🤖 machine learning

Real vs. Complex Spectral Bases for Neural Operators: The Role of Green's Function Alignment

This paper introduces the Hartley Neural Operator (HNO), a real-valued alternative to the Fourier Neural Operator (FNO), and demonstrates that the optimal spectral basis for learning PDE solution operators depends on the operator's phase content, with real Hartley bases excelling for self-adjoint elliptic problems and complex Fourier bases being superior for time-dependent, phase-carrying dynamics.

Original authors: Jason Sulskis, Sathya Ravi

Published 2026-06-24

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CC BY 4.0

Original authors: Jason Sulskis, Sathya Ravi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a computer to solve physics problems, like predicting how heat spreads through a metal plate or how waves crash on a shore. These problems are described by complex math equations called Partial Differential Equations (PDEs).

For a long time, the best way to teach computers these problems was using a tool called the Fourier Neural Operator (FNO). Think of the FNO as a translator that converts the physical problem into a "frequency language" using complex numbers (numbers with both a real part and an imaginary part). This works great, but it's a bit like carrying a heavy backpack full of duplicate items. Because the physics problems we care about usually involve real-world numbers (not imaginary ones), the FNO ends up storing a lot of redundant information just to keep the math balanced.

The authors of this paper asked: Can we build a translator that doesn't carry that extra weight?

The New Tool: The Hartley Neural Operator (HNO)

They created a new tool called the Hartley Neural Operator (HNO).

The Analogy: If the FNO is a translator who speaks a language with two dialects (Real and Imaginary) and always writes everything down twice to be safe, the HNO is a translator who speaks only one dialect (Real) and writes everything down once.
The Result: The HNO is the "exact mirror" of the FNO but uses only real numbers. It is just as powerful in terms of how many "brain cells" (parameters) it has, but it doesn't waste any space on imaginary numbers.

The Big Discovery: It Depends on the Problem

The most surprising finding isn't that one tool is simply "better" than the other. Instead, the authors discovered that the best tool depends entirely on the type of physics problem you are solving.

They found a clear rule based on something called "Phase" (which you can think of as the "rhythm" or "direction" of the wave).

1. The "Still Water" Problems (Elliptic Equations)

Examples: Heat spreading out (diffusion) or finding the shape of a stretched drumhead (Poisson/Biharmonic equations).
The Physics: These problems are "symmetrical." If you look at them from the front or the back, they look the same. They don't have a "direction" or a "beat."
The Winner: HNO (The Real-Only Tool).
Why? Because these problems are perfectly symmetrical and have no "imaginary" rhythm, the HNO fits them like a glove. It can solve them exactly with its simple, real-number math. The FNO, in this case, is like trying to solve a puzzle with a screwdriver; it can do it, but it has to waste energy pretending the imaginary parts don't exist.

2. The "Moving Wave" Problems (Time-Dependent Equations)

Examples: Sound waves traveling, wind blowing (advection), or water swirling (Navier-Stokes).
The Physics: These problems have a "beat." They move, oscillate, and carry energy from one place to another. This movement creates a "phase" (a timing shift) that requires complex numbers to describe accurately.
The Winner: FNO (The Complex Tool).
Why? You cannot describe a moving wave or a swirling vortex using only real numbers without losing the "direction" of the movement. The HNO is structurally incapable of representing this "phase." It's like trying to describe a spinning top using only a straight line; the HNO just can't do it. The FNO, with its complex numbers, captures that spin and movement perfectly.

The "Heat Equation" Borderline

There is one special case: the Heat Equation (how heat spreads).

This is a "time-dependent" problem (it happens over time), but it doesn't have a "wave" or "oscillation." It just smooths out.
Because it lacks that "phase" or "beat," it sits right on the border. Sometimes the HNO wins, sometimes the FNO wins, but they are usually very close. This proves the theory: it's not about whether the problem is "time-based," but whether it has phase.

The Bottom Line

The paper concludes that there is no single "universal winner" for solving physics problems with AI. Instead, the rule is:

Match the tool to the symmetry.
If the problem is symmetrical and still (like heat spreading or a static shape), use the Real (Hartley) tool. It's lighter and more precise.
If the problem involves movement, waves, or swirling (like sound, wind, or water), use the Complex (Fourier) tool. It's the only one that can handle the "rhythm" of the physics.

The authors tested this on many different scenarios (different starting conditions, different boundaries) and the rule held true every time. They didn't just find a new tool; they found the instruction manual for when to use which tool.

Technical Summary: Real vs. Complex Spectral Bases for Neural Operators

Problem Statement

Fourier Neural Operators (FNO) have emerged as a leading architecture for learning solution operators of partial differential equations (PDEs) by parameterizing global convolutions in the complex Fourier domain. However, for the vast majority of physical systems governed by real-valued PDE solutions, the standard Fast Fourier Transform (FFT) introduces representational redundancy. Specifically, the complex spectrum of a real signal exhibits conjugate symmetry, meaning half the spectrum is determined by the other half. Despite this, standard FNO implementations learn complex weight matrices that must implicitly discover and enforce this constraint, potentially increasing the effective search space and computational overhead without adding representational capacity.

The paper investigates whether the complex Fourier basis is the optimal spectral representation for all PDE classes or if a purely real basis might be superior for specific operator symmetries.

Methodology

The Hartley Neural Operator (HNO)

The authors introduce the Hartley Neural Operator (HNO), designed as the exact real-valued mirror of the FNO.

Transform: HNO replaces the FFT with the Discrete Hartley Transform (DHT). The DHT operates entirely in the real domain using the kernel $cas(\theta) = \cos(\theta) + \sin(\theta)$ .
Architecture: Unlike the FFT, which exploits conjugate symmetry to store only the positive-frequency half-space, the DHT of a real signal has no such symmetry; every coefficient is independent. Consequently, HNO retains the full spectrum (e.g., all four quadrants in 2D) but learns a single real multiplier per retained spectral mode.
Iso-parametric Design: To ensure a fair comparison, HNO is constructed to be iso-parametric with FNO at equal width. While FNO learns one complex weight (two real parameters) per mode over a halved spectrum, HNO learns one real weight per mode over the full spectrum. The total number of real trainable parameters remains identical between the two architectures.
Implementation: The authors implement HNO using PyTorch's torch.fft by computing the DHT via the relationship $H\{f\}(k) = \text{Re}\{F\{f\}(k)\} - \text{Im}\{F\{f\}(k)\}$ . This leverages existing GPU-accelerated FFT kernels while avoiding custom kernel development for the initial experiments.

Experimental Protocol

The study conducts a systematic evaluation across five classes of PDEs:

Elliptic: Poisson, Biharmonic.
Parabolic: Heat equation.
Hyperbolic: Wave equation.
Advective: Advection-diffusion.
Nonlinear: Burgers' equation, 2D Navier-Stokes (vorticity form).

Controls:

Initial Conditions: Three families are tested: Gaussian Random Fields (GRF), Eigenfunction Expansions, and Gaussian Bump Superpositions.
Boundary Conditions: Both periodic and homogeneous Dirichlet conditions are evaluated.
Training: Both operators are trained with identical optimizers (Adam), learning rate schedules, weight decay, gradient clipping, and network widths. Ground truth for time-dependent PDEs is generated via finite difference methods to avoid implicit bias toward Fourier-based representations.

Key Contributions

A Clean Real-Valued Mirror of FNO: The paper formulates HNO as a strictly real counterpart to FNO, eliminating complex arithmetic and conjugate symmetry constraints while maintaining identical parameter counts. This isolates the spectral basis as the sole variable in the comparison.
Basis–Operator Alignment Principle: The authors develop a theoretical framework linking the optimal spectral basis to the symmetry of the solution operator's Green's function.
- Self-Adjoint Elliptic Operators: These possess real, symmetric Green's functions. The Hartley convolution theorem simplifies to elementwise multiplication for symmetric kernels, allowing HNO to represent these operators exactly with a single real multiplier per mode. FNO, conversely, must learn to zero out its imaginary components.
- Time-Dependent Operators: These operators often carry phase (oscillation in wave equations, transport in advection). A real diagonal multiplier structurally cannot represent operators with complex symbols (phase). Thus, FNO is theoretically favored for these cases.
Operator-Phase Ordering: The paper establishes that the performance gap between HNO and FNO is not binary but monotonic, scaling with the "phase content" of the operator's propagator.
Comprehensive Benchmarks: The study provides the first benchmarks for Hartley-based operators on Burgers' and 2D Navier-Stokes equations, alongside a cost analysis showing that current performance overheads are due to the emulated DHT backend rather than the transform itself.

Results

The empirical results confirm the theoretical predictions, showing a clean division based on operator symmetry:

Elliptic PDEs (HNO Favored): On Poisson and Biharmonic equations, HNO achieves lower relative $L_2$ error than FNO. The advantage is most pronounced for the Biharmonic equation, the most strongly self-adjoint operator tested. Periodic boundaries, which preserve exact Hartley diagonalization, show larger gains than Dirichlet boundaries.
Time-Dependent PDEs (FNO Favored): FNO outperforms HNO on all time-dependent equations. The margin of victory correlates with the operator's phase content:
- Heat Equation: As a phaseless propagator ( $e^{-\nu|k|^2 t}$ ), it sits at the borderline where HNO occasionally edges ahead on smooth inputs, but FNO is generally competitive.
- Wave Equation: Oscillatory but undamped, showing an intermediate FNO advantage.
- Transport-Dominated (Advection, Burgers, Navier-Stokes): These operators exhibit the largest FNO advantage due to their strong phase-carrying symbols (e.g., $e^{-ic \cdot k t}$ ).
Initial Conditions: The choice of initial condition family modulates the magnitude of the error gap but does not change the sign of the winner. The basis preference is determined by the operator's symmetry, not the data distribution.

Significance and Claims

The paper argues against the search for a "universal" spectral basis for neural operators. Instead, it proposes a predictive rule: match the spectral basis to the symmetry of the solution operator.

Real Basis (HNO): Optimal for self-adjoint, phaseless operators (elliptic solves, diffusion). The real diagonal multiplier exactly diagonalizes the symmetric Green's function, reducing the optimization landscape and eliminating the need to learn constraints (e.g., $\text{Im}\{W\} = 0$ ).
Complex Basis (FNO): Necessary for operators carrying phase (oscillation, transport). The complex arithmetic is structurally required to represent the imaginary components of the propagator's symbol.

The authors conclude that spectral basis selection is an active design choice driven by the physics of the problem. While HNO offers a parameter-efficient and theoretically grounded alternative for elliptic problems, FNO remains superior for time-dependent, transport-dominated systems. The work suggests that future neural operator designs should adapt their spectral representations based on the detected symmetry of the target operator.