# Deep Learning with Functional Inputs

arXiv: https://arxiv.org/abs/2006.09590 · HTML: https://mlsanity.com/paper/2006.09590/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2020.

## Built on by (most cited first, 13 of 13)

- [Approximation of Functionals by Neural Network without Curse of Dimensionality](https://mlsanity.com/paper/2205.14421/) (2022)
- [Neural Networks for Scalar Input and Functional Output](https://mlsanity.com/paper/2208.05776/) (2022)
- [Noncommutative C^*-algebra Net: Learning Neural Networks with Powerful Product Structure in C^*-algebra](https://mlsanity.com/paper/2302.01191/) (2023)
- [Shape-Informed Clustering of Multi-Dimensional Functional Data via Deep Functional Autoencoders](https://mlsanity.com/paper/2509.22969/) (2025)
- [Robust Deep Neural Network Estimation for Multi-dimensional Functional Data](https://mlsanity.com/paper/2205.09604/) (2022)
- [A Functional approach for Two Way Dimension Reduction in Time Series](https://mlsanity.com/paper/2301.00357/) (2023)
- [A Functional Extension of Semi-Structured Networks](https://mlsanity.com/paper/2410.05430/) (2024)
- [FAME: Adaptive Functional Attention with Expert Routing for Function-on-Function Regression](https://mlsanity.com/paper/2510.00621/) (2025)
- [Deep learning estimation of the spectral density of functional time series on large domains](https://mlsanity.com/paper/2601.00284/) (2026)
- [Sparse Bayesian Deep Functional Learning with Structured Region Selection](https://mlsanity.com/paper/2602.20651/) (2026)
- [Shape-Based Inductive Bias for Glioma Grading from Tumor Contours](https://mlsanity.com/paper/2607.26090/) (2026)
- [Approximating Smooth Functionals with ReLU Networks](https://mlsanity.com/paper/2609.15355/) (2026)
- [Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifold](https://mlsanity.com/paper/2609.30487/) (2026)

## Abstract (arXiv)

We present a methodology for integrating functional data into deep densely connected feed-forward neural networks. The model is defined for scalar responses with multiple functional and scalar covariates. A by-product of the method is a set of dynamic functional weights that can be visualized during the optimization process. This visualization leads to greater interpretability of the relationship between the covariates and the response relative to conventional neural networks. The model is shown to perform well in a number of contexts including prediction of new data and recovery of the true underlying functional weights; these results were confirmed through real applications and simulation studies. A forthcoming R package is developed on top of a popular deep learning library (Keras) allowing for general use of the approach.