Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations

Moseley, B; Markham, A; Nissen-Meyer, T

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Journal article

Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations

Abstract:: Background. Physics-informed neural networks (PINN) demonstrated strong capabilities in solving direct and inverse problems for partial differential equations. In this study, the focus is on applying PINNs for the approximation and extrapolation of narrowband signal propagation. This effort is motivated by the potential to reduce measurement and numerical costs in applications such as acoustic and electromagnetic beacon-based navigation systems. These systems aim to map environments and track object trajectories by leveraging wave propagation data. Materials and Methods. The propagation of harmonic waves through a medium can be described using either the wave equation or the Helmholtz equation. To establish a connection between these equations, the Fourier transform is employed. PINNs are trained in the time or frequency domain to predict wave propagation characteristics such as amplitude and phase. The study compares the performance of PINNs against conventional neural networks. Results and Discussion. The study finds that PINNs exhibit superior performance over conventional neural networks when training data points are separated up to the Nyquist rate. In the time domain, PINNs accurately predict еру phase up to a distance of one cell except for the direction to the source. However, amplitude predictions are less accurate, with errors below 20% up to a distance of 0.5 cells. For larger amplitudes, the model struggles to provide reliable predictions. Training PINNs in the frequency domain requires less computational resources, but performance is worse than in the time domain. Conclusion. PINNs offer promising advantages for modeling wave propagation in narrowband signals, particularly in scenarios where measurement data is sparse or local. They can increase resolution, reduce the volume of required data, and optimize computational efficiency. Despite their limitation, there is a difference in solutions between the time and frequency domains due to the nonlinear nature of NN. Future work could address the accuracy of predictions through better network architectures or hybrid approaches

Publication status:: Published

Peer review status:: Peer reviewed

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APA Style

Moseley, B., Markham, A., & Nissen-Meyer, T. (2023). Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations. Advances in Computational Mathematics, 49(4), 62.

MLA Style

Moseley, B, et al. “Finite Basis Physics-Informed Neural Networks (FBPINNs): a Scalable Domain Decomposition Approach for Solving Differential Equations.” Advances in Computational Mathematics, vol. 49, no. 4, 2023, p. 62.

Chicago Style

Moseley, B, A Markham, and T Nissen-Meyer. 2023. “Finite Basis Physics-Informed Neural Networks (FBPINNs): a Scalable Domain Decomposition Approach for Solving Differential Equations.” Advances in Computational Mathematics 49 (4): 62.
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