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

arXiv: https://arxiv.org/abs/2205.02592 · HTML: https://mlsanity.com/paper/2205.02592/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2022.

## Built on by (most cited first, 24 of 24)

- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Physics-Informed Computer Vision: A Review and Perspectives](https://mlsanity.com/paper/2305.18035/) (2023)
- [Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regression](https://mlsanity.com/paper/2307.08107/) (2023)
- [Kronecker-Factored Approximate Curvature for Physics-Informed Neural Networks](https://mlsanity.com/paper/2405.15603/) (2024)
- [Learning in PINNs: Phase transition, diffusion equilibrium, and generalization](https://mlsanity.com/paper/2403.18494/) (2024)
- [Physics-informed Neural Network Estimation of Material Properties in Soft Tissue Nonlinear Biomechanical Models](https://mlsanity.com/paper/2312.09787/) (2023)
- [On the Generalization of PINNs outside the training domain and the Hyperparameters influencing it](https://mlsanity.com/paper/2302.07557/) (2023)
- [Achieving High Accuracy with PINNs via Energy Natural Gradients](https://mlsanity.com/paper/2302.13163/) (2023)
- [A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning](https://mlsanity.com/paper/2509.14198/) (2025)
- [Multi-Parameter Molecular MRI Quantification using Physics-Informed Self-Supervised Learning](https://mlsanity.com/paper/2411.06447/) (2024)
- [PICS in Pics: Physics Informed Contour Selection for Rapid Image Segmentation](https://mlsanity.com/paper/2311.07002/) (2023)
- [Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method](https://mlsanity.com/paper/2501.07700/) (2025)
- [FMEnets: Flow, Material, and Energy networks for non-ideal plug flow reactor design](https://mlsanity.com/paper/2505.20300/) (2025)
- [When Does AI for PDEs Yield Scientific Evidence?](https://mlsanity.com/paper/2608.22504/) (2026)
- [Quantitative Analysis of Molecular Transport in the Extracellular Space Using Physics-Informed Neural Network](https://mlsanity.com/paper/2401.12435/) (2024)
- [Adapting Physics-Informed Neural Networks for Bifurcation Detection in Ecological Migration Models](https://mlsanity.com/paper/2409.00651/) (2024)
- [U-net based prediction of cerebrospinal fluid distribution and ventricular reflux grading](https://mlsanity.com/paper/2410.04460/) (2024)
- [Improving physics-informed neural network extrapolation via transfer learning and adaptive activation functions](https://mlsanity.com/paper/2507.12659/) (2025)
- [MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries](https://mlsanity.com/paper/2603.08465/) (2026)
- [Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers](https://mlsanity.com/paper/2607.22004/) (2026)
- [Where Physics Meets Privacy: Federated PINNs for Privacy-Preserving Brain Tumor Biomechanical Modeling](https://mlsanity.com/paper/2607.26207/) (2026)
- [Forecasting Intrathecal Tracer Enhancement from Pre-Contrast Brain MRI: Direct Regression versus Flow Matching](https://mlsanity.com/paper/2609.22377/) (2026)

## Abstract (arXiv)

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.