PACMANN: Point Adaptive Collocation Method for Artificial Neural Networks

Physics-Informed Neural Networks (PINNs) are an emerging 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, adaptive collocation point sampling methods have been proposed, which have been reported to scale poorly to higher dimensions. In this work, we address this issue and present the Point Adaptive Collocation Method for Artificial Neural Networks (PACMANN). Inspired by classic optimization problems, this approach incrementally moves collocation points toward regions of higher residuals using gradient-based optimization algorithms guided by the gradient of the squared 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; the best performance is observed for the Adam optimizer. Key features of the method include its low computational cost and simplicity of integration in existing physics-informed neural network pipelines.

Deep operator network models for predicting post-burn contraction

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.

Towards Graph Neural Network Surrogates Leveraging Mechanistic Expert Knowledge for Pandemic Response

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.

Towards Model Discovery Using Domain Decomposition and PINNs

We enhance machine learning algorithms for learning model parameters in complex systems represented by ordinary differential equations (ODEs) 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, we find a better performance for the FBPINN approach compared to the vanilla PINN approach, even in cases with data from only a quasi-stationary time domain with few dynamics.

Two-level deep domain decomposition method

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.

DDU-Net: A Domain Decomposition-based CNN for High-Resolution Image Segmentation on Multiple GPUs

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/HiRes-Seg-CNN.

Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems

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.

Multifidelity domain decomposition-based physics-informed neural networks for time-dependent problems

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 are 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.

Improving Pseudo-Time Stepping Convergence for CFD Simulations With Neural Networks

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.

Learning the solution operator of two-dimensional incompressible Navier-Stokes equations using physics-aware convolutional neural networks

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.