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}.

本研究解决了泛函微分方程（FDEs）在数值分析中的高计算成本与过度简化的问题。通过结合物理启发的神经网络与圆柱近似，提出了一种新型混合方法，证明了近似形式的收敛性，并显示该方法在处理复杂泛函导数方面比传统方法更高效。实验结果表明，所提出模型在典型的$L^1$相对误差达到$\sim 10^{-3}$，为FDEs的数值分析提供了重要支持。

物理启发的神经网络在泛函微分方程中的应用：圆柱近似及其收敛保证