This study presents a novel strategy for constructing an approximator for arbitrary univariate functions. The proposed approximation utilizes the anti-derivatives of a Fourier series expansion for the presumed piecewise function, resulting in a remarkable feature that enables the simultaneous approximation of an arbitrary function and its anti-derivatives. These anti-derivatives can be employed to discover solution curves for systems of ordinary differential equations based on an optimization scheme, even in the presence of chaotic dynamics. Additionally, the anti-derivatives approximator is extended as an adaptive activation function for physics-informed neural networks, leveraging the high-order differentiability of the anti-derivatives. Systematic experiments have demonstrated the outstanding merits of the proposed anti-derivatives-based approximator, including its ability to construct regression models for volatile data and their anti-derivatives, solve differential equations, and enhance the capabilities of physics-informed neural networks.

中文翻译：

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用于增强物理信息神经网络的反导数逼近器

这项研究提出了一种为任意单变量函数构建逼近器的新策略。所提出的近似利用傅里叶级数展开的反导数来表示假定的分段函数，从而产生了一个显着的特征，可以同时逼近任意函数及其反导数。即使存在混沌动力学，这些反导数也可用于基于优化方案发现常微分方程组的解曲线。此外，反导数近似器被扩展为物理信息神经网络的自适应激活函数，利用反导数的高阶可微性。系统实验证明了所提出的基于反导数的逼近器的突出优点，包括其能够为易失性数据及其反导数构建回归模型、求解微分方程以及增强物理信息神经网络的能力。