FNO$^{\angle \theta}$: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems

TL;DR

This paper introduces an extended Fourier neural operator that incorporates complex frequency variables to better learn state and optimal control in PDE-governed systems, showing significant accuracy improvements.

Contribution

The authors extend the FNO architecture by integrating complex frequency variables based on the Ehrenpreis-Palamodov principle, enabling more accurate control learning for PDE systems.

Findings

Order of magnitude reduction in training errors.

More accurate predictions of non-periodic boundary values.

Effective learning of control in nonlinear PDEs.

Abstract

We propose an extended Fourier neural operator (FNO) architecture for learning state and linear quadratic additive optimal control of systems governed by partial differential equations. Using the Ehrenpreis-Palamodov fundamental principle, we show that any state and optimal control of linear PDEs with constant coefficients can be represented as an integral in the complex domain. The integrand of this representation involves the same exponential term as in the inverse Fourier transform, where the latter is used to represent the convolution operator in FNO layer. Motivated by this observation, we modify the FNO layer by extending the frequency variable in the inverse Fourier transform from the real to complex domain to capture the integral representation from the fundamental principle. We illustrate the performance of FNO in learning state and optimal control for the nonlinear Burgers' equation, showing order of magnitude improvements in training errors and more accurate predictions of non-periodic boundary values over FNO.