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.

翻译：近年来，物理信息神经网络（PINNs）在求解微分方程方面受到广泛关注，尽管它们存在两个基本局限，即神经网络固有的频谱偏差和多尺度现象导致的损失不平衡。本文提出一种自适应小波PINN（AW-PINN）来解决具有局部高幅值源项问题的极端损失不平衡特征。这类问题频繁出现在各种物理应用中，例如热处理、电磁学、冲击力学以及涉及局部强迫的流体动力学。所提出的框架基于残差和监督损失动态调整小波基函数。这种自适应特性使得AW-PINN能够有效处理具有高尺度特征的问题，而无需占用大量内存。此外，AW-PINN不依赖自动微分来获取损失函数中涉及的导数，从而加速了训练过程。该方法分为两个阶段：首先进行一个短暂的预训练阶段，使用固定基函数选择物理相关的小波族；随后进行自适应细化，通过调整尺度和平移来避免在整个域中填充高分辨率基函数。理论上，我们证明在一定假设下，AW-PINN允许高斯过程极限，并推导其相关的NTK结构。我们在几个具有局部高幅值源项且极端损失不平衡比高达$10^{10}:1$的挑战性偏微分方程上评估了AW-PINN。在这些偏微分方程中，包括瞬态热传导、高度局部化的泊松问题、振荡流方程以及带点电荷源的麦克斯韦方程，AW-PINN始终优于同类现有方法。