We propose the first learning scheme for functional differential equations (FDEs). FDEs play a fundamental role in physics, mathematics, and optimal control. However, the numerical analysis of FDEs has faced challenges due to its unrealistic computational costs and has been a long standing problem over decades. Thus, numerical approximations of FDEs have been developed, but they often oversimplify the solutions. To tackle these two issues, we propose a hybrid approach combining physics-informed neural networks (PINNs) with the \textit{cylindrical approximation}. The cylindrical approximation expands functions and functional derivatives with an orthonormal basis and transforms FDEs into high-dimensional PDEs. To validate the reliability of the cylindrical approximation for FDE applications, we prove the convergence theorems of approximated functional derivatives and solutions. Then, the derived high-dimensional PDEs are numerically solved with PINNs. Through the capabilities of PINNs, our approach can handle a broader class of functional derivatives more efficiently than conventional discretization-based methods, improving the scalability of the cylindrical approximation. As a proof of concept, we conduct experiments on two FDEs and demonstrate that our model can successfully achieve typical $L^1$ relative error orders of PINNs $\sim 10^{-3}$. Overall, our work provides a strong backbone for physicists, mathematicians, and machine learning experts to analyze previously challenging FDEs, thereby democratizing their numerical analysis, which has received limited attention. Code is available at \url{https://github.com/TaikiMiyagawa/FunctionalPINN}.

翻译：我们提出了首个针对函数微分方程（FDEs）的学习方案。FDEs在物理学、数学和最优控制中扮演着基础性角色。然而，由于其不切实际的计算成本，FDEs的数值分析一直面临挑战，并成为数十年来的长期难题。因此，人们开发了FDEs的数值近似方法，但这些方法往往过度简化了解。为解决这两个问题，我们提出了一种混合方法，将物理信息神经网络（PINNs）与\textit{柱面近似}相结合。柱面近似利用正交基展开函数和函数导数，并将FDEs转化为高维偏微分方程（PDEs）。为验证柱面近似在FDE应用中的可靠性，我们证明了近似函数导数和解的收敛定理。随后，通过PINNs对导出的高维PDEs进行数值求解。借助PINNs的能力，我们的方法能够比传统的基于离散化的方法更高效地处理更广泛的函数导数类别，从而提升了柱面近似的可扩展性。作为概念验证，我们在两个FDEs上进行了实验，并证明我们的模型能够成功达到PINNs典型的$L^1$相对误差量级$\sim 10^{-3}$。总体而言，我们的工作为物理学家、数学家和机器学习专家分析先前具有挑战性的FDEs提供了坚实的支撑，从而推动了其数值分析的普及，而这一领域此前受到的关注有限。代码可在\url{https://github.com/TaikiMiyagawa/FunctionalPINN}获取。