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

Taiki Miyagawa, Takeru Yokota

Keywords: Physics-informed neural network, Functional differential equation, Functional derivative

TL;DR: We propose the first learning scheme to solve functional differential equations, built on the physics-informed neural network.

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 *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.

Primary Area: Machine learning for physical sciences (for example: climate, physics)

Submission Number: 2487