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.
ABSTRACT
Despite the superior performance of deep learning in many applications, challenges remain in the area of regression on function spaces. In particular, neural networks are unable to encode function inputs compactly as each node encodes just a real value. We propose a novel idea to address this shortcoming: to encode an entire function in a single network node. To that end, we design a compact network representation that encodes and propagates functions in single nodes for the distribution regression task. Our proposed distribution regression network (DRN) achieves higher prediction accuracies while using fewer parameters than traditional neural networks.
