Non-linear Functional Modeling using Neural Networks

TL;DR

This paper introduces two novel neural network-based models, FDNN and FBNN, tailored for functional data, leveraging continuous hidden layers and basis expansions to capture complex non-linear relationships.

Contribution

It presents the first functional neural network models with continuous hidden layers and basis expansion, along with a new gradient-based optimization algorithm.

Findings

Effective in modeling complex functional data

Demonstrated superior performance in simulations

Validated on real-world datasets

Abstract

We introduce a new class of non-linear models for functional data based on neural networks. Deep learning has been very successful in non-linear modeling, but there has been little work done in the functional data setting. We propose two variations of our framework: a functional neural network with continuous hidden layers, called the Functional Direct Neural Network (FDNN), and a second version that utilizes basis expansions and continuous hidden layers, called the Functional Basis Neural Network (FBNN). Both are designed explicitly to exploit the structure inherent in functional data. To fit these models we derive a functional gradient based optimization algorithm. The effectiveness of the proposed methods in handling complex functional models is demonstrated by comprehensive simulation studies and real data examples.