Deterministic Restoration Plan

• Deterministic restoration plan is a systematic protocol that maps fixed system inputs to a unique, reproducible sequence of repair actions.
• It leverages optimization-based, dynamic programming, and model-driven approaches across domains like power grids, image, and multi-agent systems.
• The plan guarantees constraint satisfaction and optimality by integrating discrete decisions and continuous models to meet objectives such as minimal restoration time.

A deterministic restoration plan is a procedure or algorithmic protocol for transitioning a degraded, damaged, or disrupted system to a desired, restored configuration through a sequence of control, repair, or transformation actions, such that for any fixed inputs (system state, measurements, or initial conditions), the output sequence is uniquely determined—i.e., it is non-random, reproducible, and explicitly specified. These plans are developed in diverse contexts including power grid and distribution network restoration, image restoration with inverse problems or generative models, and autonomous agent-based repair strategies. Deterministic restoration plans are formally constructed to optimize specified objectives (e.g., maximal load served, minimal restoration time, maximal signal fidelity) subject to operational, physical, or logical constraints, and are characterized by algorithmic transparency, unique action sequences, and guaranteed constraints satisfaction under modeled assumptions.

1. Mathematical Formulation and Problem Classes

Deterministic restoration planning spans several applied domains. In electric power systems, these plans are generally formulated as discrete-time or continuous variable optimization problems over actuator (switch, regulator, generator, or repair crew) sequences, typically under the mixed-integer linear (MILP), mixed-integer second order cone (MISOCP), or nonlinear programming frameworks (Poudel et al., 2020, Sekhavatmanesh et al., 2018, Wei et al., 2024). Feasibility requirements commonly include constraints on system connectivity, flow equations (for currents, voltages, power), radiality, component limits, and dynamic system responses (e.g., cold load pickup or generator ramping). In image restoration, deterministic plans are derived via algorithmic integration of diffusion or flow-based models (often ODE/PDE-based) or by policy extraction from Markov decision process (MDP)-formulated sequential operations (Chen et al., 2023, Qin et al., 20 Jun 2025, Lu et al., 21 Dec 2025).

Formally, such plans result from the solution $\hat{S}$ to

$$\hat S = \arg \max_{S \in \mathcal{F}} \: \mathcal{O}(S)$$

where $S$ is a sequence of actions (e.g., switch operations, restoration tool invocations), $\mathcal{F}$ is the feasible set (determined by constraints on system behavior, radiality, capacity, or process physics), and $\mathcal{O}$ is the restoration objective (e.g., maximized load, minimized restoration time, maximized perceptual quality in images). The structure and decision variables in $\mathcal{F}$ and $\mathcal{O}$ are domain-dependent, but determinism is preserved by the fixed, algorithmically-defined decision rules and unique solution extraction for each input instance.

2. Core Principles and Restoration Workflows

Deterministic restoration plans are founded on several key principles:

• Algorithmic Uniqueness: Every input (state of damage, system measurement, degraded image) maps to a unique sequence of restorative actions.
• Constraint Satisfaction: The plan explicitly enforces physical, operational, or logical constraints at every change in system topology or configuration, as certified either by mathematical guarantees, analytical checks after each action, or by continuous enforcement via constrained optimization.
• Sequential Constructiveness: Restoration plans are typically constructed step-wise (e.g., by dynamic programming, forward greedy selection, or via scheduled optimization of switching/repair actions), ensuring no step leaves the system in an unrecoverable or unsafe intermediate state.
• Reproducibility and Adaptability: Given fixed data, plans can be regenerated verbatim, but can also be rapidly recalculated when system parameters or resource statuses change.
• Plan Synthesis: In multi-stage frameworks (e.g., (Poudel et al., 2020)), one stage produces the optimal restored configuration (topology, resource allocation); subsequent stages schedule actionable transitions, sequencing control actions so that system constraints remain satisfied throughout.

A sample workflow for power system restoration may involve: 1) identifying the restoration goal and constraints, 2) formulating a system optimality problem (maximal load or minimal time subject to all bounds), 3) solving the optimization via MILP or MISOCP, and 4) extracting and operationalizing the prescribed action sequence (Poudel et al., 2020, Sekhavatmanesh et al., 2018, Jang et al., 2020). In the context of learned or model-based restoration (e.g., image restoration), deterministic plans involve solving for a unique trajectory through a mapping (e.g., ODE integration in diffusion models), a maximum-likelihood gradient descent path, or a policy-optimized action chain (Chen et al., 2023, Qin et al., 20 Jun 2025, Lu et al., 21 Dec 2025).

3. Algorithmic Structures and Implementation Frameworks

The technical realization of deterministic restoration plans takes diverse forms:

• Optimization-Based Plans: Extensive use of MILP, MISOCP, or network flow models enables encoding of switch actions, voltage regulation, repair crew routes, or generator start-up as decision variables over system graphs. All nonlinearities, action costs, and precedence constraints are embedded explicitly; for example, in distribution network restoration, binary variables index the states (open/closed) of switches, energized buses, or active repair sites (Poudel et al., 2020, Wei et al., 2024, Sekhavatmanesh et al., 2018). Solution of the resulting program via commercial or open-source solvers yields a unique, step-by-step operational schedule, including switching order, actuator settings, and time indices.
• Algorithmic Sequencing / Recursion: Dynamic programming approaches determine optimal restoration action sequences by explicitly enumerating feasible action sets (e.g., minimal number of switching pairs for full load recovery (Yuan et al., 2017)), with cost metrics (e.g., minimal bus voltage) guiding action selection. All state evaluations are subject to full system checks (e.g., power-flow, voltage, radiality).
• Model-Based Inverse Paths: In image restoration via diffusion or normalizing flows, deterministic plans correspond to integration of the learned ODE from a degraded input to a restored sample, possibly in two-step restoration/degradation splits (Chen et al., 2023), continuous flows (Qin et al., 20 Jun 2025), or iterated deterministic denoising (Wang et al., 3 Mar 2025). The computational plan is specified completely by the model parameters and deterministic solver (e.g., Euler or Runge-Kutta).
• Agentic Policy Extraction: Where sequential decision processes are involved, as in (Lu et al., 21 Dec 2025), once a policy network is optimized (e.g., via policy gradient or actor-critic), the deterministic plan is a sequence of restoration tool invocations obtained by argmax action selection at each step and no stochastic exploration or rollback.

The following table summarizes core application domains and deterministic plan frameworks:

Domain	Planning Framework	Determinism Mechanism
Power grid	MILP/MISOCP, DP	Unique variable assignments, fixed action ordering
Image restoration	ODE/PDE integration, deterministic sampling, PnP optimization	Non-random solvers, score-based or learned flows
Multi-agent scheduling	Policy optimization, TSP assignment	Argmax policy extraction, MILP routes

4. Theoretical Guarantees and Empirical Validation

Rigorous non-asymptotic convergence and constraint satisfaction properties are essential for deterministic restoration plans:

• Optimality and Feasibility: Under standard convexity, smoothness, and Lipschitz assumptions (stated explicitly for each application), MILP and MISOCP-based plans are provably optimal within user-configured tolerances (e.g., MIP gap < 0.1%) (Poudel et al., 2020, Sekhavatmanesh et al., 2018, Wei et al., 2024). In non-convex domains, as with dynamic programming or deterministic flows, the plan is optimal within enumerated or modeled spaces, e.g., minimum switchings for full restoration (Yuan et al., 2017).
• Convergence Bounds: In diffusion-based deterministic sampling, non-asymptotic bounds on KL divergence or total variation distance between the restored distribution and the target law are proven, with explicit polynomial dependence on accuracy parameter $\varepsilon$ and model dimension (Chen et al., 2023).
• Empirical Metrics: Solution validity is confirmed via simulation of system operation at each plan step (e.g., checking voltage, frequency, line loading after each action (Jang et al., 2020)); in image restoration, empirical performance is reported on sharpness, PSNR, perceptual metrics (Qin et al., 20 Jun 2025, Wang et al., 3 Mar 2025).

5. Extensions, Limitations, and Adaptability

While deterministic restoration plans guarantee unique sequences for each instance, several practical and theoretical extensions deserve note:

• Scalability and Efficiency: For deep MILP/MISOCP models (e.g., thousands of buses or repair sites), randomized decomposition, bounding approaches, and hybrid heuristics accelerate solution times, but extracted deterministic plans remain unique for given instance and randomization seed (Chopra et al., 2022, Wei et al., 2024).
• Robustness to Uncertainty: Deterministic plans assume accurate knowledge of system status; online updating (feedback and re-optimization) incorporates state variations during restoration (Sekhavatmanesh et al., 2018, Jang et al., 2020).
• Hybrid Deterministic/Probabilistic Plans: In some image restoration paradigms, deterministic and stochastic components can be blended to optimize different trade-offs (data fidelity vs. perceptual diversity), with scalar parameters controlling this blend (e.g., the $\tau$ parameter in RDMD (Wang et al., 3 Mar 2025)).
• Human or Preference Constraints: Integration of human-aligned reward functions or semantic constraints prunes solution entropy and enforces plan reproducibility (as in preference-aligned diffusion restoration (Yao et al., 27 Jan 2026)).
• Real-Time Adaptability: Input-driven re-planning, e.g., rapid recomputation when actuator or system elements become unavailable, is supported in frameworks with explicit decision variable reassignment and deterministic selection rules (Jang et al., 2020, Poudel et al., 2020).

6. Representative Applications Across Domains

Deterministic restoration plans underpin critical decision-making in multiple high-stakes domains. In electric power systems, they provide the basis for automated service restoration modules in smart grids, distribution network reconfiguration with voltage regulation and load curtailment, and crew scheduling after extreme weather events (Poudel et al., 2020, Wei et al., 2024, Sekhavatmanesh et al., 2018). In image and signal restoration, deterministic plans are realized as ODE-based generative model inverses for diffusion models, policy extraction in agentic restoration chains, and MAP estimation in plug-and-play denoising principles (Chen et al., 2023, Lu et al., 21 Dec 2025, Chihaoui et al., 27 Mar 2025). Each context leverages the unambiguous mapping from problem instance to action sequence to enable transparency, auditability, operational safety, and performance guarantees.

7. Domain-Specific Variants and Future Directions

Distinct deterministic restoration plan types exist within and across domains:

• Discrete switching and reconfiguration plans in power grids, employing binary action variables to enforce network constraints, and sequence optimization for minimal operational cost (Poudel et al., 2020, Sekhavatmanesh et al., 2018).
• Model-based ODE/PDE restoration flows in learned generative models, including two-step restoration-degradation splittings and continuous velocity-matching in normalizing flows (Chen et al., 2023, Qin et al., 20 Jun 2025).
• Deterministic policy or tool-sequence plans in MDP or reinforcement learning frameworks, typically extracted by greedy deterministic action selection after policy convergence (Lu et al., 21 Dec 2025).
• Preference-aligned deterministic sampling using hierarchical logic and on-policy RL to minimize output entropy and inject structured constraints (Yao et al., 27 Jan 2026).

Anticipated research directions include scalable real-time algorithms for larger infrastructures, incorporation of richer semantic or preference constraints in restoration mapping, and theoretical unification of deterministic plan optimality across hybrid or multicriteria settings.