Studying First Passage Problems using Neural Networks: A Case Study in the Slit-Well Microfluidic Device

TL;DR

This paper demonstrates how deep neural networks can efficiently solve complex first passage time problems modeled by partial differential equations, providing accurate mappings from physical parameters to system metrics in microfluidic devices.

Contribution

It introduces a neural network approach to solve the time-integrated Smoluchowski equation, enabling continuous parameter-to-output mappings and handling geometry variations effectively.

Findings

Neural networks accurately solve the PDE model for first passage times.

The method provides continuous mappings from physical inputs to system metrics.

Neural networks handle geometry modifications easily, outperforming traditional methods.

Abstract

This study presents deep neural network solutions to a time-integrated Smoluchowski equation modeling the mean first passage time of nanoparticles traversing the slit-well microfluidic device. This physical scenario is representative of a broader class of parameterized first passage problems in which key output metrics are dictated by a complicated interplay of problem parameters and system geometry. Specifically, whereas these types of problems are commonly studied using particle simulations of stochastic differential equation models, here the corresponding partial differential equation model is solved using a method based on deep neural networks. The results illustrate that the neural network method is synergistic with the time-integrated Smoluchowski model: together, these are used to construct continuous mappings from key physical inputs (applied voltage and particle diameter) to key output metrics (mean first passage time and effective mobility). In particular, this capability is a unique advantage of the time-integrated Smoluchowski model over the corresponding stochastic differential equation models. Furthermore, the neural network method is demonstrated to easily and reliably handle geometry-modifying parameters, which is generally difficult to accomplish using other methods.