Paper Overview
Research Area: ML/Scientific Authors: Sai Munikoti, Ivan Vinogradov, Ksenia Pirozhenko et al. Published: 2026-04-30 arXiv: 2604.28180
Summary
In recent years, physics-informed neural networks (PINNs) have gained significant attention for solving differential equations, although they suffer from two fundamental limitations: spectral bias inherent in neural networks and loss imbalance arising from multiscale phenomena. This paper proposes an adaptive wavelet-based PINN (AW-PINN) to address the extreme loss imbalance characteristic of problems with localized high-magnitude source terms. Such problems frequently arise in various physical applications, including heat treatment, electromagnetics, shock mechanics, and fluid dynamics involving localized forcing.
The proposed framework dynamically adjusts wavelet basis functions based on residual and supervised losses. This adaptive property allows AW-PINN to effectively handle problems with high-scale features without requiring large amounts of memory. In addition, AW-PINN does not rely on automatic differentiation to obtain derivatives in the loss function, which accelerates the training process.
The method operates in two stages:
1. Pretraining stage: a short initial phase using fixed bases to select the physically relevant wavelet family. 2. Adaptive refinement stage: adjusting scales and translations without filling the entire domain with high-resolution bases.
AW-PINN was evaluated on multiple challenging partial differential equations with localized high-magnitude source terms and extreme loss imbalance ratios of up to 10^10:1.
--- *Auto-collected on 2026-05-02*