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AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decomposition

Jonah Botvinick-Greenhouse, Wael H. Ali, Mouhacine Benosman, Saviz Mowlavi

TL;DR

AB-PINNs address the challenge of solving multiscale PDEs with physics-informed neural networks by introducing adaptive, learnable subdomains defined via radial-basis-function windows and coordinated normalization. The framework adds subdomains in regions of high PDE residual (AB-PINN+), and includes a global network to learn low-frequency content, with periodicity enforced through Fourier embeddings. Across a suite of multiscale problems, AB-PINNs and AB-PINN+ demonstrate faster convergence and higher accuracy than static FBPINNs and standard PINNs, reducing reliance on extensive hyperparameter tuning. This dynamic, residual-driven domain decomposition advances PINN scalability and robustness for complex PDEs in engineering and physics.

Abstract

We introduce adaptive-basis physics-informed neural networks (AB-PINNs), a novel approach to domain decomposition for training PINNs in which existing subdomains dynamically adapt to the intrinsic features of the unknown solution. Drawing inspiration from classical mesh refinement techniques, we also modify the domain decomposition on-the-fly throughout training by introducing new subdomains in regions of high residual loss, thereby providing additional expressive power where the solution of the differential equation is challenging to represent. Our flexible approach to domain decomposition is well-suited for multiscale problems, as different subdomains can learn to capture different scales of the underlying solution. Moreover, the ability to introduce new subdomains during training helps prevent convergence to unwanted local minima and can reduce the need for extensive hyperparameter tuning compared to static domain decomposition approaches. Throughout, we present comprehensive numerical results which demonstrate the effectiveness of AB-PINNs at solving a variety of complex multiscale partial differential equations.

AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decomposition

TL;DR

AB-PINNs address the challenge of solving multiscale PDEs with physics-informed neural networks by introducing adaptive, learnable subdomains defined via radial-basis-function windows and coordinated normalization. The framework adds subdomains in regions of high PDE residual (AB-PINN+), and includes a global network to learn low-frequency content, with periodicity enforced through Fourier embeddings. Across a suite of multiscale problems, AB-PINNs and AB-PINN+ demonstrate faster convergence and higher accuracy than static FBPINNs and standard PINNs, reducing reliance on extensive hyperparameter tuning. This dynamic, residual-driven domain decomposition advances PINN scalability and robustness for complex PDEs in engineering and physics.

Abstract

We introduce adaptive-basis physics-informed neural networks (AB-PINNs), a novel approach to domain decomposition for training PINNs in which existing subdomains dynamically adapt to the intrinsic features of the unknown solution. Drawing inspiration from classical mesh refinement techniques, we also modify the domain decomposition on-the-fly throughout training by introducing new subdomains in regions of high residual loss, thereby providing additional expressive power where the solution of the differential equation is challenging to represent. Our flexible approach to domain decomposition is well-suited for multiscale problems, as different subdomains can learn to capture different scales of the underlying solution. Moreover, the ability to introduce new subdomains during training helps prevent convergence to unwanted local minima and can reduce the need for extensive hyperparameter tuning compared to static domain decomposition approaches. Throughout, we present comprehensive numerical results which demonstrate the effectiveness of AB-PINNs at solving a variety of complex multiscale partial differential equations.

Paper Structure

This paper contains 24 sections, 25 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Physics-Informed Neural Networks (PINNs)
  4. Adaptive Sampling and Reweighted Residuals
  5. Finite-Basis Physics Informed Neural Networks
  6. Modified Network Architecture
  7. Related Work
  8. Adaptive-Basis Physics-Informed Neural Networks (AB-PINNs)
  9. Learnable Subdomains
  10. Residual-Based Subdomain Addition
  11. Enforcing Periodicity in AB-PINNs
  12. Additional Implementation Details
  13. Numerical Experiments
  14. Overview of Numerical Experiments
  15. Local Adaptivity of Subdomains
  16. ...and 9 more sections

Figures (10)

  • Figure 1: Reference solutions for the test cases studied in this paper.
  • Figure 2: Solving a multiscale chirp waveform using a standard PINN, an FBPINN and an AB-PINN. In the second column the standard PINN solution is shown in yellow, while the FBPINN solution is plotted in red, the AB-PINN in blue, and the ground truth in black.
  • Figure 3: Solving a uniform sine wave using a standard PINN, an FBPINN, and an AB-PINN.
  • Figure 4: Studying the effect of the choice of reference window function on the performance of AB-PINNs for a multiscale chirp problem.
  • Figure 5: Solving the advection equation using PINNs, FBPINNs, and AB-PINNs.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Remark 1