue to Active Subspaces](https://mlsanity.com/paper/1904.13386/) (2019)

## Built on by (most cited first, 1 of 1)

- [Level set learning with pseudo-reversible neural networks for nonlinear dimension reduction in function approximation](https://mlsanity.com/paper/2112.01438/) (2021)

## Abstract (arXiv)

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.