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

• URL: http://arxiv.org/abs/2410.18153v1
• Date: Wed, 23 Oct 2024 06:16:35 GMT
• Title: Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guarantees
• Authors: Taiki Miyagawa, Takeru Yokota,
• Abstract summary: We propose the first learning scheme for functional differential equations (FDEs) FDEs play a fundamental role in physics, mathematics, and optimal control. numerical approximations of FDEs have been developed, but they often oversimplify the solutions.
• Score: 7.366405857677226
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• Abstract: 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}.

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