Investigating molecular transport in the human brain from MRI with physics-informed neural networks

TL;DR

This paper explores the use of physics-informed neural networks (PINNs) to estimate molecular diffusion in the human brain from MRI data, addressing challenges with noisy measurements and improving accuracy through various training strategies.

Contribution

It demonstrates how PINNs can be effectively applied to brain diffusion MRI data and proposes methods to enhance their robustness and accuracy in noisy conditions.

Findings

PINNs can estimate brain diffusion coefficients with accuracy comparable to finite element methods.

Training strategies like residual-based refinement improve PINN performance.

Small residuals are crucial for accurate inverse problem solutions.

Abstract

In recent years, a plethora of methods combining deep neural networks and partial differential equations have been developed. A widely known and popular example are physics-informed neural networks. They solve forward and inverse problems involving partial differential equations in terms of a neural network training problem. We apply physics-informed neural networks as well as the finite element method to estimate the diffusion coefficient governing the long term, i.e. over days, spread of molecules in the human brain from a novel magnetic resonance imaging technique. Synthetic testcases are created to demonstrate that the standard formulation of the physics-informed neural network faces challenges with noisy measurements in our application. Our numerical results demonstrate that the residual of the partial differential equation after training needs to be small in order to obtain accurate recovery of the diffusion coefficient. To achieve this, we apply several strategies such as tuning the weights and the norms used in the loss function as well as residual based adaptive refinement and exchange of residual training points. We find that the diffusion coefficient estimated with PINNs from magnetic resonance images becomes consistent with results from a finite element based approach when the residuum after training becomes small. The observations presented in this work are an important first step towards solving inverse problems on observations from large cohorts of patients in a semi-automated fashion with physics-informed neural networks.