We enhance machine learning algorithms for learning model parameters in complex systems represented by differential equations with domain decomposition methods. The study evaluates the performance of two approaches, namely (vanilla) Physics-Informed Neural Networks (PINNs) and Finite Basis Physics-Informed Neural Networks (FBPINNs), in learning the dynamics of test models with a quasi-stationary longtime behavior. We test the approaches for data sets in different dynamical regions and with varying noise level. As results, the FBPINN approach better captures the overall dynamical behavior compared to the vanilla PINN approach, even in cases with data only from a time domain with quasi-stationary dynamics.

The segmentation of ultra-high resolution images poses challenges such as loss of spatial information or computational inefficiency. In this work, a novel approach that combines encoder-decoder architectures with domain decomposition strategies to address these challenges is proposed. Specifically, a domain decomposition-based U-Net (DDU-Net) architecture is introduced, which partitions input images into non-overlapping patches that can be processed independently on separate devices. A communication network is added to facilitate inter-patch information exchange to enhance the understanding of spatial context. Experimental validation is performed on a synthetic dataset that is designed to measure the effectiveness of the communication network. Then, the performance is tested on the DeepGlobe land cover classification dataset as a real-world benchmark data set. The results demonstrate that the approach, which includes inter-patch communication for images divided into 16 × 16 non-overlapping subimages, achieves a 2 - 3% higher intersection over union (IoU) score compared to the same network without inter-patch communication. The performance of the network which includes communication is equivalent to that of a baseline U-Net trained on the full image, showing that our model provides an effective solution for segmenting ultra-high-resolution images while preserving spatial context. The code is available at https://github.com/corne00/DDU-Net.

Computational fluid dynamics (CFD) simulations of viscous fluids described by the stationary Navier–Stokes equations are considered. Depending on the Reynolds number of the flow, the Navier–Stokes equations may exhibit a highly nonlinear behavior. The system of nonlinear equations resulting from the discretization of the Navier–Stokes equations can be solved using nonlinear iteration methods, such as Newton's method. However, fast quadratic convergence is typically only obtained in a local neighborhood of the solution, and for many configurations, the classical Newton iteration does not converge at all. In such cases, so-called globalization techniques may help to improve convergence.
In this paper, pseudo-time stepping (also known as pseudo-transient continuation) is employed in order to improve nonlinear convergence. The classical algorithm is enhanced by a neural network model that is trained to predict a local pseudo-time step. Generalization of the novel approach is facilitated by predicting the local pseudo-time step separately on each element using only local information on a patch of adjacent elements as input. Numerical results for standard benchmark problems, including flow over a backward facing step geometry and Couette flow, show the performance of the machine learning-enhanced globalization approach; as the software for the simulations, the CFD Module of COMSOL Multiphysics® is employed.

In recent years, the concept of introducing physics to machine learning has become widely popular. Most physics-inclusive ML-techniques however are still limited to a single geometry or a set of parametrizable geometries. Thus, there remains the need to train a new model for a new geometry, even if it is only slightly modified. With this work we introduce a technique with which it is possible to learn approximate solutions to the steady-state Navier–Stokes equations in varying geometries without the need of parametrization. This technique is based on a combination of a U-Net-like CNN and well established discretization methods from the field of the finite difference method. The results of our physics-aware CNN are compared to a state-of-the-art data-based approach. Additionally, it is also shown how our approach performs when combined with the data-based approach.

Two-level domain decomposition preconditioners lead to fast convergence and scalability of iterative solvers. However, for highly heterogeneous problems with a rapidly varying coefficient function, the condition number of the preconditioned system generally depends on the contrast of the coefficient function. As a result, the convergence may deteriorate. Enhancing the coarse space by functions constructed from suitable local eigenvalue problems restores robust, contrast-independent convergence; these coarse spaces are often denoted as adaptive or spectral coarse spaces. However, these eigenvalue problems typically rely on nonalgebraic information such that the adaptive coarse spaces cannot be constructed from the fully assembled system matrix. In this paper, a novel algebraic adaptive coarse space which relies on the a-orthogonal decomposition of (local) finite element (FE) spaces into functions that solve the elliptic PDE with some trace and FE functions that are zero on the boundary is proposed. In particular, the basis is constructed from eigenmodes of two types of local eigenvalue problems associated with the edges of the domain decomposition. To approximate functions that solve the PDE locally, we employ a transfer eigenvalue problem which has originally been proposed for the construction of optimal local approximation spaces for multiscale methods. In addition, we make use of a Dirichlet eigenvalue problem that is a slight modification of the Neumann eigenvalue problem used in the adaptive generalized Dryja-Smith-Widlund (AGDSW) coarse space. Both eigenvalue problems rely solely on local Dirichlet matrices, which can be extracted from the fully assembled system matrix, allowing for an algebraic construction. By combining arguments from multiscale and domain decomposition methods, we derive a contrast-independent upper bound for the condition number. While we restrict ourselves here to a two-dimensional diffusion problem discretized by low-order FEs on regular meshes, the proposed framework is general, and we conjecture that the approach can be readily extended, for instance, to other elliptic problems, three dimensions, or, under mild assumptions, higher-order discretizations. The robustness of the method is confirmed numerically for a variety of heterogeneous coefficient distributions, including binary random distributions and a coefficient function constructed from the SPE10 benchmark. The results are comparable to those of the nonalgebraic AGDSW coarse space as well as for those cases where the convergence of the classical algebraic generalized Dryja-Smith-Widlund coarse space deteriorates. Moreover, the coarse space dimension is the same as or comparable to the AGDSW coarse space for all numerical experiments.

Background: Burn injuries present a significant global health challenge. Among the most severe long-term consequences are contractures, which can lead to functional impairments and disfigurement. Understanding and predicting the evolution of post-burn wounds is essential for developing effective treatment strategies. Traditional mathematical models, while accurate, are often computationally expensive and time-consuming, limiting their practical application. Recent advancements in machine learning, particularly in deep learning, offer promising alternatives for accelerating these predictions. Methods: This study explores the use of a deep operator network, a type of neural operator, as a surrogate model for finite element simulations aimed at predicting post-burn contraction across multiple wound shapes. A deep operator network was trained on three distinct initial wound shapes, with enhancements made to the architecture by incorporating initial wound shape information and applying sine augmentation to enforce boundary conditions. Findings: The performance of the trained deep operator network was evaluated on a test set including finite element simulations based on convex combinations of the three basic wound shapes. The model achieved an R² score of 0.99, indicating strong predictive accuracy and generalization. Moreover, the model provided reliable predictions over an extended period of up to one year, with speedups of up to 128-fold on the Central Processing Unit and 235-fold on the Graphical Processing Unit, compared to the numerical model. Interpretation: These findings suggest that deep operator networks can effectively serve as a surrogate for traditional finite element methods in simulating post-burn wound evolution, with potential applications in medical treatment planning.

Randomized neural networks (RaNNs), characterized by fixed hidden layers after random initialization, offer a computationally efficient alternative to fully parameterized neural networks trained using stochastic gradient descent-type algorithms. In this paper, we integrate RaNNs with overlapping Schwarz domain decomposition in two primary ways: firstly, to formulate the least-squares problem with localized basis functions, and secondly, to construct effective overlapping Schwarz preconditioners for solving the resulting linear systems. Specifically, neural networks are randomly initialized in each subdomain following a uniform distribution, and these localized solutions are combined through a partition of unity, providing a global approximation to the solution of the partial differential equation. Boundary conditions are imposed via a constraining operator, eliminating the necessity for penalty methods. Furthermore, we apply principal component analysis (PCA) within each subdomain to reduce the number of basis functions, thereby significantly improving the conditioning of the resulting linear system. By constructing additive Schwarz (AS) and restricted AS preconditioners, we efficiently solve the least-squares problems using iterative solvers such as the Conjugate Gradient (CG) and generalized minimal residual methods. Numerical experiments clearly demonstrate that the proposed methodology substantially reduces computational time, particularly for multi-scale and time-dependent PDE problems. Additionally, we present a three-dimensional numerical example illustrating the superior efficiency of employing the CG method combined with an AS preconditioner over direct methods like QR decomposition for solving the associated least-squares system.

The computational complexity and efficiency of the approximate mode component synthesis (ACMS) method is investigated for the two-dimensional heterogeneous Helmholtz equations, aiming at the simulation of large but finite-size photonic crystals. The ACMS method is a Galerkin method that relies on a non-overlapping domain decomposition and special basis functions defined based on the domain decomposition. While, in previous works, the ACMS method was realized using first-order finite elements, we use an underlying hp–finite element method. We study the accuracy of the ACMS method for different wavenumbers, domain decompositions, and discretization parameters. Moreover, the computational complexity of the method is investigated theoretically and compared with computing times for an implementation based on the open source software package NGSolve. The numerical results indicate that, for relevant wavenumber regimes, the size of the resulting linear systems for the ACMS method remains moderate, such that sparse direct solvers are a reasonable choice. Moreover, the ACMS method exhibits only a weak dependence on the selected domain decomposition, allowing for greater flexibility in its choice. Additionally, the numerical results show that the error of the ACMS method achieves the predicted convergence rate for increasing wavenumbers. Finally, to display the versatility of the implementation, the results of simulations of large but finite-size photonic crystals with defects are presented.

In this work, we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the two-dimensional heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the method. In certain, practically relevant cases, even super-algebraic convergence for the local Helmholtz problems can be achieved without oversampling.

Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.

A computational framework is presented to numerically simulate the effects of antihypertensive drugs, in particular calcium channel blockers, on the mechanical response of arterial walls. A stretch-dependent smooth muscle model by Uhlmann and Balzani is modified to describe the interaction of pharmacological drugs and the inhibition of smooth muscle activation. The coupled deformation-diffusion problem is then solved using the finite element software FEDDLib and overlapping Schwarz preconditioners from the Trilinos package FROSch. These preconditioners include highly scalable parallel GDSW (generalized Dryja–Smith–Widlund) and RGDSW (reduced GDSW) preconditioners. Simulation results show the expected increase in the lumen diameter of an idealized artery due to the drug-induced reduction of smooth muscle contraction, as well as a decrease in the rate of arterial contraction in the presence of calcium channel blockers. Strong and weak parallel scalability of the resulting computational implementation are also analyzed.

Solving partial differential equations (PDEs) is a common task in numerical mathematics and scientific computing. Typical discretization schemes, for example, finite element (FE), finite volume (FV), or finite difference (FD) methods, have the disadvantage that the computations have to be repeated once the boundary conditions (BCs) or the geometry change slightly; typical examples requiring the solution of many similar problems are time-dependent and inverse problems or uncertainty quantification.

The success and advancement of machine learning (ML) in fields such as image recognition and natural language processing has lead to the development of novel methods for the solution of problems in physics and engineering.

Monolithic fluid–structure interaction (FSI) of blood flow with arterial walls is considered, making use of sophisticated nonlinear wall models. These incorporate the effects of almost incompressibility as well as of the anisotropy caused by embedded collagen fibers. In the literature, relatively simple structural models such as Neo-Hooke are often considered for FSI with arterial walls. Such models lack, both, anisotropy and incompressibility. In this paper, numerical simulations of idealized heart beats in a curved benchmark geometry, using simple and sophisticated arterial wall models, are compared: we consider three different almost incompressible, anisotropic arterial wall models as a reference and, for comparison, a simple, isotropic Neo-Hooke model using four different parameter sets. The simulations show significant quantitative and qualitative differences in the stresses and displacements as well as the lumen cross sections. For the Neo-Hooke models, a significantly larger amplitude in the in- and outflow areas during the heart beat is observed, presumably due to the lack of fiber stiffening. For completeness, we also consider a linear elastic wall using 16 different parameter sets. However, using our benchmark setup, we were not successful in achieving good agreement with our nonlinear reference calculation.

The numerical simulation of atherosclerotic plaque growth is computationally prohibitive, since it involves a complex cardiovascular fluid-structure interaction (FSI) problem with a characteristic time scale of milliseconds to seconds, as well as a plaque growth process governed by reaction-diffusion equations, which takes place over several months. In this work we combine a temporal homogenization approach, which separates the problem in computationally expensive FSI problems on a micro scale and a reaction-diffusion problem on the macro scale, with parallel time-stepping algorithms. It has been found in the literature that parallel time-stepping algorithms do not perform well when applied directly to the FSI problem. To circumvent this problem, a parareal algorithm is applied on the macro-scale reaction-diffusion problem instead of the micro-scale FSI problem. We investigate modifications in the coarse propagator of the parareal algorithm, in order to further reduce the number of costly micro problems to be solved. The approaches are tested in detailed numerical investigations based on serial simulations.

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.

Accurate short-term predictions of phase-resolved water wave conditions are crucial for decision-making in ocean engineering. However, the initialization of remote-sensing-based wave prediction models first requires a reconstruction of wave surfaces from sparse measurements like radar. Existing reconstruction methods either rely on computationally intensive optimization procedures or simplistic modelling assumptions that compromise the real-time capability or accuracy of the subsequent prediction process. We therefore address these issues by proposing a novel approach for phase-resolved wave surface reconstruction using neural networks based on the U-Net and Fourier neural operator (FNO) architectures. Our approach utilizes synthetic yet highly realistic training data on uniform one-dimensional grids, that is generated by the high-order spectral method for wave simulation and a geometric radar modelling approach. The investigation reveals that both models deliver accurate wave reconstruction results and show good generalization for different sea states when trained with spatio-temporal radar data containing multiple historic radar snapshots in each input. Notably, the FNO demonstrates superior performance in handling the data structure imposed by wave physics due to its global approach to learn the mapping between input and output in Fourier space.

Multilevel extensions of overlapping Schwarz domain decomposition preconditioners of Generalized Dryja-Smith-Widlund (GDSW) type are considered in this paper. The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner, which can be constructed algebraically from the fully assembled stiffness matrix. The FROSch software, which belongs to the ShyLU package of the Trilinos software library, provides parallel implementations of different variants of GDSW preconditioners. The coarse problem can limit the parallel scalability of two-level GDSW preconditioners. As a remedy, in the past, three-level GDSW approaches have been proposed, which can significantly extend the range of scalability. Here, a multilevel extension of the GDSW preconditioner is introduced and analyzed. Finally, parallel results for the implementation in FROSch for up to 40 000 cores of the SuperMUC-NG supercomputer at Leibniz Supercomputing Centre (LRZ) and to 48 000 cores of the JUWELS supercomputer at Jülich Supercomputing Centre (JSC) are presented.

The generalized Dryja–Smith–Widlund (GDSW) preconditioner is a two-level overlapping Schwarz domain decomposition (DD) preconditioner that couples a classical one-level overlapping Schwarz preconditioner with an energy-minimizing coarse space. When used to accelerate the convergence rate of Krylov subspace iterative methods, the GDSW preconditioner provides robustness and scalability for the solution of sparse linear systems arising from the discretization of a wide range of partial different equations. In this paper, we present FROSch (Fast and Robust Schwarz), a domain decomposition solver package which implements GDSW-type preconditioners for both CPU and GPU clusters. To improve the solver performance on GPUs, we use a novel decomposition to run multiple MPI processes on each GPU, reducing both solver’s computational and storage costs and potentially improving the convergence rate. This allowed us to obtain competitive or faster performance using GPUs compared to using CPUs alone. 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.The solver has a wide variety of algorithmic and implementation choices, which poses both opportunities and challenges for its GPU implementation. We conduct a thorough experimental study with different solver options including the exact or inexact solution of the local overlapping subdomain problems on a GPU. 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 2× using GPUs, while the GPU acceleration of the numerical setup time depend on the solver options and the local matrix sizes.

Numerical simulations of Greenland and Antarctic ice sheets involve the solution of large-scale highly nonlinear systems of equations on complex shallow geometries. This work is concerned with the construction of Schwarz preconditioners for the solution of the associated tangent problems, which are challenging for solvers mainly because of the strong anisotropy of the meshes and wildly changing boundary conditions that can lead to poorly constrained problems on large portions of the domain. Here, two-level generalized Dryja-Smith-Widlund (GDSW)-type Schwarz preconditioners are applied to different land ice problems, i.e., a velocity problem, a temperature problem, as well as the coupling of the former two problems. We employ the message passing interface (MPI)- parallel implementation of multilevel Schwarz preconditioners provided by the package FROSch (fast and robust Schwarz) from the Trilinos library. The strength of the proposed preconditioner is that it yields out-of-the-box scalable and robust preconditioners for the single physics problems. To the best of our knowledge, this is the first time two-level Schwarz preconditioners have been applied to the ice sheet problem and a scalable preconditioner has been used for the coupled problem. The preconditioner for the coupled problem differs from previous monolithic GDSW preconditioners in the sense that decoupled extension operators are used to compute the values in the interior of the subdomains. Several approaches for improving the performance, such as reuse strategies and shared memory OpenMP parallelization, are explored as well. In our numerical study we target both uniform meshes of varying resolution for the Antarctic ice sheet as well as nonuniform meshes for the Greenland ice sheet. We present several weak and strong scaling studies confirming the robustness of the approach and the parallel scalability of the FROSch implementation. Among the highlights of the numerical results are a weak scaling study for up to 32 K processor cores (8 K MPI ranks and 4 OpenMP threads) and 566 M degrees of freedom for the velocity problem as well as a strong scaling study for up to 4 K processor cores (and MPI ranks) and 68 M degrees of freedom for the coupled problem.