Alexander Heinlein

recent publications (full list)

1. A short note on solving partial differential equations using convolutional neural networks

   Grimm, Viktor, Heinlein, Alexander, and Klawonn, Axel

   Submitted November 2022

   Abstract Bib Preprint

   The approach of using physics-based machine learning to solve PDEs has recently become very popular. A recent approach to solve PDEs based on CNNs uses finite difference stencils to include the residual of the partial differential equation into the loss function. In this work, the relation between the network training and the solution of a respective finite difference linear system of equations using classical numerical solvers is discussed. It turns out that many beneficial properties of the linear equation system are neglected in the network training. Finally, numerical results which underline the benefits of classical numerical solvers are presented.

   @techreport{Grimm:2022:SHS,
     author = {Grimm, Viktor and Heinlein, Alexander and Klawonn, Axel},
     title = {A short note on solving partial differential
     equations using convolutional neural networks},
     note = {Submitted November 2022},
     preprint = {http://kups.ub.uni-koeln.de/id/eprint/64227},
     keywords = {submitted, recent},
     bibtex_show = {true}
   }

2. Finite basis physics-informed neural networks as a Schwarz domain decomposition method

   Dolean, Victorita, Heinlein, Alexander, Mishra, Siddhartha, and Moseley, Ben

   Submitted November 2022

   Abstract Bib Preprint

   Physics-informed neural networks (PINNs) [4, 10] are an approach for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to use a neural network to approximate the solution to the PDE and to incorporate the residual of the PDE as well as boundary conditions into its loss function when training it. This provides a simple and mesh-free approach for solving problems relating to PDEs. However, a key limitation of PINNs is their lack of accuracy and efficiency when solving problems with larger domains and more complex, multi-scale solutions. In a more recent approach, finite basis physics-informed neural networks (FBPINNs) [8] use ideas from domain decomposition to accelerate the learning process of PINNs and improve their accuracy. In this work, we show how Schwarz-like additive, multiplicative, and hybrid iteration methods for training FBPINNs can be developed. We present numerical experiments on the influence of these different training strategies on convergence and accuracy. Furthermore, we propose and evaluate a preliminary implementation of coarse space correction for FBPINNs.

   @techreport{Dolean:2022:FBP,
     author = {Dolean, Victorita and Heinlein, Alexander and Mishra, Siddhartha and Moseley, Ben},
     title = {Finite basis physics-informed neural networks as a Schwarz domain decomposition method},
     note = {Submitted November 2022},
     preprint = {https://arxiv.org/abs/2211.05560},
     keywords = {submitted, recent},
     bibtex_show = {true}
   }

3. Reduced order fluid modeling with generative adversarial networks

   Kemna, Mirko, Heinlein, Alexander, and Vuik, Cornelis

   Submitted October 2022

   Abstract Bib Preprint

   Surrogate models based on convolutional neural networks (CNNs) for computational fluid dynamics (CFD) simulations are investigated. In particular, the flow field inside two-dimensional channels with a sudden expansion and an obstacle is predicted using an image representation of the geometry as the input. Generative adversarial neural networks (GANs) have been shown to excel at such image-to-image translation tasks. This motivates the focus of this work on investigating the specific effect of adversarial training on model performance. Numerical results show that the overall accuracy of the GANs is generally lower compared to an identical generator model trained directly on the ground truth using an L1 data loss. On the other hand, GAN predictions are often visually more convincing and exhibit a lower continuity residual.

   @techreport{Kemna:2022:FST,
     author = {Kemna, Mirko and Heinlein, Alexander and Vuik, Cornelis},
     title = {Reduced order fluid modeling with generative adversarial networks},
     note = {Submitted October 2022},
     preprint = {},
     keywords = {submitted, recent},
     bibtex_show = {true}
   }

4. First steps towards modeling the interaction of cardiovascular agents and smooth muscle activation in arterial walls

   Ramesh, Sharan Nurani, Uhlmann, Klemens, Saßmannshausen, Lea, Rheinbach, Oliver, Klawonn, Axel, Heinlein, Alexander, and Balzani, Daniel

   Submitted October 2022

   Abstract Bib Preprint

   Numerical simulation of the response of healthy and pathological arteries to cardiovascular agents can provide valuable information to the physician in the treatment of diseases such as hypertension, atherosclerosis, and the Marfan syndrome. Here, we provide a first step towards a computational framework to model the effects of antihypertensive agents on the mechanical response of arterial walls. A material model is developed by extending an existing formulation for wall tissue to incorporate the effects of calcium-ion channel blockers. The resulting coupled deformation-diffusion problem is then solved using the finite element method. Simulation results with drug activity show that, indeed, an increased lumen diameter due to reduced contraction is obtained. Additionally, a decrease in the rate of arterial contraction is observed, which is also consistent with expected behavior. Finally, we compare results for an implicit or explicit treatment of the the deformation-diffusion coupling, and we observe that both coupling schemes yield comparable results for a wide range of time step sizes.

   @techreport{Ramesh:2022:FST,
     author = {Ramesh, Sharan Nurani and Uhlmann, Klemens and Sa{\ss}mannshausen, Lea and Rheinbach, Oliver and Klawonn, Axel and Heinlein, Alexander and Balzani, Daniel},
     title = {First steps towards modeling the interaction of cardiovascular agents and smooth muscle activation in arterial walls},
     note = {Submitted October 2022},
     preprint = {},
     keywords = {submitted, recent},
     bibtex_show = {true}
   }

5. An Experimental Study of Two-level Schwarz Domain-Decomposition Preconditioners on GPUs

   Yamazaki, Ichitaro, Heinlein, Alexander, and Rajamanickam, Sivasankaran

   Submitted October 2022

   Abstract Bib Preprint

   Generalized Dryja–Smith–Widlund (GDSW) preconditioners are two-level overlapping Schwarz domain decomposition (DD) preconditioners that couple a classical one-level overlapping Schwarz preconditioner with an energy-minimizing coarse space. When used together with Krylov subspace methods, GDSW preconditioners are robust and scalable for solving sparse linear systems arising from the discretization of a wide range of partial different equations. However, the solver has a wide variety of algorithmic and implementation choices, which poses several challenges for its GPU implementation, and thus, its GPU implementation has not been presented so far. To address this gap, in this paper, we present FROSch (Fast and Robust Schwarz), a domain decomposition solver package which implements GDSW algorithms for both CPU and GPU clusters. In order to improve the solver performance on GPUs, we use a novel decomposition to run multiple MPI processes on each GPU, reducing both computational and storage costs of the DD solver, and potentially improving the solver convergence rate. This allows the GPU implementation to be competitive or faster than the CPU implementations. We demonstrate the performance of FROSch on the Summit supercomputer with NVIDIA V100 GPUs, where we used NVIDIA Multi-Process Service (MPS) to implement our decomposition strategy. We conduct a thorough experimental study with different solver options including the exact or inexact solution of the local overlapping subdomain problems based on the sparse direct or incomplete factorizations on a GPU, respectively. We also discuss the effect of using the iterative variant of the incomplete LU factorization and sparse-triangular solve as the approximate local solver and using lower precision for computing the whole FROSch preconditioner. Overall, the solve time was reduced by factors of about 2x using GPUs, while the effects on the numerical setup time depend on the solver options and the local matrix sizes.

   @techreport{Yamazaki:2022:EST,
     author = {Yamazaki, Ichitaro and Heinlein, Alexander and Rajamanickam, Sivasankaran},
     title = {An Experimental Study of Two-level Schwarz Domain-Decomposition Preconditioners on GPUs},
     note = {Submitted October 2022},
     preprint = {},
     keywords = {submitted, reviewed, recent},
     bibtex_show = {true}
   }