提出自适应小波激活函数，提升物理信息神经网络的训练稳定性和精度。
A Family of Adaptive Activation Functions for Mitigating Failure Modes in Physics-Informed Neural Networks
- 用可学习的小波函数结合双曲正切或软指数函数构建新激活函数。
- 在四类偏微分方程上测试，比传统激活函数更鲁棒、更准确。
- 适合需要高稳定性与精确求解的科学计算场景，如流体模拟、结构力学。
物理信息神经网络（PINNs）是一种强大且灵活的学习框架，在近年受到广泛关注，并在众多科学与工程问题中表现出色。与此同时，小波因其出色的逼近能力被广泛用作高效的计算工具。针对标准PINNs中常见的失效模式，本文提出一类新型自适应小波基激活函数。该方法通过将可训练的小波函数与可训练或固定的双曲正切、软指数函数相结合，显著提升了训练稳定性与模型表达能力。在PINN框架内共设计五种不同的激活函数，并在四类典型的偏微分方程（PDEs）上进行系统评估。通过柱状图对比显示，新方法在鲁棒性与精度方面均优于传统激活函数。此外，还通过与基准PINNs、基于Transformer的架构PINNsFormer及其他深度学习模型的直接比较，验证了该方法的有效性与通用性。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks(PINNs) are a powerful and flexible learning framework that has gained significant attention in recent years. It has demonstrated strong performance across a wide range of scientific and engineering problems. In parallel, wavelets have been extensively used as efficient computational tools due to their strong approximation capabilities. Motivated by the common failure modes observed in standard PINNs, this work introduces a novel family of adaptive wavelet-based activation functions. The proposed activation functions significantly improve training stability and expressive power by combining trainable wavelet functions with either trainable or fixed hyperbolic tangent and softplus functions. Five distinct activation functions are developed within the PINN framework and systematically evaluated across four representative classes of partial differential equations (PDEs). Comprehensive comparisons using bar plots demonstrate improved robustness and accuracy compared to traditional activation functions. Furthermore, the proposed approach is validated through direct comparisons with baseline PINNs, transformer-based architectures such as PINNsFormer, and other deep learning models, highlighting its effectiveness and generality.
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