Alexander Heinlein

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.

The approach of using physics-based machine learning to solve PDEs has recently become very popular. A recent approach to solve PDEs based on CNNs uses finite difference stencils to include the residual of the partial differential equation into the loss function. In this work, the relation between the network training and the solution of a respective finite difference linear system of equations using classical numerical solvers is discussed. It turns out that many beneficial properties of the linear equation system are neglected in the network training. Finally, numerical results which underline the benefits of classical numerical solvers are presented.

Physics-informed neural networks (PINNs) [4, 10] are an approach for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to use a neural network to approximate the solution to the PDE and to incorporate the residual of the PDE as well as boundary conditions into its loss function when training it. This provides a simple and mesh-free approach for solving problems relating to PDEs. However, a key limitation of PINNs is their lack of accuracy and efficiency when solving problems with larger domains and more complex, multi-scale solutions. In a more recent approach, finite basis physics-informed neural networks (FBPINNs) [8] use ideas from domain decomposition to accelerate the learning process of PINNs and improve their accuracy. In this work, we show how Schwarz-like additive, multiplicative, and hybrid iteration methods for training FBPINNs can be developed. We present numerical experiments on the influence of these different training strategies on convergence and accuracy. Furthermore, we propose and evaluate a preliminary implementation of coarse space correction for FBPINNs.

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.

For the robustness and numerical scalability of domain decomposition-based linear solvers, the incorporation of a coarse level, which provides global transport of information, is crucial. State-of-the-art spectral, or adaptive, methods can generate the basis functions of the coarse space, which are adapted to the specific properties of the target problem, and yield provably robust convergence for certain classes of problems. However, their construction is computationally expensive and requires non-algebraic information. To improve the practicability of the solver, in this paper, we design a hierarchical math-informed local Graph Neural Network (GNN) to generate effective coarse-space basis functions. Our GNN uses only the local subdomain matrices available as the input to the algebraic linear solvers. This approach has several advantages including: 1) it is algebraic; 2) it is local and therefore as scalable as the classical domain decomposition solvers; and 3) the cost for training, inference, and generating data sets is much lower than that needed for approaches relying on the global matrix. To study the potential of our GNN architecture, we present numerical results with homogeneous and heterogeneous 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.

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.