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Related Concept Videos

Wave Parameters01:10

Wave Parameters

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The simplest mechanical waves are associated with simple harmonic motion and repeat themselves for several cycles. These simple harmonic waves can be modeled using a combination of sine and cosine functions. Consider a simplified surface water wave that moves across the water's surface. Unlike complex ocean waves, in surface water waves, water moves vertically, oscillating up and down, whereas the disturbance of the wave moves horizontally through the medium. If a seagull is floating on the...
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An efficient wavelet-based physics-informed neural network for multiscale problems.

Himanshu Pandey1, Anshima Singh2, Ratikanta Behera1

  • 1Department of Computational and Data Sciences, Indian Institute of Science, Bangalore, 560012, India.

Neural Networks : the Official Journal of the International Neural Network Society
|March 24, 2026
PubMed
Summary
This summary is machine-generated.

Physics-informed neural networks (PINNs) are enhanced by wavelets (W-PINNs) to efficiently solve complex differential equations. This new approach reduces training time and improves accuracy for problems with abrupt features.

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Area of Science:

  • Computational Science and Engineering
  • Applied Mathematics
  • Artificial Intelligence

Background:

  • Physics-informed neural networks (PINNs) integrate physical laws (differential equations) into deep learning for complex problems, especially with limited data.
  • Standard PINNs struggle with differential equations exhibiting rapid oscillations, steep gradients, or singular behavior, limiting their applicability.
  • Existing methods often require automatic differentiation (AD) and prior knowledge of solution behavior, increasing computational cost and complexity.

Purpose of the Study:

  • To introduce an efficient wavelet-based physics-informed neural network (W-PINN) for solving challenging differential equations.
  • To enhance the training efficiency and accuracy of PINNs for problems with multiscale characteristics and abrupt features.
  • To develop a PINN framework that avoids reliance on automatic differentiation and prior solution information.

Main Methods:

  • Representing the differential equation's solution in wavelet space using a family of localized wavelets.
  • Developing a W-PINN architecture that enables training within the wavelet domain, mitigating pronounced multiscale characteristics.
  • Eliminating the need for automatic differentiation (AD) in the loss function calculation and prior knowledge of solution behavior.

Main Results:

  • W-PINNs significantly reduce the degrees of freedom required to represent complex solutions while preserving essential dynamics.
  • Training within the wavelet domain proves more efficient, especially for problems with localized non-linear information and abrupt features.
  • The W-PINN model demonstrates comparable or improved accuracy and significantly reduced training times compared to standard PINNs, validated across diverse problems.

Conclusions:

  • The proposed W-PINN framework offers an efficient and accurate solution for a wide range of differential equations, particularly those with multiscale and singularly perturbed behaviors.
  • By leveraging wavelets and avoiding AD, W-PINNs overcome key limitations of traditional PINNs, enhancing their practical applicability.
  • The study confirms the efficacy of W-PINNs through theoretical analysis (NTK) and practical demonstrations on benchmark problems like FHN, Helmholtz, Maxwell, and Allen-Cahn equations.