# Neural Networks for Scalar Input and Functional Output

arXiv: https://arxiv.org/abs/2208.05776 · HTML: https://mlsanity.com/paper/2208.05776/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 60 references that are in the map.
- First posted on arXiv 10 Aug 2022.

## Builds on (most cited first, 13 of 13)

- [DeepSurv: Personalized Treatment Recommender System Using A Cox Proportional Hazards Deep Neural Network](https://mlsanity.com/paper/1606.00931/) (2016)
- [LSTM: A Search Space Odyssey](https://mlsanity.com/paper/1503.04069/) (2015)
- [Towards Causal Representation Learning](https://mlsanity.com/paper/2102.11107/) (2021)
- [Functional Regression](https://mlsanity.com/paper/1406.4068/) (2014)
- [Deep Learning for Functional Data Analysis with Adaptive Basis Layers](https://mlsanity.com/paper/2106.10414/) (2021)
- [Deep Learning with Functional Inputs](https://mlsanity.com/paper/2006.09590/) (2020)
- [A Non-linear Function-on-Function Model for Regression with Time Series Data](https://mlsanity.com/paper/2011.12378/) (2020)
- [Multi-dimensional Functional Principal Component Analysis](https://mlsanity.com/paper/1510.04439/) (2015)
- [Generalized Functional Additive Mixed Models](https://mlsanity.com/paper/1506.05384/) (2015)
- [Spatio-Temporal Functional Neural Networks](https://mlsanity.com/paper/2009.05665/) (2020)
- [Functional Nonlinear Learning](https://mlsanity.com/paper/2206.11424/) (2022)
- [Neural Networks as Functional Classifiers](https://mlsanity.com/paper/2010.04305/) (2020)
- [A Deep Latent-Variable Model Application to Select Treatment Intensity in Survival Analysis](https://mlsanity.com/paper/1811.12323/) (2018)

## Built on by (most cited first, 4 of 4)

- [Functional Autoencoder for Smoothing and Representation Learning](https://mlsanity.com/paper/2401.09499/) (2024)
- [Deep learning estimation of the spectral density of functional time series on large domains](https://mlsanity.com/paper/2601.00284/) (2026)
- [Deep Shape Regression for Planar Curves with Multimodal Covariates](https://mlsanity.com/paper/2607.19600/) (2026)
- [Shape-Based Inductive Bias for Glioma Grading from Tumor Contours](https://mlsanity.com/paper/2607.26090/) (2026)

## Abstract (arXiv)

The regression of a functional response on a set of scalar predictors can be a challenging task, especially if there is a large number of predictors, or the relationship between those predictors and the response is nonlinear. In this work, we propose a solution to this problem: a feed-forward neural network (NN) designed to predict a functional response using scalar inputs. First, we transform the functional response to a finite-dimensional representation and construct an NN that outputs this representation. Then, we propose to modify the output of an NN via the objective function and introduce different objective functions for network training. The proposed models are suited for both regularly and irregularly spaced data, and a roughness penalty can be further applied to control the smoothness of the predicted curve. The difficulty in implementing both those features lies in the definition of objective functions that can be back-propagated. In our experiments, we demonstrate that our model outperforms the conventional function-on-scalar regression model in multiple scenarios while computationally scaling better with the dimension of the predictors.