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

Alexander Heinlein, Amanda A. Howard, Damien Beecroft, Panos Stinis

TL;DR

This work tackles multiscale time-dependent PDEs by addressing PINN limitations from spectral bias through a novel combination of multifidelity stacking PINNs and domain-decomposition-based finite-basis PINNs (FBPINNs). The method introduces time-domain decomposition with hierarchical, multifidelity networks (stacking FBPINNs) to learn high-frequency multiscale features more efficiently, achieving superior accuracy on challenging tests such as a pendulum, a two-frequency model, and the Allen-Cahn equation. It also extends to operator learning via multifidelity stacking DeepONets (FB-DONs), enabling rapid surrogate solutions for varying initial conditions. The framework demonstrates improved accuracy and efficiency over traditional PINNs and prior stacking approaches, with detailed training parameters and clear potential as surrogate solvers and multi-physics integrations across scales.

Abstract

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.

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

TL;DR

This work tackles multiscale time-dependent PDEs by addressing PINN limitations from spectral bias through a novel combination of multifidelity stacking PINNs and domain-decomposition-based finite-basis PINNs (FBPINNs). The method introduces time-domain decomposition with hierarchical, multifidelity networks (stacking FBPINNs) to learn high-frequency multiscale features more efficiently, achieving superior accuracy on challenging tests such as a pendulum, a two-frequency model, and the Allen-Cahn equation. It also extends to operator learning via multifidelity stacking DeepONets (FB-DONs), enabling rapid surrogate solutions for varying initial conditions. The framework demonstrates improved accuracy and efficiency over traditional PINNs and prior stacking approaches, with detailed training parameters and clear potential as surrogate solvers and multi-physics integrations across scales.

Abstract

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.
Paper Structure (15 sections, 13 equations, 6 figures, 2 tables)

This paper contains 15 sections, 13 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Physics-informed neural networks
  4. Multifidelity stacking PINNs
  5. Domain decomposition-based neural network architectures
  6. Stacking FBPINNs
  7. Domain decomposition in time
  8. Results
  9. Pendulum
  10. Multiscale problem
  11. Allen-Cahn equation
  12. Extension to DeepONets
  13. Discussion
  14. Acknowledgments
  15. Training parameters

Figures (6)

  • Figure 1: Multilevel overlapping domain decomposition of $\Omega$ with $L=3$ levels.
  • Figure 2: Window functions $\omega_j$ for $l = 2$ and $T = 1$.
  • Figure 3: Stacking FBPINN results for the pendulum problem: Left: Stacking FBPINN results for an illustrative example of $s_1$ (top) and $s_2$ (bottom) as a function of time for the pendulum problem up to five stacking FBPINN levels. Right: Pendulum relative $\ell_2$ training errors comparing the work in the current paper (solid line) with the approach from howard_stacked_2023 (dashed lines).
  • Figure 4: Stacking FBPINN results for the multiscale problem: Left: Stacking FBPINN results for the single fidelity level 0 and the first four stacking FBPINN levels. Right: Multiscale relative $\ell_2$ training errors comparing the work in the current paper with howard_stacked_2023.
  • Figure 5: Stacking FBPINN results for the Allen-Cahn equation. Left: Stacking FBPINN results for the single fidelity level 0 and the first two stacking FBPINN levels. Right: Line plots of the results from the stacking FBPINN at $t=0.25$ (top) and $t=0.75$ (bottom).
  • ...and 1 more figures