arXiv subjects

Sarah Helfert

Publications and source records attributed to Sarah Helfert.

3 recordsLinked to original sources

arXiv2026

Adaptive Domain Decomposition for Finite Basis Physics-Informed Neural Networks

In this work, we present adaptive domain decomposition finite basis physics-informed neural networks (ADD-FBPINNs). The proposed method integrates Partition of Unity Networks (POUNets), a data-driven approach for learning domain partitions, with finite Basis physics-informed neural networks (FBPINNs) to perform more efficient domain decomposition and achieve more accurate approximations. FBPINNs improve upon the accuracy of standard physics-informed neural networks (PINNs) through user-defined subdivisions of the computational domain. By adapting the domain decomposition based on the solution's features using a separable POUNet, the framework can automatically learn partitions that are stable and can be geometrically interpretable. We test this approach on the two-dimensional Helmholtz equation, two-dimensional Poisson equation with multiscale features, and viscous Burgers equation and observe improved accuracy compared with FBPINNs with fixed domain decomposition.

cs.LG↗

arXiv2026

Numerical stability revisited: A family of benchmark problems for the analysis of explicit stochastic differential equation integrators

We revisit the numerical stability of four well-established explicit stochastic integration schemes through a new generic benchmark stochastic differential equation designed to assess asymptotic statistical accuracy and stability properties. This one-parameter benchmark equation is derived from a general one-dimensional first-order SDE using spatio-temporal nondimensionalization and is employed to evaluate the performance of the (1) Euler-Maruyama, (2) Milstein, (3) Stochastic Heun, and (4) three-stage Runge-Kutta schemes. Our findings reveal that lower-order schemes can outperform higher-order ones over a range of time step sizes, depending on the benchmark parameters and application context. The theoretical results are validated through a series of numerical experiments, and we discuss their implications for more general applications, including a nonlinear example. Our results suggest that the insights obtained from the linear benchmark problem provide reliable guidance for time-stepping strategies when simulating nonlinear SDEs.

math.NA↗

arXiv2025

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.

cs.LG↗