Real vs. Complex Spectral Bases for Neural Operators: The Role of Green's Function Alignment
Researchers propose Hartley Neural Operators as a computationally efficient alternative to Fourier Neural Operators for learning PDE solution maps. By replacing complex FFT with the real Discrete Hartley Transform, HNO eliminates redundant conjugate symmetry in real-valued problems while maintaining parameter parity with FNO. The shift trades complex arithmetic for doubled spectral resolution at identical model width, offering potential efficiency gains for scientific machine learning without architectural compromise. This work addresses a fundamental inefficiency in how neural operators represent real physical systems, relevant to anyone scaling operator learning for climate, materials, or fluid dynamics applications.
Modelwire context
ExplainerThe paper's core insight is that real-valued PDEs don't need complex arithmetic at all. Fourier Neural Operators use FFT on complex numbers, which stores redundant conjugate pairs for real outputs. Hartley swaps this for a real-only transform, doubling spectral resolution per parameter without adding model width.
This sits in the optimization layer of neural operator training. The convergence bounds work from late June tightened how we think about training efficiency in non-smooth settings, but that's about the solver itself. Hartley's contribution is architectural: it removes a representational inefficiency upstream of optimization. The two papers address different bottlenecks in the same pipeline (what you compute vs. how you train it), but neither directly depends on the other.
If teams report wall-clock speedups of 1.5x or better on standard benchmarks (turbulence, heat equation, Navier-Stokes) using HNO at the same parameter count as FNO within the next six months, the efficiency claim holds. If speedups are under 1.2x or require wider models to match FNO accuracy, the real-arithmetic advantage was overstated.
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MentionsFourier Neural Operators · Hartley Neural Operator · Discrete Hartley Transform · FFT
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