Smooth Saturation Functions

📐 Definition

A representative smooth saturation is

s(x) = s_{\max}\,\tanh\!\left(\frac{x}{s_{\max}}\right)

where $s_{\max} > 0$ sets the asymptotic magnitude.

Domain and Codomain

For real inputs, $s:\mathbb{R}\to(-s_{\max},s_{\max})$ .

⚙️ Key Properties

Smoothness, Parity, and Monotonicity

$s$ is $C^\infty$ , odd, and strictly increasing.

Derivative

s'(x) = \operatorname{sech}^2\!\left(\frac{x}{s_{\max}}\right)

peaks at the origin and decays as $|x|$ grows. Approaches $\pm s_{\max}$ exponentially for large $\pm x$ .

🎯 Special Cases and Limits

As $s_{\max}\to\infty$ , $s(x)$ approaches the identity map.
Scaling $s_{\max}$ rescales the saturation plateau and the transition width.
Other smooth saturations (e.g. arctangent or rational forms) share the same bounded, differentiable character.

Hyperbolic tangent is the unit-plateau case; soft clamping introduces asymmetric bounds and logistic transitions.

Usage in Oakfield

Oakfield primarily uses tanh-shaped saturations as simple, robust nonlinearities:

Analytic warp (TANH profile) provides a smooth bounded warp option intended to avoid singularities/poles present in other analytic profiles.
Remainder (TANH nonlinearity) uses tanh as a pre-difference transform to reduce sensitivity to outliers and large spikes.
Visualization and diagnostics often apply tanh-like compression to keep displays stable under wide dynamic range.

Historical Foundations

📜 Saturating Nonlinearities

Smooth saturation functions are modern numerical modeling primitives that replace hard clamps with differentiable alternatives, often built from classical special functions like $\tanh$ .

🌍 Modern Perspective

They are commonly used to preserve differentiability while enforcing boundedness, improving robustness in optimization and PDE simulations.

📚 References

NIST Digital Library of Mathematical Functions (DLMF), §4
Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations I