Lyapunov-Informed Parameter Selection

• Lyapunov-informed parameter selection is a methodology that uses Lyapunov exponents, regularized losses, and inequalities to automate tuning in non-stationary, chaotic systems.
• The strategy integrates adaptive training algorithms, real-time tuning via local Lyapunov estimation, and data-driven approaches to enhance convergence and performance.
• Applications span control systems and physics-informed networks, providing robust stability guarantees while reducing error and computational overhead.

A Lyapunov-informed parameter selection strategy comprises a family of methodologies and algorithmic frameworks that use Lyapunov-function theory—especially Lyapunov exponents, Lyapunov-regularized losses, and constructive Lyapunov inequalities—to systematically inform or automate the tuning of free parameters in learning systems, control laws, and iterative optimization schemes. These strategies enable adaptive, context-sensitive parameterization while retaining rigorous stability or performance guarantees, especially in regimes characterized by non-stationarity, parameter variations, or chaotic dynamics. The following sections survey the principal formulations and mechanisms underlying Lyapunov-informed parameter selection.

1. Dynamical Foundations: Lyapunov Exponents and the Onset of Chaos

The Lyapunov exponent quantifies the average exponential rate of divergence (or contraction) between nearby trajectories in a dynamical system. Formally, for a parametrized map $x_{t+1} = F(x_t, w)$ (with $w$ as parameters), the (maximum) Lyapunov exponent over a finite horizon $T$ is computed via QR-based analysis of the product of Jacobians $J_t = \partial F(x_t, w)/\partial x_t$, yielding

$$\lambda_{\text{max}} \approx \frac{1}{T}\sum_{t=1}^T \ln r_t,$$

where $r_t$ are the $R$-diagonal entries in the QR decomposition. The sign of $\lambda_{\text{max}}$ determines the regime: $\lambda_{\text{max}} > 0$ indicates chaos; $\lambda_{\text{max}} < 0$ contraction; $\lambda_{\text{max}} \approx 0$ maximal responsiveness at the "edge of chaos" (Benati et al., 15 Jun 2025).

Parametric control and learning systems exploit these insights by regulating parameters so that the Lyapunov exponent hovers near a target value (often zero), optimizing both adaptability and stability near regime boundaries. In root-finding (e.g., INVM$^\alpha$ schemes), analytical and empirical Lyapunov exponents delineate parameter regions with distinct stability characteristics, enabling automated, real-time selection for robust convergence (Shams et al., 20 Jan 2026).

2. Lyapunov-Regularized Losses and Adaptive Training Algorithms

Lyapunov-informed learning algorithms extend classical loss functions with Lyapunov-regularization terms that explicitly depend on the estimated Lyapunov exponent or related contraction metrics. A canonical formulation for neural systems is

$$L^{(n)} = L_{\mathrm{data}} + \alpha^{(n)} | \lambda_{\text{max}}^{(n)} - \lambda^* |,$$

where $L_{\mathrm{data}}$ is the nominal prediction error, $\alpha^{(n)}$ dynamically regulates the influence of the regularizer, and $\lambda^*$ is the edge-of-chaos target (usually zero) (Benati et al., 15 Jun 2025).

The resulting weight update at each iteration incorporates both the data-loss and Lyapunov-gradient:

$$w \leftarrow w - \eta_w [ \nabla_w L_{\mathrm{data}} + \alpha \cdot \operatorname{sign}(\lambda_{\text{max}}) \nabla_w \lambda_{\text{max}} ].$$

The regularizer strength $\alpha$ can itself be adapted online via a feedback law:

$$\alpha \leftarrow \alpha + \eta_\alpha ( |\lambda_{\text{max}}| - \epsilon ),$$

where $\epsilon$ defines a dead-zone for local exploration. Schedules for the learning rate $\eta_w$ can additionally depend on the estimated Lyapunov exponent, promoting stability when the system becomes too chaotic, and encouraging exploration when over-stabilized.

Empirically, this framework delivers significant post-regime shift performance gains (e.g., up to $96\%$ reduction in MSE for non-stationary Lorenz tasks), outperforming classical $L_1/L_2$ penalties and dropout under abrupt regime changes (Benati et al., 15 Jun 2025).

3. Data-Driven Selection: Genetic Algorithms and Machine Learning Approaches

Lyapunov-informed parameter selection can be formulated as a search or regression problem. For continuous-time nonlinear systems, candidate Lyapunov functions parameterized by tunable coefficients (or controller gains $K$ embedded in $V(x;K)$) are optimized using stochastic search methods (e.g., genetic algorithms). The optimization enforces universal Lyapunov decrease conditions, with the fitness function counting the violation rate of $V(x) > 0$ and $\dot{V}(x)<0$ across sampled domains (Zenkin et al., 2023).

In control--especially quantum and nonlinear settings--machine learning models (e.g., feedforward or general regression neural networks) are trained offline to map state-feature vectors to Lyapunov control parameters or control-scheme selections. The heavy numerical optimization is performed during dataset creation; at runtime, the inference is immediate and low-cost. This paradigm achieves near-optimal fidelity and maintains $\dot{V} \leq 0$, obtaining both performance and formal stability guarantees (Hou et al., 2018).

4. Real-Time Parameter Tuning via Local Lyapunov Estimation

Lyapunov-informed parameter selection in iterative solvers and dynamical algorithms benefits from local and sliding-window estimation of Lyapunov exponents. A practical methodology uses kNN-driven micro-series analysis: from short time-series fragments (windows), a local largest Lyapunov exponent is estimated via the slope of the log-geometric mean absolute error (ln-GMAE) across prediction horizons, fitted by piecewise linear regression (Shams et al., 20 Jan 2026).

With a sequence of Lyapunov profiles available, parameters (such as the tuning parameter $\alpha$ in parallel root-finders) are chosen or adapted so that the exponent drops below zero after brief transients and remains predominantly negative. If persistent instability is detected (e.g., $\lambda_1 > \theta$ for M windows), the parameter is adjusted (e.g., decreased by $\Delta \alpha$). Empirical results confirm close correspondence between theoretical stability diagrams and kNN-LLE empirical maps, with dramatic gains in convergence speed and resource utilization.

5. Lyapunov-Informed Scheduling in Nonlinear and Parameter-Varying Control

In control-affine, nonlinear parameter-varying (NPV) systems, parameter-dependent Control Lyapunov Functions (PD-CLFs) define admissibility regions for both the scheduling parameter $\theta$ and the control law. The min-norm controller is derived by solving a robust quadratic program:

$$u^*(x,\theta) = \arg\min_{u}\|u\|^2 \quad \text{s.t.}\quad L_fV(x,\theta)+L_gV(x,\theta)u + \frac{\partial V}{\partial\theta}(x,\theta)v \leq -\alpha(\|x\|),\;\forall v\in \operatorname{vert}(V)$$

(Zhao, 10 Feb 2025).

Synthesis of $V(x,\theta)$ and the associated controller (for polynomial $f,g$) is tractable via convex sum-of-squares (SOS) programming; the certified region of stabilization $\Omega_V = \{(x,\theta): V(x,\theta)\leq 1\}$ serves as a real-time admissibility certificate: $\theta(t)$ is scheduled online so that $(x(t),\theta(t))\in \Omega_V$, thus guaranteeing closed-loop stability under parameter variation and input constraints.

6. Extensions to Physics-Informed Learning and Thermodynamic Neural Controllers

Lyapunov exponents can be used to inform time-weighting schemes in physics-informed neural networks (PINNs). By leveraging a theoretical bound on final-time error via Grönwall-type arguments, the optimal temporal weighting profile is shown to be $$\rho^*(t) \propto \exp\left(-\int_0^t \lambda(\tau)d\tau\right)$$, where $\lambda(t)$ is a (locally estimated) Lyapunov exponent (Turinici, 2024). This weighting automatically allocates computational effort to stages where errors are amplified by local instability (chaotic regions), yielding self-tuning, principled improvement in convergence and accuracy across chaotic, periodic, and stable regimes—with no need for ad hoc hyperparameters.

In adaptive neural controllers with stochastic (Langevin-type) dynamics, Lyapunov-informed inequalities precisely dictate admissible classes of the generalized temperature law $T(x, \hat{\theta}, t)$, which mediates the diffusion term in weight updates. The bounds on the derivative of the Lyapunov function $\mathcal{L}V_L(z) \leq -b_2\|z\|^2 + b_1$ translate directly into design constraints linking the temperature gain $k_T$ and learning rate $\gamma$ to convergence and exploration-exploitation trade-offs. Annealing schedules, constant or state-dependent $T$, and explicit inequalities are all admissible provided the Lyapunov bound holds; empirical performance is validated by substantial reductions in tracking/approximation error (Akbari et al., 20 Aug 2025).

7. Theoretical Guarantees and Empirical Validation

Lyapunov-informed parameter selection strategies provide robust theoretical guarantees: either explicit $\dot V < 0$ (as in classical Lyapunov analysis), uniform ultimate boundedness in probability (for stochastic systems), or explicit exponential contraction rates. Practical performance is substantiated by quantitative studies:

• In regime-shifting chaotic neural forecasting, optimal Lyapunov regularization reduces post-shift MSE by $96\%$ compared to vanilla or dropout baselines (Benati et al., 15 Jun 2025).
• For parallel solvers, kNN-LLE-tuned parameter regimes reduce CPU time by $>90\%$, decrease memory footprint, and boost computational order of convergence from $3$–$4$ to $5$ (Shams et al., 20 Jan 2026).
• In parameter-varying control, PD-CLF/SOS-synthesized gain scheduling certifies the region of attraction and ensures closed-loop stabilization, with real-time eligibility certificates (Zhao, 10 Feb 2025).
• In physics-informed learning and stochastic neural adaptation, Lyapunov-based weighting and update schedules yield substantial error reductions and reliability gains across diverse dynamical regimes (Turinici, 2024, Akbari et al., 20 Aug 2025).

Together, these methods demonstrate that Lyapunov-informed parameter selection offers a unified, data-driven yet provably stable approach for controlling adaptation, stability, exploration, and continual learning in a variety of high-dimensional, non-stationary, and nonlinear systems.