# An efficient wavelet-based physics-informed neural network for multiscale problems

arXiv: https://arxiv.org/abs/2409.11847 · HTML: https://mlsanity.com/paper/2409.11847/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 44 references that are in the map.
- First posted on arXiv 18 Sep 2024.

## Builds on (most cited first, 16 of 16)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Towards Understanding the Spectral Bias of Deep Learning](https://mlsanity.com/paper/1912.01198/) (2019)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations](https://mlsanity.com/paper/2205.09332/) (2022)
- [Stochastic projection based approach for gradient free physics informed learning](https://mlsanity.com/paper/2209.13724/) (2022)
- [A practical PINN framework for multi-scale problems with multi-magnitude loss terms](https://mlsanity.com/paper/2308.06672/) (2023)
- [Physics-informed neural wavefields with Gabor basis functions](https://mlsanity.com/paper/2310.10602/) (2023)

## Built on by (most cited first, 2 of 2)

- [A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization](https://mlsanity.com/paper/2604.21411/) (2026)
- [Conditional Flow Matching for Single-Neuron Electrophysiology: Capturing Multimodal Responses Across Stimuli](https://mlsanity.com/paper/2610.06520/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) are a class of deep learning models that utilize physics in the form of differential equations to address complex problems, including those with limited data availability. However, solving differential equations with rapid oscillations, steep gradients, or singular behavior remains challenging for PINNs. To address this, we propose an efficient wavelet-based physics-informed neural network (W-PINN) that learns solutions in wavelet space. Here, we represent the solution using localized wavelets. This framework represents the solution of a differential equation with significantly fewer degrees of freedom while retaining the dynamics of complex physical phenomena. The proposed architecture enables training to search for solutions within the wavelet domain, where multiscale characteristics are less pronounced compared to the physical domain. This facilitates more efficient training for such problems. Furthermore, the proposed model does not rely on automatic differentiation for derivatives in the loss function and does not require prior information regarding the behavior of the solution, such as the location of abrupt features. The removal of AD significantly reduces training time while maintaining accuracy. Thus, through a strategic fusion of wavelets with PINNs, W-PINNs capture localized nonlinear information, making them well-suited for problems with abrupt behavior, such as singularly perturbed and other multiscale problems. We further analyze the convergence behavior of W-PINN through a comparative study using Neural Tangent Kernel theory. The efficiency and accuracy of the proposed model are demonstrated across various problems, including the FitzHugh--Nagumo (FHN) model, Helmholtz equation, Maxwell equation, Allen--Cahn equation, and lid-driven cavity flow, along with other highly singularly perturbed nonlinear differential equations.