Alexander Heinlein

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 to several forward and inverse problems, including one with a low-regularity solution and 3D Navier Stokes, 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.

During the COVID-19 crisis, mechanistic models have been proven fundamental to guide evidence-based decision making. However, time-critical decisions in a dynamically changing environment restrict the time available for modelers to gather supporting evidence. As infectious disease dynamics are often heterogeneous on a spatial or demographic scale, models should be resolved accordingly. In addition, with a large number of potential interventions, all scenarios can barely be computed on time, even when using supercomputing facilities. We suggest to combine complex mechanistic models with data-driven surrogate models to allow for on-the-fly model adaptations by public health experts. We build upon a spatially and demographically resolved infectious disease model and train a graph neural network for data sets representing early phases of the pandemic. The resulting networks reached an execution time of less than a second, a significant speedup compared to the metapopulation approach. The suggested approach yields potential for on-the-fly execution and, thus, integration of disease dynamics models in low-barrier website applications. For the approach to be used with decision-making, datasets with larger variance will have to be considered.

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.

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.

We present a new iteration bound for the preconditioned conjugate gradient (PCG) method that accurately captures convergence for systems with clustered eigenspectra, where the classical condition number-based bound is too pessimistic. By using the extremal values of each subcluster in the spectral distribution, our bound is orders of magnitude sharper than the classical bound. We demonstrate its effectiveness on a high-contrast elliptic PDE preconditioned with a two-level overlapping Schwarz method, where it successfully distinguishes between the performance of different (algebraic) preconditioners. A key contribution is theoretical evidence that, for certain high-contrast problems, PCG convergence can make simpler coarse spaces competitive. Conversely, more complex preconditioners are not always necessary. Finally, we show that our bound can be estimated effectively from Ritz values computed during early PCG iterations.

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.

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.

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.

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.