Fourier Feature Embeddings

Fourier feature embeddings transform coordinates or other inputs into sine-and-cosine components, helping models represent fine detail, periodicity, and rapidly varying functions that ordinary networks may learn slowly. Methods range from fixed random Fourier features for kernel approximation to trainable, multiscale, spatially adaptive, or problem-informed bases, often paired with neural networks or attention models. They are studied in implicit image and 3D representations, physics-informed PDE solvers, kernel learning, causal discovery, and temporal or geometric representations, with evaluation on reconstruction, PDE, kernel, and sequence benchmarks. A central challenge is choosing frequencies and feature complexity: richer or adaptive spectra can improve accuracy and convergence, but must remain computationally efficient and avoid unstable optimization or poor behavior beyond observed data.