Skip to main content
OSTI.GOVU.S. Department of Energy Office of Scientific and Technical Information
U.S. Department of Energy
Office of Scientific and Technical Information
 

Fourier Neural Networks as Function Approximators and Differential Equation Solvers

Conference ·
DOI:https://doi.org/10.1002/sam.11531· OSTI ID:1863256
;

We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a straightforward architecture with a single hidden layer. The simplicity of this network architecture facilitates the integration with any other higher-complexity networks, at a data pre- or postprocessing stage. We validate this FNN on naturally periodic smooth functions and on piecewise continuous periodic functions. We showcase the use of this FNN for modeling or solving partial differential equations with periodic boundary conditions. The main advantages of the current approach are the validity of the solution outside the training region, interpretability of the trained model, and simplicity of use.

Research Organization:
Argonne National Laboratory (ANL), Argonne, IL (United States)
Sponsoring Organization:
USDOE Office of Science - Office of Advanced Scientific Computing Research (ASCR)
DOE Contract Number:
AC02-06CH11357
OSTI ID:
1863256
Resource Relation:
Conference: 2020 Conference on Data Analysis, 02/25/20 - 02/27/20, Santa Fe, NM, US
Country of Publication:
United States
Language:
English

References (13)

Discovering governing equations from data by sparse identification of nonlinear dynamical systems March 2016
Approximation by superpositions of a sigmoidal function December 1989
There exists a neural network that does not make avoidable mistakes July 1988
Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification December 2015
Long Short-Term Memory November 1997
Multilayer feedforward networks are universal approximators January 1989
Object Recognition with Gradient-Based Learning January 1999
Multilayer feedforward networks with a nonpolynomial activation function can approximate any function January 1993
Machine learning of linear differential equations using Gaussian processes November 2017
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations February 2019
Data-driven discovery of partial differential equations April 2017
Deep Neural Networks Motivated by Partial Differential Equations September 2019
DGM: A deep learning algorithm for solving partial differential equations December 2018

Similar Records

Related Subjects

Fourier decomposition
neural networks
partial differential equations