DeepONet vs Fourier Neural Operator for EM Simulation

DeepONet excels at learning continuous mappings between function spaces, making it a strong candidate for parametric EM design where boundary conditions and source excitations vary. Its branch-trunk architecture decouples the input function encoding from the output location evaluation, allowing a single trained model to predict fields for any input function within the training distribution. For example, in a microstrip line impedance sweep, a DeepONet can achieve <2% relative error in predicting S-parameters across a 5:1 impedance range after training on just 500 full-wave simulations, demonstrating strong generalization from sparse data.

Fourier Neural Operator (FNO) takes a fundamentally different approach by learning operators directly in Fourier space. By parameterizing integral kernels in the frequency domain, FNO achieves mesh-invariant resolution and captures global dependencies efficiently. This results in exceptional performance for problems governed by wave equations and periodic structures. In a frequency-selective surface (FSS) design task, FNO demonstrated a 3x speedup over DeepONet for predicting transmission coefficients across a 10 GHz bandwidth, while maintaining <1% error. However, this spectral approach can struggle with highly irregular geometries where the Fourier basis is less efficient.

The key trade-off: If your priority is generalization across diverse geometries and boundary conditions with a single trained model, choose DeepONet. Its architecture naturally handles varying input functions without retraining. If you prioritize raw inference speed and accuracy on grid-like or periodic structures common in antenna arrays and metamaterials, choose FNO. For teams running large parametric sweeps over frequency and geometry, consider a hybrid approach: use FNO for the bulk of regular structures and DeepONet for edge cases with complex, non-periodic boundary conditions.

DeepONet excels at learning explicit mappings between function spaces, making it highly effective when the input and output functions are clearly defined, such as mapping a specific source excitation to the resulting field solution. Its architecture, which separates the branch and trunk networks, allows for rapid inference across a massive number of sensor points once trained. For example, in a parametric study of a microstrip filter, a trained DeepONet can predict the full S-parameter response for 10,000 new geometric variations in under a second, a task that would take a full-wave solver several hours.

Fourier Neural Operator (FNO) takes a fundamentally different approach by learning operators directly in Fourier space, which provides exceptional mesh invariance and zero-shot super-resolution. This results in a model that can be trained on low-resolution simulation data and then evaluated at a much higher resolution without retraining. In a benchmark for scattering parameter prediction, FNO demonstrated a 3x improvement in generalization accuracy when moving from a coarse training mesh to a fine evaluation mesh compared to a standard CNN, making it ideal for designs requiring adaptive mesh refinement.

The key trade-off: If your priority is ultra-fast, real-time inference for a fixed sensor layout with many different input functions, choose DeepONet. If you prioritize mesh invariance, the ability to train on cheap, coarse data and deploy on high-fidelity grids, and superior generalization across resolutions, choose FNO. For CTOs building a digital twin platform that must handle diverse, user-defined geometries without constant retraining, FNO's architecture provides a more robust foundation for a general-purpose EM surrogate.