Energy-Aware Symbolic Load Model

• The paper presents a model that mathematically links system demand, workload, and migration costs to energy consumption while enforcing strict operational and QoS constraints.
• It integrates interpretable machine learning and symbolic regression to derive transparent equations for optimization in data centers, wireless networks, and cloud management.
• The model demonstrates practical efficiency with applications yielding up to 50% energy savings and over 2× performance gains under dynamic scheduling conditions.

An energy-aware symbolic load model is a mathematical, often closed-form or algorithmically-symbolic representation linking resource demand or utilization to energy consumption, while explicitly incorporating operational, workload, system state, and migration costs. Such models are fundamental in data center scheduling, power system simulation, dynamic scaling of cloud computing, inference-serving optimization, and next-generation grid modeling. They combine workload-dependent demand/response characteristics, explicit energy cost terms, service-level constraints, and, increasingly, interpretable equation forms derived from machine learning or symbolic regression, enabling both efficiency and transparency in energy-sensitive deployments.

1. Symbolic Formulation and Core Components

Energy-aware symbolic load models synthesize system state, demand, and control actions into structured equations designed to minimize energy under operational constraints. For instance, in clustered cloud management, the canonical model incorporates:

• Sets: physical servers $S$: $\{1,\ldots,n\}$; applications/VMs $A$: $\{1,\ldots,m\}$.
• Workload Demand: $\lambda_i(t)$—CPU-cycle demand for VM $i$; bounded growth per interval $\Delta_i$ such that $\lambda_i(t+\tau)-\lambda_i(t)\leq\Delta_i$.
• Server Utilization: Load $L_k=\sum_{i\in A}x_{i,k}\lambda_{i}(t+\tau)$; normalized $b_k=L_k/C_k$, $C_k$ = service rate.
• Energy Functions: $$E_k(b_k)=P_k^\mathrm{idle}+(P_k^\mathrm{peak}-P_k^\mathrm{idle})b_k$$ for active, $E_k^\mathrm{sleep}$ for sleeping.
• VM Migration Costs: $M_{i,k\rightarrow j}$ per VM migration.

The complete objective is:

$$\min_{x,y,z} \sum_{k\in S}\big[ y_k E_k(b_k) + (1-y_k) E_k^\mathrm{sleep} \big] + \sum_{i,k,j} M_{i,k\rightarrow j} z_{i,k\rightarrow j}$$

Subject to:

• Each VM placed exactly once.
• Servers only active if hosting a VM.
• No server overloaded: $b_k \leq 1$.
• Strict QoS, typically $$b_k \leq \beta_\mathrm{QoS}=1-1/(C_kR_\mathrm{max})$$.
• Demand does not grow too rapidly.
• Migration decisions consistent with placement.

Such symbolic models provide precise mappings from resource assignments to energy and performance, enabling optimal packing, concentration, and balancing (Paya et al., 2014).

2. Interpretable Machine Learning and Symbolic Regression Approaches

Recent advances have augmented energy-aware symbolic load modeling with interpretable regression and learning-based frameworks:

• Kolmogorov–Arnold Networks (KAN): These models actively learn activation functions along network edges, deriving free-form symbolic equations that capture nonlinear relationships. The KAN uses ridge-sum transforms and B-spline basis expansions for each neuron edge, constructing mappings of voltage $V(t)$ and current $I(t)$ to active/reactive power $P(t), Q(t)$ in symbolic, human-readable form. Resultant equations can match physical ZIP models or adapt to composite/dg blends (Dorji et al., 27 Aug 2025).
• Actor–Critic Symbolic Regression (ACSR): This RL-based method grows a trainable expression tree using a constrained operator set. The actor samples operators, constructing a symbolic template, while the critic evaluates and rewards performance. Final models yield explicit expressions such as:

$$\hat{P}(V, E) = c_1 e^{c_2 V} + c_3 V^2 + c_4 \ln(E + c_5)$$

incorporating voltage $V$, cumulative energy $E$, and learned coefficients, enabling dynamic adaptation to renewable penetration and operational conditions (Lin et al., 10 Mar 2025).

These methodologies address the need for both accuracy and interpretability beyond classic polynomial (ZIP) and ANN models.

3. Energy-Aware Models in Cloud and Communication Systems

In cloud and wireless network contexts, energy-aware symbolic load models are adapted to different resource and constraint structures:

• Packing Problem Formulations: The VS-CIF-P model describes the optimal assignment of computational tasks to bins (servers) with energy costs, capacities, and allowable fragmentation. The model supports item splitting, bin class constraints, and both offline (CIFFD) and online (CFFf) approximation algorithms. Proven NP-completeness informs algorithm selection and worst-case guarantees, with empirical ratios typically $\leq$1.1 compared to optimal (Carli et al., 2014).
• Wireless OFDMA Networks: Transmission energy minimization is formulated as a load-coupled, interference-aware mapping. Fixed-point equations establish cell load $\rho_i$ and transmission energy $E_\mathrm{tot}$, with association and RU allocation variables. The optimization is NP-hard; partial optimality and iterative assignment algorithms enable tractable, high-performance solutions (You et al., 2016).

These symbolic models facilitate targeted control interventions, such as load concentration, task migration, or relay selection, directly minimizing energy under stringent resource and QoS constraints.

4. Application to AI Inference and Heterogeneous Workloads

In AI serving, especially for LLM inference, energy-aware symbolic models encode energy and runtime as explicit functions of workload (input/output token count):

For each model $K$,

$$e_K(\ell_\mathrm{in}, \ell_\mathrm{out}) = \alpha_{K,0} \ell_\mathrm{in} + \alpha_{K,1} \ell_\mathrm{out} + \alpha_{K,2} \ell_\mathrm{in} \ell_\mathrm{out}$$

$$r_K(\ell_\mathrm{in}, \ell_\mathrm{out}) = \beta_{K,0} \ell_\mathrm{in} + \beta_{K,1} \ell_\mathrm{out} + \beta_{K,2} \ell_\mathrm{in} \ell_\mathrm{out}$$

Empirically, these workload-dependent predictors explain $>$96% of variance, are rapidly recalibratable, and form the basis for joint energy/accuracy scheduling via integer optimization and frontier tracing (Wilkins et al., 2024). The interaction term ($\ell_\mathrm{in} \ell_\mathrm{out}$) is crucial for accurate estimation.

5. Performance Guarantees, Theoretical Properties, and Adaptivity

Energy-aware symbolic load models are often formulated as NP-hard optimization problems, motivating the development of approximation algorithms with provable guarantees:

• Packing-based models: 2-approximation offline (CIFFD), (4/3 + $o(1)$)-approximation online (CFFf) for energy cost, rapid convergence, and practical performance well below theoretical worst-case (Carli et al., 2014).
• OFDMA relay selection: Partial optimality conditions support efficient assignment policies with substantial energy savings over naïve selection (You et al., 2016).
• Cloud migration scheduling: Empirically, energy-aware concentration yields 2.25$\times$ efficiency gains in homogeneous clusters and 30–50% savings over always-on deployment with SLA constraint satisfaction (Paya et al., 2014).

Advanced symbolic regression setups (KAN, ACSR) are trained via Bayesian optimization or RL policy gradients, automatically adapting model order and functional form to evolving renewable penetration and fault scenarios. As energy-metrics shift, coefficients evolve, capturing dynamic system response without manual reparameterization (Dorji et al., 27 Aug 2025, Lin et al., 10 Mar 2025).

6. Interpretability, Integration, and Domain-Specific Extensions

The trend toward free-form, symbolic energy-aware load models reflects a broader demand for explainability in critical infrastructure modeling:

• Symbolic Equation Modeling: Automatically discovered equations (via KAN or RL-SR) facilitate physical diagnosis and scenario analysis, supporting integration and feedback control.
• Cumulative Energy Tracking: Explicit models (e.g., $E(t) = \int P(\tau)d\tau$) and direct $E$ dependence introduce energy-awareness into both predictive and prescriptive modeling, enabling operational decisions that modulate state-of-charge, recoverability, or renewable inertia response (Dorji et al., 27 Aug 2025, Lin et al., 10 Mar 2025).
• Scalability: These methods exhibit competitive computational scaling compared to ANN and polynomial fit baselines, with training times typically $<$1500s on commodity hardware and lower inference complexity due to small symbolic tree depth.

A plausible implication is that as distributed energy systems, cloud, and AI workloads continue to diversify, symbolic energy-aware load models—combining domain knowledge, explicit optimization, and interpretable machine learning—will form the backbone of operational orchestration frameworks, offering both optimality and physical transparency.