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Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guarantees

23 Oct 2024arXiv:2410.18153archive 2025-07-28

Taiki Miyagawa, Takeru Yokota

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 L¹ relative error orders of PINNs ∼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}.

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