Learned Routing Network

• Learned Routing Networks are data-driven models that map network states and traffic demands to optimized routing decisions using neural architectures like GNNs, MLPs, and RL agents.
• They employ continuous relaxations and differentiable programming techniques to address the non-differentiability of classical routing methods, achieving significant throughput and latency improvements.
• Their applications span traffic engineering, adaptive SDN routing, and multi-modal model selection, with ongoing research focusing on scalability, stability, and real-time optimization.

A learned routing network is an algorithmic or neural system that maps inputs (such as network state, traffic demand, or multi-modal query features) to optimized routing decisions by leveraging data-driven models, most often neural networks or other trainable architectures. The term encompasses frameworks for packet routing in computer networks, traffic engineering, multi-agent coordination, multimodal query arbitration, and model selection in large-model deployments. Learned routing networks differ from classical routing by utilizing high-capacity models (e.g., GNNs, MLPs, RL agents) to synthesize complex dependencies and optimize for throughput, latency, cost, reliability, or accuracy, often in scenarios characterized by volatility or large action/state spaces.

1. Mathematical Foundations and Classical Formulations

The prototypical learned routing problem in network engineering is cast as an optimization over routing policies that minimize maximum link utilization or maximize throughput under demand uncertainty. For instance, the MinMaxLoad problem is given by

$w_* = \arg\min_{w>0} \max_{k=1\dots n_e} \rho_k(w)$

where $w$ is the link-weight vector, $\rho_k(w)$ the utilization of link $k$ under shortest-path routing, and $P(w)$ the path-incidence matrix derived from the weighted graph (Rusek et al., 2022).

Challenges arise from the discrete nature of shortest-path selection (via Dijkstra) and the non-differentiability of the max function. Modern approaches replace these with continuous relaxations (e.g., softmax-approximated max, GNN-driven “soft” path predictions), enabling gradient-based optimization:

$$L(w) = \widetilde{\max}(\hat\rho(w)),\quad \hat\rho(w) = \hat{P}(w)^T d \oslash c$$

where $\hat{P}(w)$ is a GNN-predicted soft routing matrix and $\widetilde{\max}$ is a log-sum-exp softmax.

Other learned routing networks cast routing as sequential decision-making, with RL agents operating on network graph state representations and optimizing cumulative reward (e.g., throughput, delay, reliability) (Abrol et al., 7 Feb 2024, Manfredi et al., 2020). In model selection, learned routers output $m^* = \arg\max_m u(x,m)$ for query $x$ and candidate models $m$, where $u(x,m)$ encodes joint utility (Li, 19 May 2025).

2. Network Architectures and Surrogate Models

Learned routing networks employ architectural choices matching domain constraints and optimization goals:

• Graph Neural Networks (GNNs): Used extensively for encoding topology, link state, and node features, enabling per-pair or per-path soft routing predictions (Rusek et al., 2022, Liu et al., 2021). Encode-process-decode GNNs with stacked message-passing blocks infer link-belonging probabilities $\hat{p}_{u,v}(w)$ per (u,v) pair.
• Feedforward Neural Networks (MLP, DNN): Map traffic demand histories and network state to per-OD-path scores or splitting ratios for tunnel selection, as in DOTE for stochastic WAN routing (Perry et al., 2023).
• Deep Graph Convolutional Neural Networks (DGCNN): Learn adaptive policies from both topological and flow attributes for SDN/next-gen routing (Abrol et al., 7 Feb 2024).
• Attentive, Matrix-Factorization, and Graph Routers: In multimodal settings or model selection, routers combine semantic embeddings, modality encodings, and user preferences to output routing scores or model probabilities by MLP, attention, or GNN mechanisms (Saini et al., 9 Nov 2025, Li, 19 May 2025).
• Reinforcement Learning Agents: Routing as an (multi-)agent MDP, often with decentralized policies and communication modules for coordination (Sykora et al., 2020, Mai et al., 2021).

3. Learning, Optimization, and Training Procedures

The fundamental training objectives include:

• Supervised Learning: Where ground-truth shortest-paths, routing labels, or optimal model choices are available, neural routers or value networks are trained by cross-entropy, MSE, or regression over utility/performance (Azzouni et al., 2017, Li, 19 May 2025).
• Imitation Learning: Model-free networks imitate optimal or heuristic routing policies, for instance by matching baseline heuristic outputs in SDN (Azzouni et al., 2017).
• Policy Gradient or TD Learning (RL): In reinforcement settings, policies are optimized for cumulative reward using Q-learning, SARSA, or REINFORCE, often in environments with stochastic traffic, topology changes, or packet-level dynamics (Abrol et al., 7 Feb 2024, Manfredi et al., 2020).
• Differentiable Programming: The use of autodiff tools (e.g., JAX) to backpropagate through surrogate routing loss, enabling rapid optimization of link weights or routing decisions (Rusek et al., 2022, Perry et al., 2023).
• Multi-Agent Training: Collaborative RL with multiple agents sharing function blocks and reward signals, aiming for both specialization and transfer across tasks (Rosenbaum et al., 2017, Collier et al., 2020).

Tables of typical metrics (as found in the literature):

Method	Routing Objective	Training Target(s)
Routing by Backprop	MinMaxLoad, OSPF	Surrogate $L(w)$ via GNN
DGCNN-RL	Throughput, Delay	Q-values via Deep Q-Learning
NeuRoute	Throughput, Heuristic	Cross-entropy on BH paths
Expert Model Routing	Utility/Cost Trade-off	CE loss + cost penalty

4. Practical Applications and Deployment

Learned routing networks have been deployed or validated in:

• Traffic Engineering: Routing by Backprop achieves $\approx$25% improvement over default OSPF configurations and serves as a high-quality initializer for TE solvers, yielding an additional $\approx$30% reduction in max-load when bootstrapped (Rusek et al., 2022).
• Adaptive SDN Routing: DGCNN-RL increases throughput by up to 7.8% and reduces delay by up to 16.1% over OSPF in high-load scenarios; RLSR-Routing demonstrates superior load balancing and faster convergence via Q-table reuse and segment routing support (Abrol et al., 7 Feb 2024, Wumian et al., 23 Sep 2024).
• Wireless Networks and IoT: Packet-centric and relational RL agents generalize across topologies, congestion regimes, and link failure patterns, outperforming shortest-path and backpressure (Manfredi et al., 2020). Single-graph supervised training with domain-guided features yields universally near-optimal policies on random graphs (Chen et al., 2023).
• Multi-Agent Mobility and Coordination: MARVIN and Graph Attention RL methods outperform VRP/ILP solvers on mapping and fleet coordination tasks by leveraging value iteration and communication modules (Sykora et al., 2020, Mai et al., 2021).
• Multimodal Model Routing and Expert Selection: Frameworks that arbitrate between LLMs and other expert models cut API cost by >67% versus monolithic deployments, matching or exceeding answer quality, with multi-agent orchestration and feature-based gating (Saini et al., 9 Nov 2025).
• LLM Routing and Model Arbitration: Surprisingly, non-parametric kNN routers match or outperform complex learned routers in utility-prediction across a spectrum of tasks and modalities—with lower sample complexity and robust performance (Li, 19 May 2025).

5. Computational Characteristics and Performance Metrics

Execution time and scalability vary with architecture and application:

• RBB's surrogate TE optimizer executes in 36–322 ms per step on moderate-topology networks (Tesla V100 GPU); three GD iterations suffice for convergence (Rusek et al., 2022).
• DGCNN policies with prioritized replay converge in ≈500 episodes on mid-size graphs. Online inference step ≈30 ms (Abrol et al., 7 Feb 2024). NeuRoute's TRU achieves $>$99.95% path-match accuracy at ~30 ms per inference—over an order of magnitude speedup over Dijkstra (Azzouni et al., 2017).
• RL routing agents generalize zero-shot between topologies and traffic regimes given suitable feature design and training (Chen et al., 2023, Manfredi et al., 2020).
• Multi-modal routing networks reduce mean latency by ~18% and boost throughput by ~20% compared to always-premium baselines; only 4% of queries sent to expensive LLMs in optimized policy (Saini et al., 9 Nov 2025).
• Learned drift-plus-penalty optimizers (Neural-B variants) obtain 10–15% queue reductions over classical backpressure and SPBP kernels, and retain Lyapunov throughput optimality (Rashwan et al., 11 Sep 2025).

6. Limitations, Generalization, and Prospects

Learned routing networks encounter:

• Scalability constraints in very large topologies, particularly for packet-level RL or multi-agent communication modules.
• Dependence on Measurement Fidelity: Some policies require real-time global state information or accurate telemetry—practical assessment needed for deployment in WAN, SDN controllers, or next-gen satellite LEO networks (Liu et al., 2021, Abrol et al., 7 Feb 2024).
• Domain Drift and Adaptivity: Policies can be sensitive to unmodeled dynamics (e.g., sudden traffic regime changes, link failures) unless designed for continual or meta-learning updates (Azzouni et al., 2017).
• Sample Complexity: Non-parametric methods like kNN excel at low data volumes and strong locality; parametric routers (MLP, GNN) are more data-hungry but may eventually outperform in large datasets (Li, 19 May 2025).
• Interpretability and Stability: Piecewise-linear parametrizations equipped with explicit stability constraints (Foster–Lyapunov) offer interpretable, provably stabilizing templates, training in seconds vs. hours for unconstrained deep RL (Wu et al., 27 Aug 2024).

A plausible implication is that the choice of routing model should be informed by topological scale, measurement capabilities, traffic volatility, and real-time requirements, leveraging interpretable, efficient parameterizations where stability is critical, and using more expressive but heavier models (GNNs, RL) where dynamic optimization for complex objectives is feasible.

7. Impact and Continuing Research Directions

Learned routing networks have shifted the computational paradigm in routing and traffic engineering:

• The integration of GNNs, RL, and differentiable programming enables sub-second, continuously adaptive decisions aligned with fluctuating demand, latency, and reliability requirements.
• Multi-modal expert model routing and cost-aware arbitration frameworks enable large-scale resource-efficient AI deployments.
• Foundational analysis of sample complexity and locality underscores the need for benchmarking against simple baselines before escalating model complexity (Li, 19 May 2025).
• Ongoing work pursues hierarchical GNNs, decentralized learning, continual adaptation, and the synthesis of stability guarantees in fully trainable architectures.

Further advances are expected in scalability (hundreds to thousands of nodes), generalization to unseen topologies/types, and rigorous integration with operational requirements in SDN, satellite, IoT, and multi-agent AI systems.