Alexander Heinlein

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.

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.

Multiscale problems are challenging for neural network-based discretizations of differential equations, such as physics-informed neural networks (PINNs) and opera-tor networks. This can be (partly) attributed to the so-called spectral bias of neural net-works. To improve the performance of PINNs for time-dependent problems, a combination of multifidelity stacking PINNs and domain decomposition-based finite basis PINNsis employed. Inparticular, to learn thehigh-fidelity part of the multifidelity model, a do-main decomposition in time is employed. The performance is investigated for a pendulum and a two-frequency problem as well as the Allen–Cahn equation. It can be observed that the domain decomposition approach clearly improves the PINN and stacking PINN approaches. Finally, it is demonstrated that the FBPINN approach can be extended tomultifidelity physics-informed deep operator networks.

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\times16 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.

Abstract. 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.

Computational fluid dynamics (CFD) simulations of viscous fluids described by the 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-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 through 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.

Monolithic preconditioners applied to the linear systems arising during the solution of the discretized incompressible Navier–Stokes equations are typically more robust than preconditioners based on incomplete block factorizations. Lower number of iterations and a reduced sensitivity to parameters like velocity and viscosity can significantly outweigh the additional cost for their setup. Different monolithic preconditioning techniques are introduced and compared to a selection of block preconditioners. In particular, two-level additive overlapping Schwarz methods (OSM) are used to set up monolithic preconditioners and to approximate the inverses arising in the block preconditioners. GDSW-type (Generalized Dryja–Smith–Widlund) coarse spaces are used for the second level. These highly scalable, parallel preconditioners have been implemented in the solver framework FROSch (Fast and Robust Overlapping Schwarz), which is part of the software library Trilinos. The new GDSW-type coarse space GDSW* is introduced; combining it with other techniques results in a robust algorithm. The block preconditioners PCD (Pressure Convection–Diffusion), SIMPLE (Semi-Implicit Method for Pressure Linked Equations), and LSC (Least-Squares Commutator) are considered to various degrees. The OSM for the monolithic as well as the block approach allows the optimized combination of different coarse spaces for the velocity and pressure component, enabling the use of tailored coarse spaces. The numerical and parallel performance of the different preconditioning methods for finite element discretizations of stationary as well as time-dependent incompressible fluid flow problems is investigated and compared. Their robustness is analyzed for a range of Reynolds and Courant-Friedrichs-Lewy (CFL) numbers with respect to a realistic problem setting.

Machine learning methods often struggle with real-world applications in science and engineering due to an insufficient amount or quality of training data. In this work, the example of subsurface porous media flow is considered; this corresponds to advection-diffusion processes under heterogeneous flow conditions, i.e., for spatially varying material parameters, and a large number of spatially distributed source terms. This challenge comes at high computing costs for classical simulation methods due to the required high spatio-temporal resolution and large domains. Machine learning-based surrogate models seem to offer a computationally efficient alternative. However, faced with real-world data-limitations, purely data-driven approaches face difficulties in predicting the advection process, which is highly sensitive to input variations and involves long-range interactions. Therefore, in this work, a Local-Global Convolutional Neural Network (LGCNN) approach is introduced, that combines a lightweight numerical surrogate for the global transport process with convolutional neural networks (CNNs) for the remaining local processes. With the LGCNN, we model a city-wide subsurface temperature field, involving a heterogeneous groundwater flow field and one hundred groundwater heat pump injection points forming interacting heat plumes over long distances. In order to first systematically analyze the method, random subsurface input fields are employed. Then, the model is trained on a single cut-out from a real-world subsurface map of the Munich region in Germany. Our model scales to larger cut-outs without retraining by accounting for the global effects with numerical physics models. All datasets, our code, and trained models are published for reproducibility.