Solving the inverse problem of time independent Fokker–Planck equation with a self supervised neural network method

Abstract The Fokker–Planck equation (FPE) has been used in many important applications to study stochastic processes with the evolution of the probability density function (pdf). Previous studies on FPE mainly focus on solving the forward problem which is to predict the time-evolution of the pdf fro...

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Autores principales:	Wei Liu, Connie Khor Li Kou, Kun Hee Park, Hwee Kuan Lee
Formato:	article
Lenguaje:	EN
Publicado:	Nature Portfolio 2021
Materias:	Medicine R Science Q
Acceso en línea:	https://doaj.org/article/317ebbfd45d840cdabcb45e40112c67c
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spelling	oai:doaj.org-article:317ebbfd45d840cdabcb45e40112c67c2021-12-02T16:06:44ZSolving the inverse problem of time independent Fokker–Planck equation with a self supervised neural network method10.1038/s41598-021-94712-52045-2322https://doaj.org/article/317ebbfd45d840cdabcb45e40112c67c2021-07-01T00:00:00Zhttps://doi.org/10.1038/s41598-021-94712-5https://doaj.org/toc/2045-2322Abstract The Fokker–Planck equation (FPE) has been used in many important applications to study stochastic processes with the evolution of the probability density function (pdf). Previous studies on FPE mainly focus on solving the forward problem which is to predict the time-evolution of the pdf from the underlying FPE terms. However, in many applications the FPE terms are usually unknown and roughly estimated, and solving the forward problem becomes more challenging. In this work, we take a different approach of starting with the observed pdfs to recover the FPE terms using a self-supervised machine learning method. This approach, known as the inverse problem, has the advantage of requiring minimal assumptions on the FPE terms and allows data-driven scientific discovery of unknown FPE mechanisms. Specifically, we propose an FPE-based neural network (FPE-NN) which directly incorporates the FPE terms as neural network weights. By training the network on observed pdfs, we recover the FPE terms. Additionally, to account for noise in real-world observations, FPE-NN is able to denoise the observed pdfs by training the pdfs alongside the network weights. Our experimental results on various forms of FPE show that FPE-NN can accurately recover FPE terms and denoising the pdf plays an essential role.Wei LiuConnie Khor Li KouKun Hee ParkHwee Kuan LeeNature PortfolioarticleMedicineRScienceQENScientific Reports, Vol 11, Iss 1, Pp 1-11 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Medicine R Science Q
spellingShingle	Medicine R Science Q Wei Liu Connie Khor Li Kou Kun Hee Park Hwee Kuan Lee Solving the inverse problem of time independent Fokker–Planck equation with a self supervised neural network method
description	Abstract The Fokker–Planck equation (FPE) has been used in many important applications to study stochastic processes with the evolution of the probability density function (pdf). Previous studies on FPE mainly focus on solving the forward problem which is to predict the time-evolution of the pdf from the underlying FPE terms. However, in many applications the FPE terms are usually unknown and roughly estimated, and solving the forward problem becomes more challenging. In this work, we take a different approach of starting with the observed pdfs to recover the FPE terms using a self-supervised machine learning method. This approach, known as the inverse problem, has the advantage of requiring minimal assumptions on the FPE terms and allows data-driven scientific discovery of unknown FPE mechanisms. Specifically, we propose an FPE-based neural network (FPE-NN) which directly incorporates the FPE terms as neural network weights. By training the network on observed pdfs, we recover the FPE terms. Additionally, to account for noise in real-world observations, FPE-NN is able to denoise the observed pdfs by training the pdfs alongside the network weights. Our experimental results on various forms of FPE show that FPE-NN can accurately recover FPE terms and denoising the pdf plays an essential role.
format	article
author	Wei Liu Connie Khor Li Kou Kun Hee Park Hwee Kuan Lee
author_facet	Wei Liu Connie Khor Li Kou Kun Hee Park Hwee Kuan Lee
author_sort	Wei Liu
title	Solving the inverse problem of time independent Fokker–Planck equation with a self supervised neural network method
title_short	Solving the inverse problem of time independent Fokker–Planck equation with a self supervised neural network method
title_full	Solving the inverse problem of time independent Fokker–Planck equation with a self supervised neural network method
title_fullStr	Solving the inverse problem of time independent Fokker–Planck equation with a self supervised neural network method
title_full_unstemmed	Solving the inverse problem of time independent Fokker–Planck equation with a self supervised neural network method
title_sort	solving the inverse problem of time independent fokker–planck equation with a self supervised neural network method
publisher	Nature Portfolio
publishDate	2021
url	https://doaj.org/article/317ebbfd45d840cdabcb45e40112c67c
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Solving the inverse problem of time independent Fokker–Planck equation with a self supervised neural network method

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