Alexander Heinlein

Viktor Grimm, Alexander Heinlein, Axel Klawonn

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.

Adrienne M. Propp, Mauro Perego, Eric C. Cyr et al.

Accurate yet efficient surrogate models are essential for large-scale simulations of partial differential equations (PDEs), particularly for uncertainty quantification (UQ) tasks that demand hundreds or thousands of evaluations. We develop a physics-inspired graph neural network (GNN) surrogate that operates directly on unstructured meshes and leverages the flexibility of graph attention. To improve both training efficiency and generalization properties of the model, we introduce a domain decomposition (DD) strategy that partitions the mesh into subdomains, trains local GNN surrogates in parallel, and aggregates their predictions. We then employ transfer learning to fine-tune models across subdomains, accelerating training and improving accuracy in data-limited settings. Applied to ice sheet simulations, our approach accurately predicts full-field velocities on high-resolution meshes, substantially reduces training time relative to training a single global surrogate model, and provides a ripe foundation for UQ objectives. Our results demonstrate that graph-based DD, combined with transfer learning, provides a scalable and reliable pathway for training GNN surrogates on massive PDE-governed systems, with broad potential for application beyond ice sheet dynamics.

Corné Verburg, Alexander Heinlein, Eric C. Cyr

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.

Yuhan Wu, Jan Willem van Beek, Victorita Dolean et al.

Deep learning-based hybrid iterative methods (DL-HIMs) integrate classical numerical solvers with neural operators, utilizing their complementary spectral biases to accelerate convergence. Despite this promise, many DL-HIMs stagnate at false fixed points where neural updates vanish while the physical residual remains large, raising questions about reliability in scientific computing. In this paper, we provide evidence that performance is highly sensitive to training paradigms and update strategies, even when the neural architecture is fixed. Through a detailed study of a DeepONet-based hybrid iterative numerical transferable solver (HINTS) and an FFT-based Fourier neural solver (FNS), we show that significant physical residuals can persist when training objectives are not aligned with solver dynamics and problem physics. We further examine Anderson acceleration (AA) and demonstrate that its classical form is ill-suited for nonlinear neural operators. To overcome this, we introduce physics-aware Anderson acceleration (PA-AA), which minimizes the physical residual rather than the fixed-point update. Numerical experiments confirm that PA-AA restores reliable convergence in substantially fewer iterations. These findings provide a concrete answer to ongoing controversies surrounding AI-based PDE solvers: reliability hinges not only on architectures but on physically informed training and iteration design.

Victorita Dolean, Serge Gratton, Alexander Heinlein et al.

This study presents a two-level Deep Domain Decomposition Method (Deep-DDM) augmented with a coarse-level network for solving boundary value problems using physics-informed neural networks (PINNs). The addition of the coarse level network improves scalability and convergence rates compared to the single level method. Tested on a Poisson equation with Dirichlet boundary conditions, the two-level deep DDM demonstrates superior performance, maintaining efficient convergence regardless of the number of subdomains. This advance provides a more scalable and effective approach to solving complex partial differential equations with machine learning.

Coen Visser, Alexander Heinlein, Bianca Giovanardi

Physics-Informed Neural Networks (PINNs) have emerged as a tool for approximating the solution of Partial Differential Equations (PDEs) in both forward and inverse problems. PINNs minimize a loss function which includes the PDE residual determined for a set of collocation points. Previous work has shown that the number and distribution of these collocation points have a significant influence on the accuracy of the PINN solution. Therefore, the effective placement of these collocation points is an active area of research. Specifically, available adaptive collocation point sampling methods have been reported to scale poorly in terms of computational cost when applied to high-dimensional problems. In this work, we address this issue and present the Point Adaptive Collocation Method for Artificial Neural Networks (PACMANN). PACMANN incrementally moves collocation points toward regions of higher residuals using gradient-based optimization algorithms guided by the gradient of the PINN loss function, that is, the squared PDE residual. We apply PACMANN for forward and inverse problems, and demonstrate that this method matches the performance of state-of-the-art methods in terms of the accuracy/efficiency tradeoff for the low-dimensional problems, while outperforming available approaches for high-dimensional problems. Key features of the method include its low computational cost and simplicity of integration into existing physics-informed neural network pipelines. The code is available at https://github.com/CoenVisser/PACMANN.

Marc Salvadó-Benasco, Aymane Kssim, Alexander Heinlein et al.

A multi-preconditioned LBFGS (MP-LBFGS) algorithm is introduced for training finite-basis physics-informed neural networks (FBPINNs). The algorithm is motivated by the nonlinear additive Schwarz method and exploits the domain-decomposition-inspired additive architecture of FBPINNs, in which local neural networks are defined on subdomains, thereby localizing the network representation. Parallel, subdomain-local quasi-Newton corrections are then constructed on the corresponding local parts of the architecture. A key feature is a novel nonlinear multi-preconditioning mechanism, in which subdomain corrections are optimally combined through the solution of a low-dimensional subspace minimization problem. Numerical experiments indicate that MP-LBFGS can improve convergence speed, as well as model accuracy over standard LBFGS while incurring lower communication overhead.

Anouk Zandbergen, Tycho van Noorden, Alexander Heinlein

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.

Matthias Mayr, Alexander Heinlein, Christian Glusa et al.

Trilinos is a community-developed, open-source software framework that facilitates building large-scale, complex, multiscale, multiphysics simulation code bases for scientific and engineering problems. Since the Trilinos framework has undergone substantial changes to support new applications and new hardware architectures, this document is an update to ``An Overview of the Trilinos project'' by Heroux et al. (ACM Transactions on Mathematical Software, 31(3):397-423, 2005). It describes the design of Trilinos, introduces its new organization in product areas, and highlights established and new features available in Trilinos. Particular focus is put on the modernized software stack based on the Kokkos ecosystem to deliver performance portability across heterogeneous hardware architectures. This paper also outlines the organization of the Trilinos community and the contribution model to help onboard interested users and contributors.

Alexander Heinlein, Taniya Kapoor

Approximating the solutions of boundary value problems governed by partial differential equations with neural networks is challenging, largely due to the difficult training process. This difficulty can be partly explained by the spectral bias, that is, the slower convergence of high-frequency components, and can be mitigated by localizing neural networks via (overlapping) domain decomposition. We combine this localization with the Gauss-Newton method as the optimizer to obtain faster convergence than gradient-based schemes such as Adam; this comes at the cost of solving an ill-conditioned linear system in each iteration. Domain decomposition induces a block-sparse structure in the otherwise dense Gauss-Newton system, reducing the computational cost per iteration. Our numerical results indicate that combining localization and Gauss-Newton optimization is promising for neural network-based solvers for partial differential equations.

Julia Pelzer, Corné Verburg, Alexander Heinlein et al.

Machine learning methods often struggle with real-world applications in science and engineering due to limited or low-quality training data. In this work, the example of groundwater flow with heat transport is considered; this corresponds to an advection-diffusion process under heterogeneous flow conditions, that is, spatially distributed material parameters and heat sources. Classical numerical simulations are costly and challenging due to high spatio-temporal resolution requirements and large domains. While often computationally more efficient, purely data-driven surrogate models face difficulties, particularly in predicting the advection process, which is highly sensitive to input variations and involves long-range spatial interactions. Therefore, in this work, a Local-Global Convolutional Neural Network (LGCNN) approach is introduced. It combines a lightweight numerical surrogate for the transport process (global) with convolutional neural networks for the groundwater velocity and heat diffusion processes (local). With the LGCNN, a city-wide subsurface temperature field is modeled, involving a heterogeneous groundwater flow field and one hundred groundwater heat pump injection points forming interacting heat plumes over long distances. The model is first systematically analyzed based on random subsurface input fields. Then, the model is trained on a handful of cut-outs from a real-world subsurface map of the Munich region in Germany, and it scales to larger cut-outs without retraining. All datasets, our code, and trained models are published for reproducibility.

Alexander Heinlein, Johannes Taraz

Operator learning has the potential to strongly impact scientific computing by learning solution operators for differential equations, potentially accelerating multi-query tasks such as design optimization and uncertainty quantification by orders of magnitude. Despite proven universal approximation properties, deep operator networks (DeepONets) often exhibit limited accuracy and generalization in practice, which hinders their adoption. Understanding these limitations is therefore crucial for further advancing the approach. This work analyzes performance limitations of the classical DeepONet architecture. It is shown that the approximation error is dominated by the branch network when the internal dimension is sufficiently large, and that the learned trunk basis can often be replaced by classical basis functions without a significant impact on performance. To investigate this further, a modified DeepONet is constructed in which the trunk network is replaced by the left singular vectors of the training solution matrix. This modification yields several key insights. First, a spectral bias in the branch network is observed, with coefficients of dominant, low-frequency modes learned more effectively. Second, due to singular-value scaling of the branch coefficients, the overall branch error is dominated by modes with intermediate singular values rather than the smallest ones. Third, using a shared branch network for all mode coefficients, as in the standard architecture, improves generalization of small modes compared to a stacked architecture in which coefficients are computed separately. Finally, strong and detrimental coupling between modes in parameter space is identified.

Agatha Schmidt, Henrik Zunker, Alexander Heinlein et al.

During the COVID-19 crisis, mechanistic models have guided evidence-based decision making. However, time-critical decisions in a dynamical environment limit the time available to gather supporting evidence. We address this bottleneck by developing a graph neural network (GNN) surrogate of a spatially and demographically resolved mechanistic metapopulation simulator. This combined approach advances classical machine learning approaches which are often black box. Our design of experiments spans outbreak and persistent-threat regimes, up to three contact change points, and age-structured contact matrices on a 400-node spatial graph. We benchmark multiple GNN layers and identify an ARMAConv-based architecture that offers a strong accuracy-runtime trade-off. Across 30-90 day horizons and up to three contact change points, the surrogate attains 10-27 % mean absolute percentage error (MAPE) while delivering (near) constant runtime with respect to the forecast horizon. Our approach accelerates evaluation by up to 28,670 times compared with the mechanistic model, allowing responsive decision support in time-critical scenarios and straightforward web integration. These results show how GNN surrogates can translate complex metapopulation models into immediate, reliable tools for pandemic response.

Alexander Heinlein, Amanda A. Howard, Damien Beecroft et al.

Multiscale problems are challenging for neural network-based discretizations of differential equations, such as physics-informed neural networks (PINNs). This can be (partly) attributed to the so-called spectral bias of neural networks. To improve the performance of PINNs for time-dependent problems, a combination of multifidelity stacking PINNs and domain decomposition-based finite basis PINNs is employed. In particular, to learn the high-fidelity part of the multifidelity model, a domain 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 to multifidelity physics-informed deep operator networks.

Amirhossein Mollaali, Christian Bolivar Moya, Amanda A. Howard et al.

This paper explores uncertainty quantification (UQ) methods in the context of Kolmogorov-Arnold Networks (KANs). We apply an ensemble approach to KANs to obtain a heuristic measure of UQ, enhancing interpretability and robustness in modeling complex functions. Building on this, we introduce Conformalized-KANs, which integrate conformal prediction, a distribution-free UQ technique, with KAN ensembles to generate calibrated prediction intervals with guaranteed coverage. Extensive numerical experiments are conducted to evaluate the effectiveness of these methods, focusing particularly on the robustness and accuracy of the prediction intervals under various hyperparameter settings. We show that the conformal KAN predictions can be applied to recent extensions of KANs, including Finite Basis KANs (FBKANs) and multifideilty KANs (MFKANs). The results demonstrate the potential of our approaches to improve the reliability and applicability of KANs in scientific machine learning.

Selma Husanovic, Ginger Egberts, Alexander Heinlein et al.

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. This study explores the use of a deep operator network (DeepONet), a type of neural operator, as a surrogate model for finite element simulations, aimed at predicting post-burn contraction across multiple wound shapes. A DeepONet was trained on three distinct initial wound shapes, with enhancement made to the architecture by incorporating initial wound shape information and applying sine augmentation to enforce boundary conditions. The performance of the trained DeepONet was evaluated on a test set including finite element simulations based on convex combinations of the three basic wound shapes. The model achieved an $R^2$ 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 CPU and 235-fold on GPU, compared to the numerical model. These findings suggest that DeepONets can effectively serve as a surrogate for traditional finite element methods in simulating post-burn wound evolution, with potential applications in medical treatment planning.

Amanda A. Howard, Bruno Jacob, Sarah Helfert et al.

Kolmogorov-Arnold networks (KANs) have attracted attention recently as an alternative to multilayer perceptrons (MLPs) for scientific machine learning. However, KANs can be expensive to train, even for relatively small networks. Inspired by finite basis physics-informed neural networks (FBPINNs), in this work, we develop a domain decomposition method for KANs that allows for several small KANs to be trained in parallel to give accurate solutions for multiscale problems. We show that finite basis KANs (FBKANs) can provide accurate results with noisy data and for physics-informed training.

Svenja Ehlers, Marco Klein, Alexander Heinlein et al.

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.