FNO∠θ: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems
Authors: Zhexian Li, Ketan Savla
Organizations: Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA. · Sonny Astani Department of Civil and Environmental Engineering, the Daniel Epstein Department of Industrial and Systems Engineering, and the Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA.
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.
Figures & tables
Fig. 1 : Comparison of relative mean squared errors (MSE) over all training data during the training phase of FNO and FNO ∠θ for the Burgers’ equation ( 16 ). FNO ∠θ achieves order of magnitude improvement over FNO.
Fig. 2 : Learning state operator: comparison of the absolute error between the neural operator output and the training data for a particular boundary condition. The error for FNO is concentrated on the boundaries, whereas FNO ∠θ generates more accurate boundary values.
Fig. 3 : Learning optimal control operator: comparison of the absolute error between the neural operator output and the training data for a particular boundary condition. The error for FNO is scattered across the domain, whereas the error for FNO ∠θ is mostly smaller.