Inverse resolution of spatially varying diffusion coefficient using physics-informed neural networks

Abstract

Resolving the diffusion coefficient is a key element in many biological and engineering systems, including pharmacological drug transport and fluid mechanics analyses. Additionally, these systems often have spatial variation in the diffusion coefficient that must be determined, such as for injectable drug-eluting implants into heterogeneous tissues. Unfortunately, obtaining the diffusion coefficient from images in such cases is an inverse problem with only discrete data points. The development of a robust method that can work with such noisy and ill-posed datasets to accurately determine spatially varying diffusion coefficients is of great value across a large range of disciplines. Here, we developed an inverse solver that uses physics-informed neural networks (PINNs) to calculate spatially varying diffusion coefficients from numerical and experimental image data in varying biological and engineering applications. The residual of the transient diffusion equation for a concentration field is minimized to find the diffusion coefficient. The robustness of the method as an inverse solver was tested using both numerical and experimental datasets. The predictions show good agreement with both the numerical and experimental benchmarks; an error of less than 6.31% was obtained against all numerical benchmarks, while the diffusion coefficient calculated in experimental datasets matches the appropriate ranges of other reported literature values. Our work demonstrates the potential of using PINNs to resolve spatially varying diffusion coefficients, which may aid a wide-range of applications, such as enabling better-designed drug-eluting implants for regenerative medicine or oncology fields.