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

arXiv:2604.05187v1 Announce Type: new 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 t

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Zhexian Li, Ketan Savla · · 1 min read · 16 views
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arXiv:2604.05187v1 Announce Type: new 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.

Executive Summary

The article introduces an innovative extension of the Fourier Neural Operator (FNO) architecture, termed FNO$^{\angle \theta}$, designed to enhance the learning of state and optimal control for distributed parameter systems governed by partial differential equations (PDEs). By leveraging the Ehrenpreis-Palamodov fundamental principle, the authors demonstrate that state and optimal control of linear PDEs with constant coefficients can be represented as complex-domain integrals. The proposed modification extends the frequency variable in the inverse Fourier transform from the real to the complex domain, aligning the FNO layer with this integral representation. Empirical validation on the nonlinear Burgers' equation reveals significant improvements in training accuracy and boundary value prediction compared to standard FNO, marking a substantial advancement in operator learning for control systems.

Key Points

  • ▸ Theoretical foundation: Uses the Ehrenpreis-Palamodov fundamental principle to justify complex-domain integral representations of PDE states and controls.
  • ▸ Architectural innovation: Extends FNO layers by incorporating complex frequency variables, improving expressive capacity for non-periodic boundary conditions.
  • ▸ Empirical validation: Demonstrates order-of-magnitude reductions in training errors and superior accuracy in predicting non-periodic boundary values using the nonlinear Burgers' equation as a test case.

Merits

Theoretical Rigor

The article grounds its contributions in deep mathematical principles, particularly the Ehrenpreis-Palamodov theorem, ensuring robustness and generalizability for linear PDEs with constant coefficients.

Architectural Novelty

The extension of FNO to the complex domain addresses a critical limitation in standard FNO—its reliance on real-frequency representations—thereby expanding its applicability to broader classes of PDEs.

Empirical Performance

The reported improvements in training accuracy and boundary value prediction for the nonlinear Burgers' equation underscore the practical utility of FNO$^{\angle \theta}$.

Demerits

Scope Limitations

The theoretical framework is explicitly limited to linear PDEs with constant coefficients, and its applicability to nonlinear or variable-coefficient systems remains unverified.

Complexity Overhead

The extension to complex-domain frequency variables may introduce computational overhead, particularly in high-dimensional systems or real-time applications.

Validation Scope

While the Burgers' equation is a canonical test case, broader validation across diverse PDE systems (e.g., Navier-Stokes, wave equations) is needed to fully assess generalizability.

Expert Commentary

The authors present a compelling and theoretically grounded extension of Fourier Neural Operators that addresses a longstanding limitation in the field: the restriction to real-frequency representations. By invoking the Ehrenpreis-Palamodov fundamental principle, they not only provide a rigorous mathematical justification for their approach but also align it with classical harmonic analysis. This is a notable contribution, as it bridges a gap between abstract mathematical theory and practical machine learning architecture design. The empirical results, while promising, should be interpreted with caution: the Burgers' equation, while nonlinear, lacks the complexity of fully coupled systems (e.g., Navier-Stokes in turbulent regimes). Nonetheless, the order-of-magnitude improvements in training errors suggest that complex-domain extensions could become a paradigm in operator learning. Future work should explore the scalability of FNO$^{\angle \theta}$ to higher-dimensional systems and its robustness in noisy or uncertain environments. Additionally, the computational complexity of complex-domain operations warrants further study to ensure practical deployability in real-time systems.

Recommendations

  • ✓ Expand empirical validation to include a broader range of PDE systems, particularly nonlinear and variable-coefficient equations, to assess generalizability and robustness.
  • ✓ Investigate the computational efficiency of FNO$^{\angle \theta}$, including its memory footprint and inference speed, to determine feasibility for real-time applications.
  • ✓ Develop hybrid architectures that combine FNO$^{\angle \theta}$ with traditional numerical solvers (e.g., finite element methods) to leverage the strengths of both approaches in hybrid modeling frameworks.
  • ✓ Explore the integration of uncertainty quantification techniques (e.g., Bayesian neural networks) to assess the reliability of FNO$^{\angle \theta}$ predictions in high-stakes applications.
  • ✓ Collaborate with domain experts in fluid dynamics, structural mechanics, or climate science to co-design benchmarks and validate the method in practical scenarios.

Sources

Original: arXiv - cs.LG

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