🤖 AI Summary
This work addresses the redundancy of complex Fourier bases in neural operators when learning solution operators for real-valued partial differential equations (PDEs). The authors propose the Hartley Neural Operator (HNO), which replaces the complex-valued FFT in Fourier Neural Operators (FNOs) with a purely real discrete Hartley transform, adaptively selecting the optimal spectral basis according to the symmetry and phase characteristics of the underlying PDE operator while maintaining the same number of parameters. They establish, for the first time, a theoretical link between spectral basis choice and the symmetry of Green’s functions, and formulate a guideline for selecting real or complex spectral bases based on operator type—elliptic versus time-dependent. Experiments demonstrate that HNO significantly outperforms FNO on self-adjoint elliptic problems such as Poisson’s equation, whereas FNO excels in phase-sensitive dynamic problems like wave propagation and Navier–Stokes, with performance differences monotonically correlated with the operator’s phase content.
📝 Abstract
Fourier Neural Operators (FNO) learn solution operators of partial differential equations by parameterizing global convolutions in the complex Fourier domain. For real-valued PDE solutions, the complex FFT carries representational redundancy through conjugate symmetry. We introduce the Hartley Neural Operator (HNO), the exact real-valued mirror of FNO: it replaces the FFT with the purely real Discrete Hartley Transform and learns a single real multiplier per retained spectral mode, with no complex arithmetic. Because the real Hartley spectrum is not halved by conjugate symmetry, HNO retains twice as many frequency corners as FNO but one real weight where FNO carries a complex pair, so the two operators are iso-parametric at equal width and differ only in spectral basis. Our central thesis is that the best basis is a property of the operator. Self-adjoint elliptic operators (Poisson, biharmonic) have real, symmetric Green's functions that the real Hartley multiplier diagonalizes exactly, and HNO is favored there. Time-dependent operators carry phase, from oscillation in the wave equation to transport in advection, Burgers, and Navier-Stokes, which a real diagonal multiplier cannot represent, so FNO is favored there, and increasingly so with the operator's phase content, leaving the phaseless heat equation as the borderline case. Training both operators identically and benchmarking across PDE classes, initial-condition families, and boundary conditions, we find an elliptic-versus-time-dependent split that is monotone in operator phase content and matches the Green's-function theory we develop. Rather than a universal winner, our findings give a predictive rule: match the spectral basis to the symmetry of the solution operator.