State-Inclusive Logistic Lifting (SILL) Overview

• State-Inclusive Logistic Lifting (SILL) is a method that augments state vectors with multivariate logistic functions to form a finite-dimensional, interpretable Koopman approximation.
• The lifting mechanism combines constant, linear, and conjunctive logistic elements to map m-dimensional states into a higher-dimensional feature space with controllable sharp transitions.
• SILL achieves uniform finite approximate closure with exponential error decay, offering scalable surrogate modeling for complex nonlinear systems.

State-Inclusive Logistic Lifting (SILL) is a structured approach for constructing finite-dimensional approximations to nonlinear dynamical systems within the Koopman operator framework. SILL achieves this by augmenting the measurement or state vector with a set of multivariate logistic (conjunctive logistic) functions, forming a dictionary with advantageous closure, approximation, and interpretability properties. SILL has been demonstrated to facilitate accurate, scalable, and interpretable surrogate models for nonlinear systems, particularly in the context of data-driven operator learning and Extended Dynamic Mode Decomposition (EDMD) (Johnson et al., 2022, Johnson et al., 2022, Johnson et al., 2017).

1. Mathematical Formulation of the SILL Dictionary

Let $y \in \mathbb{R}^m$ denote the measurement vector, often identified directly with the state $x\in\mathbb{R}^n$ of a nonlinear system $\dot x = f(x)$. The SILL dictionary $\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$ is defined as an affine augmentation combined with a bank of multivariate logistic functions:

• Constant function: $\psi_0(y) = 1$
• Linear terms: $\psi_i(y) = y_i$, for $i=1,\ldots,m$
• Multivariate logistics (“conjunctive logistics”): for $k=1,\ldots,N_L$,

$$\Lambda(y; \mu_k, \alpha_k) = \prod_{i=1}^m \lambda(y_i; \mu_{k i}, \alpha_{k i}), \quad \lambda(y_i; \mu_{k i}, \alpha_{k i}) = \frac{1}{1 + \exp(-\alpha_{k i}(y_i - \mu_{k i}))}$$

These multivariate functions are parameterized by center vectors $\mu_k\in\mathbb{R}^m$ and positive steepness vectors $x\in\mathbb{R}^n$0.

The complete SILL lift is given by:

$x\in\mathbb{R}^n$1

where $x\in\mathbb{R}^n$2.

2. Lifting Mechanism and Geometry

The SILL construction lifts the original $x\in\mathbb{R}^n$3-dimensional state space into a higher-dimensional feature space:

• The constant $x\in\mathbb{R}^n$4 and coordinate terms $x\in\mathbb{R}^n$5 provide the affine component, preserving the original state in $x\in\mathbb{R}^n$6.
• Each $x\in\mathbb{R}^n$7 transitions sharply from $x\in\mathbb{R}^n$8 to $x\in\mathbb{R}^n$9 along the hyperplane $\dot x = f(x)$0, with sharpness tuned by $\dot x = f(x)$1. The conjunction over $\dot x = f(x)$2 coordinates yields regions in state-space where specific thresholds are crossed.
• A bank of such conjunctive logistics, with centers $\dot x = f(x)$3 arranged on a suitable grid or covering, partitions state-space into regions with distinct activation patterns in the lifted coordinates.

This construction produces a high-dimensional feature mapping where nonlinear interactions in $\dot x = f(x)$4 become (approximately) linear in $\dot x = f(x)$5 (Johnson et al., 2022, Johnson et al., 2022, Johnson et al., 2017).

3. Homogeneity and Approximate Closure Under the Koopman Generator

The key structural property of SILL is homogeneity: all nonlinearity in the dictionary arises from the conjunctive logistic family. This yields two crucial consequences:

• Product Structure: The product of any two conjunctive logistics, $\dot x = f(x)$6, approaches a single conjunctive logistic $\dot x = f(x)$7 as $\dot x = f(x)$8, where the maximum is component-wise. This is formalized as approximate closure—products of basis elements remain (with exponential accuracy) within the span of the original dictionary.
• Uniform Finite Approximate Closure: For any compact domain and $\dot x = f(x)$9, a sufficiently large $\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$0 ensures that the error in replacing $\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$1 with $\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$2 is uniformly below $\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$3.

For the dynamics, assume $\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$4 can be approximated by a linear combination of SILL lifts:

$\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$5

The time derivative of each lift $\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$6 yields bilinear forms that, after the product approximation above, can be re-expressed as linear combinations of SILL elements plus an exponentially small residual:

$\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$7

where $\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$8 decays exponentially with $\psi(y)\colon \mathbb{R}^m \to \mathbb{R}^N$9 and with measurement dimension $\psi_0(y) = 1$0. This property allows for uniform finite $\psi_0(y) = 1$1-closure across the state space (Johnson et al., 2022).

4. Error Bounds and Parameter Tuning

The overall closure error can be explicitly bounded. For centers arranged on a mesh with resolution $\psi_0(y) = 1$2 and steepness $\psi_0(y) = 1$3, the closure error for each bilinear term can be bounded as $\psi_0(y) = 1$4 as either $\psi_0(y) = 1$5 or $\psi_0(y) = 1$6. Summing over all terms,

$\psi_0(y) = 1$7

Thus, increasing $\psi_0(y) = 1$8 or refining the grid ($\psi_0(y) = 1$9) allows tuning the regression and closure error arbitrarily small. The two principal tuning parameters are:

• Steepness $\psi_i(y) = y_i$0: Controls logistic sharpness; larger values yield sharper transitions, reduced closure error, but risk numerical ill-conditioning.
• Mesh Spacing $\psi_i(y) = y_i$1: Finer meshes (smaller $\psi_i(y) = y_i$2) lead to better function approximation and lower closure error, at increased computational cost.

The combined limits $\psi_i(y) = y_i$3, $\psi_i(y) = y_i$4 drive the SILL surrogate arbitrarily close to the exact Koopman generator (Johnson et al., 2022, Johnson et al., 2017).

5. Comparison with Other Dictionaries and Numerical Performance

Traditional monomial or polynomial dictionaries ($\psi_i(y) = y_i$5) lack finite closure for nonlinear problems: e.g., derivatives $\psi_i(y) = y_i$6 are not spanned by finite sets. Gaussian RBF bases similarly lack closure, with product approximations depending on pairwise distances and suffering only algebraic error decay.

By contrast, SILL achieves:

• Exact product closure in the steepness limit ($\psi_i(y) = y_i$7)
• Exponential decay of closure error in both steepness and measurement dimension
• Computational scalability, with per-evaluation cost $\psi_i(y) = y_i$8
• Geometric interpretability: each dictionary element’s center $\psi_i(y) = y_i$9 marks a threshold in state space; $i=1,\ldots,m$0 indicates transition sharpness

Benchmark studies on systems such as the Van der Pol oscillator and the bistable toggle switch demonstrate that SILL yields high-fidelity Koopman generator approximations, matching empirical trajectories and capturing dynamical behaviors, often with fewer parameters than neural network or deepDMD representations (Johnson et al., 2022, Johnson et al., 2017).

6. Algorithmic Implementation and Extensions

• Dictionary Construction: Centers $i=1,\ldots,m$1 can be arranged on uniform grids or through data clustering. Steepness $i=1,\ldots,m$2 can be constant or tuned by cross-validation.
• EDMD with SILL: Compute SILL features $i=1,\ldots,m$3 for sample points, estimate time derivatives for the dictionary, and solve a least squares problem to fit the linear generator $i=1,\ldots,m$4.
• deepDMD with SILL: Treat $i=1,\ldots,m$5 as a parameterized differentiable layer, optimize both $i=1,\ldots,m$6 and the parameters of $i=1,\ldots,m$7 using stochastic gradient methods.
• Hybrid (augSILL) Extensions: SILL has been augmented with additional basis types (e.g., radial basis functions) to form heterogeneous dictionaries, achieving accuracy and scaling comparable to deepDMD with an order of magnitude reduction in parameters, while preserving interpretability (Johnson et al., 2022).

7. Applications and Theoretical Implications

SILL provides a principled and interpretable alternative to black-box dictionary learning for Koopman operator approximation. Its approximate closure and explicit error control make it suitable for data-driven analysis of high-dimensional, nonlinear systems. SILL bridges the gap between analytic operator theory and empirical machine learning pipelines, combining theoretical guarantees with computational efficiency.

The success of SILL theory and practice in achieving uniform finite approximate closure provides a hypothesis for the empirical effectiveness of neural-network-based dictionaries in deepDMD: both achieve near closure but with differing degrees of interpretability and parameter efficiency (Johnson et al., 2022).

Notably, SILL-based models demonstrate subspace-invariance, allowing construction of low-dimensional Koopman surrogates for complex nonlinear flows, and facilitating analysis, control, or model reduction in applications ranging from biomechanics to synthetic biology (Johnson et al., 2017, Johnson et al., 2022).