An adaptive wavelet-based PINN for problems with localized high-magnitude source
2026-04-30 • Machine Learning
Machine Learning
AI summaryⓘ
The authors developed a new version of physics-informed neural networks called AW-PINN to better solve differential equations that have very uneven challenges, especially when parts of the problem have very large, localized forces. Their method adapts wavelet functions during training to focus computational effort where it's needed and speeds up the process by avoiding some costly calculations. They first do a quick setup to pick the best wavelets and then fine-tune these adaptively to handle tricky areas without wasting resources. Tests on various difficult equations showed that their approach worked better than other similar methods.
Physics-Informed Neural NetworksWaveletsSpectral BiasLoss ImbalancePartial Differential EquationsAdaptive MethodsGaussian ProcessesNeural Tangent KernelLocalized Source TermsAutomatic Differentiation
Authors
Himanshu Pandey, Ratikanta Behera
Abstract
In recent years, physics-informed neural networks (PINNs) have gained significant attention for solving differential equations, although they suffer from two fundamental limitations, namely, 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, such as thermal processing, electro-magnetics, impact mechanics, and fluid dynamics involving localized forcing. The proposed framework dynamically adjusts the wavelet basis function based on residual and supervised loss. This adaptive nature makes AW-PINN handle problems with high-scale features effectively without being memory-intensive. Additionally, AW-PINN does not rely on automatic differentiation to obtain derivatives involved in the loss function, which accelerates the training process. The method operates in two stages, an initial short pre-training phase with fixed bases to select physically relevant wavelet families, followed by an adaptive refinement that adapts scales and translations without populating high-resolution bases across entire domains. Theoretically, we show that under certain assumptions, AW-PINN admits a Gaussian process limit and derive its associated NTK structure. We evaluate AW-PINN on several challenging PDEs featuring localized high-magnitude source terms with extreme loss imbalances having ratios up to $10^{10}:1$. Across these PDEs, including transient heat conduction, highly localized Poisson problems, oscillatory flow equations, and Maxwell equations with a point charge source, AW-PINN consistently outperforms existing methods in its class.