Paper Overview
Field: 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 enables 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: an initial short pretraining phase using fixed bases to select a physically relevant wavelet family, followed by an adaptive refinement phase that adjusts scale and translation without padding the entire domain with high-resolution bases. The authors evaluated AW-PINN on multiple challenging PDEs with localized high-magnitude source terms and extreme loss imbalance ratios (up to 10^10:1).
Key Contributions
- Adaptive wavelet basis functions tuned via residual and supervised losses
- Memory-efficient handling of high-scale features
- Training speedup by avoiding automatic differentiation in the loss
- Two-stage design: wavelet family selection, then adaptive refinement
- Validation on PDEs with loss imbalance ratios up to 10^10:1