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 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.

Numerical Solution of the Fokker-Planck Equation Using Orthogonal Collocation

“…The Fokker-Planck Equation (FPE) was developed to describe the Brownian motion of particles, represent the variation of probability of a random function in space and time, and study stochastic processes [1]. As mentioned by Liu et al (2021) [2], during a stochastic process, the temporal evolution of variables is affected by random fluctuations. As a consequence, it is impossible to obtain a deterministic trajectory.…”

Self-Validated Physics-Embedding Network: A General Framework for Inverse Modelling

“…An alleviation to this problem is injecting physical knowledge within the modelling framework. Examples of such approaches include the use of PDE-based regularization [25,26,27,28] to reflect the physical conservations the model should obey, or the results from complex physical models to constrain the training direction of the model [29,30], so that a deeper correlation between the ML and physical models can be built. Nevertheless, still no one can assure data-driven models act exactly as physical models, especially in corner cases or unseen cases.…”