Variable-Order Models: Adaptive Dynamics

Updated 1 December 2025

Variable-order models are mathematical frameworks where operator orders change with system variables, enabling context-dependent and adaptive dynamics.
They generalize fixed-order methods by incorporating variable memory effects and dependency lengths, which improve modeling of heterogeneous, nonlocal, and evolving systems.
Numerical and learning-based methods, such as adaptive integrators and spectral collocation, are key to efficiently solving the complex equations underlying variable-order models.

A variable-order model is a mathematical or computational structure in which the order of a key operator—such as a derivative, a Markov dependency, or a logical quantifier—may itself vary as a function of other system variables, time, space, or learned features. Such models generalize both classical integer-order and constant-order fractional models, introducing heightened adaptability, nonlocality, and the potential to represent systems with heterogeneous, time-evolving, or context-dependent dynamics. The variable-order paradigm appears prominently in fractional calculus (variable-order derivatives and integrals), dynamical systems (variable-order neural and Markov models), and logic (variable-order logic for complexity and descriptive frameworks).

1. Foundational Definitions and Mathematical Principles

Classical operators—differentiation, integration, and finite dependence—are associated with a fixed order: a $k$ -th derivative, an order- $K$ Markov chain, or a $k$ -ary logic quantifier. Variable-order models replace the fixed order by a function of time, space, features, or learned contexts.

Variable-Order Fractional Derivatives: The Caputo variable-order derivative with order $\alpha(t)\in(0,1]$ is

${}_{t_0}D_t^{\alpha(t)} x(t) = \frac{1}{\Gamma(1-\alpha(t))} \int_{t_0}^{t} \frac{d x(\tau)}{d\tau} (t-\tau)^{-\alpha(t)} d\tau.$

When the order is allowed to depend on both time and other variables (e.g., hidden features in a neural network), one writes $\alpha(h(t),t)$ , yielding maximal expressivity for data-driven memory adaptation (Cui et al., 20 Mar 2025).

Variable-Order Markov Chains: A variable-order Markov chain of maximal order $K$ assigns to each path $P_H=(s_{n-K},\ldots,s_{n-1})$ a possibly different dependency length $k\leq K$ , adapting context length to local data support or inferred dependencies (Secchini et al., 24 Jan 2025, Begleiter et al., 2011).
Variable-Order Logic: Variable-Order Logic extends classical higher-order logic by introducing order variables (quantifying over the level in the logical type hierarchy), yielding the ability to natively embed the analytical hierarchy and surpass the expressive limits of fixed-order logics (Milchior, 2014).
Variable-Order Sobolev Spaces: In function spaces, variable-order is encoded via $s(x,y)$ and $p(x,y)$ , delivering spaces $W^{s(x,y),p(x,y)}$ in which the smoothness and integrability exponents are functions over the domain or domain pairs (Sakuma, 8 Aug 2024).

Variable order is almost always implemented through a non-constant, typically continuous, order function $\alpha$ (or $s$ , $p$ , etc.), and requires significant care in ensuring well-posedness, invertibility, and consistent embedding relationships.

2. Variable-Order Fractional Calculus: Operators and Applications

Operator Construction

The three primary classes of variable-order fractional derivatives are the Caputo, Riemann-Liouville, and combined Caputo types, with variable order typically a function $\alpha(t,\tau)$ parametrized over both current ( $t$ ) and history ( $\tau$ ) variables. For example, the left Caputo-VO derivative is

${}^C_a D_t^{\alpha(t,\tau)} x(t) = \int_a^t \frac{1}{\Gamma(1-\alpha(t, \tau))} (t-\tau)^{-\alpha(t,\tau)} x'(\tau)\, d\tau.$

See (Almeida et al., 2018, Tavares et al., 2017, Odzijewicz et al., 2012) for detailed constructions and their higher-dimensional analogues.

Analytical Properties and Variational Calculus

Fractional Integration by Parts (Variable Order) Integration by parts identities hold for variable order under suitable regularity and allow for rigorous derivation of Euler–Lagrange equations in the fractional calculus of variations, including for double integrals and multi-field systems (Odzijewicz et al., 2012, Almeida et al., 2018).
Memory Adaptation: The variable order directly controls the memory kernel, interpolating between local ( $\alpha=1$ ) and strong nonlocal ( $\alpha\to 0$ ) behavior, thus enabling adaptive modeling of dissipative, viscoelastic, or aging systems (Giusti et al., 2023, Tavares et al., 2017, Odzijewicz et al., 2012).

Numerical Methods

Spectral collocation (e.g., Laguerre and Bernstein polynomial bases) admits efficient recurrences for variable-order actions, with error estimates often showing exponential convergence for analytic data (Zaky et al., 2018, Rayal et al., 2023).
Discretizations of variable-order PDEs, including diffusive and oscillatory equations, require time- and space-adaptive convolution weights and careful stability analysis (Rayal et al., 2023, Angstmann et al., 26 Apr 2025).

3. Variable-Order Markov and Neural Models

Markov Chains and Sequence Prediction

Definition: Variable-order Markov models adapt the context length used for prediction according to observed sample support or information gain. This is implemented using context trees, context mixing (e.g., CTW), and adaptive pruning (Begleiter et al., 2011, Secchini et al., 24 Jan 2025).
Overfitting and Model Selection: High-order Markov models are susceptible to variance-induced overfitting; variable-order approaches employ cross-validation, divergence scoring, and classifier-based pruning (e.g., DIVOP) to select informative higher-order contexts (Secchini et al., 24 Jan 2025).

Neural Networks with Variable Order

Neural Variable-Order FDEs (NvoFDE): A neural ODE/FDE framework where both the dynamics and the order of the fractional derivative are learned as functions of hidden features, e.g.,

${}_{t_0}D_t^{\,\alpha(h(t),t)}\,h(t) = f_{\theta}(h(t), t),\quad \alpha(\cdot)\in(0,1].$

The order-net is parameterized as a neural network, typically enforcing $\alpha\in(0,1]$ via a sigmoid (Cui et al., 20 Mar 2025).

Graph Diffusion and Deep Learning: Variable-order dynamics are embedded in diffusion models on graphs, where the memory kernel adjusts to node or feature context, resulting in measurable performance improvements (1–3 percentage points in node classification, 2–3 points in ROC-AUC on heterophilic graphs) relative to fixed-order baselines (Cui et al., 20 Mar 2025).

Reduced Order and Adaptive Model Extraction

SSNNO / R-SSNNO: Ordered state-space neural networks facilitate systematic model order reduction; states with smallest variance are truncated, yielding an aggressively reduced yet high-performing dynamical model for control and estimation (Augustine et al., 14 Jun 2024).

4. Practical Implementation and Computational Considerations

Timestepping and Variable-Order Integrators: Modern VSVO (Variable Stepsize Variable Order) schemes enable cost-efficient dynamic adjustment of order and step-size for stiff ODEs, employing filter-based “one-solve” updates, error control, and SIMD-friendly implementations (DeCaria et al., 2018).
Numerical Schemes for Variable-Order PDEs: Finite-difference, wavelet, and spectral methods are adapted for variable-order convolutional weights, often requiring precomputation or efficient recurrence (Rayal et al., 2023, Zaky et al., 2018, Angstmann et al., 26 Apr 2025).
Neural Model Differentiability: Variable-order integrators and CDEs for deep learning are designed to be fully differentiable for end-to-end training, enabling joint optimization of dynamic and order parameters (Cui et al., 20 Mar 2025).
Parameter Interpretation: The learned or specified order function directly encodes physical or representational properties—local order controls memory decay, spatial order captures heterogeneity or trapping, and context-order in Markov models encodes dependency depth (Giusti et al., 2023, Angstmann et al., 26 Apr 2025, Secchini et al., 24 Jan 2025).

5. Applications Across Scientific Domains

Anomalous Transport and Diffusion: Spatially and temporally variable-order fractional diffusion models—derived systematically from CTRWs and DTRWs with heterogeneous waiting time exponents—account for spatially inhomogeneous trapping, nonstationary subdiffusive behavior, and multiscale diffusion (Angstmann et al., 26 Apr 2025, Zheng et al., 2019).
Viscoelasticity and Material Memory: Viscoelastic models with time-varying order (e.g., Maxwell model with Scarpi’s variable-order derivative) model systems with evolving microstructure; the order function can itself follow a physical evolution law (relaxation or fractional-relaxation) (Giusti et al., 2023).
Fracture Mechanics: Variable-order derivatives provide a mechanism for dynamic crack propagation, roughening, and branching by encoding local strain history via order evolution laws, yielding mesh-insensitive and evolutionary fracture descriptions (Patnaik et al., 2020).
Fractional Calculus of Variations: Variable-order operators enter generalized variational principles (including Herglotz-type), yielding Euler–Lagrange equations whose nonlocal memory kernels are themselves adaptive; applications span generalized control, optimal design, and non-conservative mechanics (Almeida et al., 2018, Tavares et al., 2017, Odzijewicz et al., 2012).
Embedding Theory and Nonlocal PDEs: Variable-order Sobolev spaces have compact embedding properties even up to critical exponents (with logarithmic pinching), supporting the analysis and solution of variable-order nonlinear Choquard and p-Laplacian equations (Sakuma, 8 Aug 2024).

6. Challenges, Open Questions, and Future Directions

Analytical Theory: Existence and uniqueness theory for variable-order FDEs and PDEs are often markedly more subtle than for constant-order cases; mapping properties and asymptotic analysis remain incompletely characterized (Zheng et al., 2019, Giusti et al., 2023).
Numerical Error Analysis: Rigorous convergence and stability results for variable-order numerical schemes are considerably less mature, particularly in high dimensions or under strong smoothness variation (Zaky et al., 2018, Rayal et al., 2023).
Model Selection and Overfitting: For Markov and neural models, model order selection must account for both variance (noise) and bias (missed memory) in high-dimensional data; cross-validation and data-driven pruning strategies are essential to prevent misidentification of spurious dependencies (Secchini et al., 24 Jan 2025, Begleiter et al., 2011).
Interpretability and Physical Grounding: While variable order offers exceptional expressive power, the interpretability of learned order functions—especially in deep learning contexts—remains an open challenge; physics- or domain-informed parameterizations offer a promising direction but are not yet standard (Cui et al., 20 Mar 2025, Giusti et al., 2023).
Expressivity in Logic: Variable-order logic strictly extends the expressive power of second-order logic; characterizations of computational complexity classes and equivalence with analytical hierarchy provide profound connections to computability, but the descriptive complexity landscape for variable-order quantification is still under exploration (Milchior, 2014).

7. Summary Table: Variable-Order Model Classes

Model Type	Order Function	Core Application
Caputo/RL FDEs	$\alpha(t), \alpha(x), \alpha(h(t),t)$	Memory-adaptive diffusion, viscoelasticity, ML
Markov/Context Tree	$k = k(x, \mathrm{data})$	Adaptive sequence prediction, network transport
Neural FDE/CDE	$\alpha(h(t), t)$ (learned)	Graph neural diffusion, sequence modeling
Sobolev Spaces	$s(x,y), p(x,y)$	Critical embedding, nonlocal PDEs
Logic (VOL)	order variable $r$	Analytical hierarchy, descriptive complexity

In all cases, the distinctive property of variable-order models is their capacity for nonstationary, spatially or contextually heterogeneous adaptation, whether in memory, dependency, or descriptive power. This flexibility is mathematically nontrivial and computationally potent, yet demands commensurate advances in theory, numerics, and interpretability for full exploitation in scientific and engineering domains (Cui et al., 20 Mar 2025, Secchini et al., 24 Jan 2025, Giusti et al., 2023, Begleiter et al., 2011, Milchior, 2014).