将傅里叶神经算子扩展至复频域，提升对偏微分方程状态与最优控制的学习精度。
FNO$^{\angle θ}$: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems
- 在反傅里叶变换中引入复频率变量，捕捉PDE解的积分表示机制。
- 在非线性Burgers方程上训练误差降低一个数量级，非周期边界预测更准。
- 适合需高精度求解复杂分布参数系统的研究人员，如控制工程与科学计算领域。
我们提出一种扩展的傅里叶神经算子（FNO）架构，用于学习由偏微分方程（PDE）支配的系统状态及线性二次加性最优控制。基于Ehrenpreis-Palamodov基本原理，我们证明：任意常系数线性PDE的状态与最优控制均可表示为复域上的积分形式，其被积函数包含与反傅里叶变换相同的指数项。受此启发，我们将FNO层中的频率变量从实数域拓展至复数域，以捕获该基本原理中的积分表达式。我们在非线性Burgers方程上验证了该方法性能，结果显示训练误差降低一个数量级，且对非周期边界值的预测更加准确，显著优于原始FNO。
原文摘要 · Abstract (English)
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.
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