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

Zapf, Bastian; Haubner, Johannes; Kuchta, Miroslav; Ringstad, Geir; Eide, Per Kristian; Mardal, Kent‐André

doi:10.48550/arxiv.2205.02592

Cited by 3 publications

(5 citation statements)

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“…During the preparation of this paper, a few new studies appeared [38,39,40,41,42,43,44] that also proposed modified versions of RAR or PDF-based resampling. Most of these methods are special cases of the proposed RAD and RAR-D, and our methods can achieve better performance.…”

Section: Related Work and Our Contributionsmentioning

confidence: 99%

“…During the preparation of this paper, a few new papers appeared [38,39,40,41,42,43,44] that also proposed similar methods. Here, we summarize the similarities and differences between these studies.…”

Section: Comparison With Related Workmentioning

confidence: 99%

“…• Zapf et al [38] (in May 2022) proposed a modified version of RAR-G, where some points with small PDE residual are removed while adding points with large PDE residual. They show that compared with RAR, this reduces the computational cost, but the accuracy keeps similar.…”

Section: Comparison With Related Workmentioning

confidence: 99%

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A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

Wu¹,

Zhu²,

Tan³

et al. 2022

Preprint

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Physics-informed neural networks (PINNs) have shown to be an effective tool for solving both forward and inverse problems of partial differential equations (PDEs). PINNs embed the PDEs into the loss of the neural network using automatic differentiation, and this PDE loss is evaluated at a set of scattered spatio-temporal points (called residual points). The location and distribution of these residual points are highly important to the performance of PINNs. However, in the existing studies on PINNs, only a few simple residual point sampling methods have mainly been used. Here, we present a comprehensive study of two categories of sampling for PINNs: non-adaptive uniform sampling and adaptive nonuniform sampling. We consider six uniform sampling methods, including (1) equispaced uniform grid, (2) uniformly random sampling, (3) Latin hypercube sampling, (4) Halton sequence, (5) Hammersley sequence, and (6) Sobol sequence. We also consider a resampling strategy for uniform sampling. To improve the sampling efficiency and the accuracy of PINNs, we propose two new residual-based adaptive sampling methods: residual-based adaptive distribution (RAD) and residual-based adaptive refinement with distribution (RAR-D), which dynamically improve the distribution of residual points based on the PDE residuals during training. Hence, we have considered a total of 10 different sampling methods, including six non-adaptive uniform sampling, uniform sampling with resampling, two proposed adaptive sampling, and an existing adaptive sampling. We extensively tested the performance of these sampling methods for four forward problems and two inverse problems in many setups. Our numerical results presented in this study are summarized from more than 6000 simulations of PINNs. We show that the proposed adaptive sampling methods of RAD and RAR-D significantly improve the accuracy of PINNs with fewer residual points for both forward and inverse problems. The results obtained in this study can also be used as a practical guideline in choosing sampling methods.

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Section: Related Work and Our Contributionsmentioning

confidence: 99%

Section: Comparison With Related Workmentioning

confidence: 99%

See 1 more Smart Citation

A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

Wu¹,

Zhu²,

Tan³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Another approach was proposed in the literature as structure-constrained low rank approximation (SLR) method based on wiener filtering regularization for image denoising [33]. The elliptic partial differential equations-based approach was appeared in [32] where the performance of the optimization problem was investigated for the appropriate choice of the regularization parameters. According to the available literature, the modeling of the effective regularizers play the crucial part in the image denoising and other ill-posed imaging optimization problems.…”

Section: Related Workmentioning

confidence: 99%

Image Driven Isotropic Diffusivity and Complementary Regularization Approach for Image Denoising Problem

Pirzada,

Amur,

Arain

et al. 2022

VFAST trans. math.

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We present the idea of image driven isotropic diffusivity along with complementary regularization for image denoising problem. The method is based on the optimization of a quadratic function in L2 norm. The minimizationof the energy functional leads to the Partial Differential equation (PDE)-based problem. We are looking for a steady state solution of equivalent time dependent problem. We discretize the problem with standard ﬁnite differences. The steady-state numerical solution of the time dependent problem leads to the iterative procedure, which allow to compute a regularized version of the solution as a denoised image. We have applied our designed model on synthetic as well as real images. The numerous experiments have been conducted to analyse the performance of the method for the different choices of scaling parameters. From the quality of the obtained results and comparative study it is observed that the proposed model performs well as compared to well existing methods.

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“…Recently, physics-informed neural networks (PINNs) have gained popularity due to the novel approach for solving forward [1][2][3][4] and inverse problems [5][6][7] involving PDEs using neural networks (NNs). Unlike conventional numerical techniques for solving PDEs, PINNs are non-data-driven meshless models that satisfy the prescribed initial (IC) and Llion Evans, Michelle Tindall, and Perumal Nithiarasu have contributed equally to this work.…”

Section: Introductionmentioning

confidence: 99%

Stiff-PDEs and Physics-Informed Neural Networks

Sharma

Evans

Tindall

et al. 2023

Arch Computat Methods Eng

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In recent years, physics-informed neural networks (PINN) have been used to solve stiff-PDEs mostly in the 1D and 2D spatial domain. PINNs still experience issues solving 3D problems, especially, problems with conflicting boundary conditions at adjacent edges and corners. These problems have discontinuous solutions at edges and corners that are difficult to learn for neural networks with a continuous activation function. In this review paper, we have investigated various PINN frameworks that are designed to solve stiff-PDEs. We took two heat conduction problems (2D and 3D) with a discontinuous solution at corners as test cases. We investigated these problems with a number of PINN frameworks, discussed and analysed the results against the FEM solution. It appears that PINNs provide a more general platform for parameterisation compared to conventional solvers. Thus, we have investigated the 2D heat conduction problem with parametric conductivity and geometry separately. We also discuss the challenges associated with PINNs and identify areas for further investigation.

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

Cited by 3 publications

References 0 publications

A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

Image Driven Isotropic Diffusivity and Complementary Regularization Approach for Image Denoising Problem

Stiff-PDEs and Physics-Informed Neural Networks

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