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Self-Supervised Tuning of Low-Fidelity Optimization Models via High-Fidelity Feedback

Gonzalo E. Constante-Flores, André H. Quisaguano-Paredes, Asha Ramanujam, Rishabh Gupta, Can Li

arXiv:2610.28851v1math.OCcs.LG

TL;DR

Low-fidelity optimization models are computationally efficient but can mismatch high-fidelity formulations, while prior calibration approaches may require expensive bilevel optimization. This paper tunes LF model parameters using HF operational feedback without HF solution labels, improving objective quality and feasibility across three applications and generalizing to unseen conditions.

  • Problem

    LF models can exhibit systematic mismatches with HF formulations, while existing calibration approaches may require expensive bilevel optimization and HF solution labels.

    Source: §1 · p. 2

  • Method

    The framework embeds a parameterized LF optimization problem as a differentiable layer and trains it using HF operational feedback without HF-optimal solution labels.

    Source: §1 · p. 2, §4 · p. 23

  • Results

    Across three applications, self-supervised training improves HF performance and feasibility over untuned LF models and generalizes to unseen operating conditions.

    Source: §1 · p. 3, §4 · p. 23

  • Takeaways & Limitations

    The learned parameterizations retain the LF model structure while adapting its parameters to operating conditions, with generalization to unseen systems reported for network applications.

    Source: §4 · pp. 22–23

    The learned LF decisions are restricted to the model family's reachable decision set, which can be substantially smaller than its feasible decision set.

    Source: §2.2 · p. 8

Abstract

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Low-fidelity (LF) optimization models are widely used in large-scale operations because they can be solved efficiently and reliably, but they often exhibit systematic errors relative to high-fidelity (HF) formulations that better capture physical and operational constraints. Existing learning-based approaches typically rely on supervised or semi-supervised training with HF solutions, which are expensive to compute and may not align with HF operational objectives. We propose a self-supervised framework that parameterizes LF optimization models directly from HF feedback. The low-fidelity problem is embedded as a differentiable optimization layer and trained end-to-end, while a HF correction problem evaluates its solution and provides gradient information through its dual variables, without requiring HF solution labels. We evaluate the framework on optimal power flow, optimal gas flow, and pooling problems, spanning different sources of low- and high-fidelity mismatch. Across all three applications, the self-supervised approach improves objective quality and feasibility over untuned LF models and generally outperforms semi-supervised training. The learned parameterizations also generalize to unseen operating conditions and, for network applications with up to 30,000 nodes, to systems not encountered during training.

1. Introduction

Low-fidelity optimization models trade accuracy for computational efficiency, yet systematic mismatches can yield suboptimal or infeasible decisions. The paper proposes tuning LF model parameters through high-fidelity operational feedback and reports improved objective quality and feasibility across three applications.

1. Introduction

  • LF models trade modeling accuracy for computational efficiency, and their approximations or omitted constraints can produce systematic mismatches that lead to suboptimal or infeasible decisions.For power systems, DC-OPF simplifies AC-OPF physics, including by assuming fixed voltage magnitudes and negligible line losses.Source: §1 · pp. 1–2
  • Prior calibration approaches can preserve LF structure but may require repeated, computationally expensive bilevel solves, while related methods use labels or direct nonlinear-objective gradients.The proposed framework instead learns context-dependent corrections to a structured LF model and evaluates decisions with a distinct HF model.Source: §1 · pp. 2–4
  • The framework embeds the LF problem as a differentiable optimization layer and trains its parameters against HF operational feedback, avoiding HF-optimal solution labels and costly bilevel procedures.The HF problem evaluates the LF decisions and returns objective and sensitivity information for gradient-based training.Source: §1 · p. 2
  • Across all three applications, self-supervised training outperforms untuned LF models and semi-supervised baselines in HF objective value, feasibility, scalability, and generalization to unseen operating conditions.The experiments cover DC approximations of nonlinear AC power-flow physics, gas-flow relaxations, and McCormick-based pooling approximations.Source: §1 · pp. 2–3

2. Proposed Framework

The section presents a framework for parameterizing and tuning low-fidelity models using high-fidelity evaluation, distinguishing the goals of matching high-fidelity decisions and optimizing high-fidelity performance. It describes alternative training objectives and graph neural networks for structure-aware corrections in network-flow problems.

2.1. Problem Setup

  • A context-dependent map tunes selected LF parameters, and a slack-augmented HF evaluation loss measures operational performance and corrections needed to reconcile LF and HF decisions.The tunable components are specified in advance; the evaluation loss is the HF problem’s optimal value.Source: §2.1 · p. 6

2.2. Approximation Capacity of the Parameterized LF Model

  • The LF reachable decision set can be smaller than the feasible-decision set, imposing an architecture-independent limit that no correction map or training procedure can overcome.The framework learns optimization-model parameters rather than decisions directly, so induced decisions must remain optimizers of the parameterized LF model.Source: §2.2 · pp. 8, 11
  • The local Jacobian of shared LF decisions with respect to tunable parameters identifies attainable first-order directions; the linearized residual diagnoses discrepancy alignment but is not the exact approximation error.The selected parameters affect this local capacity, while nonlinear changes may yield additional directions away from the nominal correction.Source: §2.2 · pp. 8–9

2.3. Supervised Learning

  • Supervised learning generates sample-specific correction labels by calibrating the LF model to reproduce HF reference decisions, then trains a correction map on those labels.The calibration problems are solved independently across samples.Source: §2.3 · p. 11

2.4. Semi-Supervised Learning

  • Semi-supervised learning avoids correction-label generation by training the correction map end-to-end through the tuned LF optimization problem to reproduce shared HF-optimal decision components.It still requires HF reference decisions for training contexts.Source: §2.4 · p. 12

2.5. Self-Supervised Learning

  • Self-supervised learning trains directly on the HF evaluation loss, using HF constraint multipliers to propagate sensitivities without differentiating through HF solver iterations.The loss includes both HF operational performance and feasibility corrections; training requires solving both LF and HF problems.Source: §2.5 · pp. 12–13

2.6. Interpretation of the Learning Objectives

  • Semi-supervised learning minimizes Euclidean error from an HF-optimal shared decision, whereas self-supervised learning evaluates the LF decision by its HF evaluation value.The self-supervised objective is closer to the empirical objective defined for the framework.Source: §2.6 · p. 13, §2.5 · p. 13
  • When HF-optimal decisions are outside the LF-reachable set, semi-supervised learning selects the closest attainable decision, while self-supervised learning minimizes HF evaluation loss.The closest reachable decision need not provide the best HF performance; training can also be harder when the HF evaluation is nonconvex, nonsmooth, locally flat, or sensitive to local solution branches.Source: §2.6 · pp. 14–15
  • Locally, semi-supervised learning evaluates attainable discrepancy directions by Euclidean distance, whereas self-supervised learning uses an HF-curvature-weighted quadratic form.The criteria are locally aligned when the curvature matrix acts approximately as a scalar multiple of the identity over relevant decision directions; otherwise, they prioritize different discrepancy components.Source: §2.6 · pp. 14–15

2.7. Special Considerations for Network Flow Problems

  • For network flows, a GNN can generate node-wise, edge-wise, or global LF-parameter corrections while respecting graph locality and permutation symmetries.Shared parameters can transfer across networks with different sizes when node and edge features retain consistent meanings, and may improve sample efficiency and generalization to unseen conditions or topologies.Source: §2.7 · pp. 15–16

3. Results and Discussion

Across OPF, optimal gas flow, and pooling experiments, the self-supervised approach improves HF objective quality and feasibility over untuned LF models, with more consistent gains than semi-supervised training. The experiments also examine generalization to unseen conditions and computational performance at scale.

3. Results and Discussion

  • Evaluation uses 1,000 unseen test instances; the HF objective reference is a high-quality solver value, not necessarily a certified global optimum.Feasibility is summarized by application-specific mean constraint violation and its P90 across samples.Source: §3 · pp. 16–17

3.1.1. Results

  • Mean relative objective error below 0.4% and P90 feasibility violation below 0.5 MW; self-supervised OPF performs best across systems.The approach also reduces HF corrective actions relative to the untuned model, with especially pronounced gains on larger systems.Source: §3.1.1 · pp. 17–18, Table 1 · p. 18, Figure 2 · p. 18
  • On the 500- to 30,000-bus OPF systems, inductive learning from the 500-bus system achieves relative errors below 2% and consistently beats the untuned model.Figure 3 presents relative errors across systems for transductive and inductive learning.Source: §3.1.1 · p. 18, Figure 3 · p. 19
  • For the 13,659-bus OPF system, the proposed approach is approximately 26× faster than AC-OPF; on 30,000 buses, it remains approximately 3.5× faster under GPU acceleration.The inference-time comparison is reported in seconds; the 30,000-bus comparison uses GPU implementations for both approaches.Source: §3.1.1 · pp. 18–19, Table 2 · p. 19

3.2.1. Results

  • Mean objective errors are 0.032% on GasLib-40-E and 0.457% on GasLib-135-F; self-supervised OGF also substantially reduces P90 feasibility violations.Semi-supervised training provides essentially no improvement over the untuned model on GasLib-40-E and only moderate improvement on GasLib-135-F.Source: Table 3 · p. 20, §3.2.1 · p. 20
  • On unseen GasLib-135-F, inductive self-supervised learning achieves 0.669% mean relative error versus 1.961% for semi-supervised learning.The inductive result is below 1% and close to the transductive self-supervised error of 0.457%.Source: §3.2.1 · p. 21, Figure 5 · p. 21

3.3.1. Results

  • Across all five pooling cases, self-supervised training yields mean objective errors between 0.355% and 2.304% and substantially smaller feasibility violations than other approaches.Semi-supervised training improves on the untuned model in Cases 1–4 but performs worse in Case 5.Source: §3.3.1 · pp. 21–22, Table 4 · p. 22, Figure 6 · p. 22

4. Conclusion

The framework tunes low-fidelity optimization models using high-fidelity feedback while preserving their structure and computational advantages. Experiments show improved high-fidelity performance and feasibility across three applications, with generalization to unseen conditions and, for network applications, unseen systems.

4. Conclusion

  • The method embeds the low-fidelity model and high-fidelity evaluation problem in training to learn parameters against the operational objective without high-fidelity solution labels.The low-fidelity model’s structure and computational advantages are preserved while its parameters are adjusted as a function of operating conditions.Source: §4 · pp. 22–23
  • Across power flow, gas network optimization, and pooling, self-supervised training improves high-fidelity performance and feasibility across distinct sources of model mismatch.Learned parameterizations also generalize to unseen operating conditions and, for network applications, to systems not encountered during training.Source: §4 · p. 23
  • High-fidelity models can provide useful training signals without being solved to global optimality to generate labels.Future work will investigate discrete decisions, stochastic and multistage formulations, and learning which low-fidelity parameters to tune.Source: §4 · p. 23

Appendix A: Common Computational and Training Details

The appendix describes a shared training pipeline that maps operating conditions and static problem data to low-fidelity model parameters, solves the resulting model, and evaluates its decisions with the high-fidelity model. It also explains differentiation through the low-fidelity model, high-fidelity sensitivities, and the parameterized DC-OPF.

Appendix A: Common Computational and Training Details

  • A GNN maps operating conditions and static problem data to low-fidelity parameters; the solved low-fidelity decisions are then evaluated with the corresponding high-fidelity model.Self-supervised training uses the high-fidelity evaluation objective and dual-based sensitivities, whereas semi-supervised training uses offline high-fidelity reference solutions as decision targets.Source: Appendix A · p. 26

A.1. Differentiation Through the LF Models

  • Implicit differentiation through low-fidelity optimality conditions computes parameter sensitivities, which are propagated through the GNN using automatic differentiation.Small regularization of the internal linear systems improves numerical stability without materially affecting the low-fidelity solution or computed sensitivities.Source: §A.1 · p. 26

A.2. HF Sensitivities

  • High-fidelity objective sensitivities come from dual multipliers for constraints coupling the low- and high-fidelity decisions, so the backward pass does not differentiate through high-fidelity solver iterations.Differentiation is required through the low-fidelity optimization problem, but not through internal HF NLP or QCP solver iterations.Source: §A.2 · p. 26

B.1. HF and LF Models

  • The AC-OPF high-fidelity model captures nonlinear power-flow physics and operating limits, while the convex quadratic DC-OPF low-fidelity model uses linearized active-power flows and omits active-power losses.The active-power generation dispatch is passed from the DC-OPF to the high-fidelity evaluation, which includes corrective redispatch and penalizes those corrective actions.Source: §B.1 · pp. 26–27

B.2. Parameterized DC-OPF Model

  • A GNN predicts positive bus-specific demand scaling coefficients for the DC-OPF, adjusting model parameters rather than directly predicting generator dispatch.This parameterization is intended to compensate implicitly for effects omitted by the DC approximation, particularly active-power losses.Source: §B.2 · p. 27

B.3. Benchmark Systems and Instance Generation

The OPF experiments cover systems from 500 to 30,000 buses and unseen operating conditions, with inductive tests applying a GNN trained on the 500-bus system to larger networks. The appendix also details the graph representation, training procedures, and large-scale implementation.

B.3. Benchmark Systems and Instance Generation

  • OPF benchmark systems range from 500 to 30,000 buses, with active demand perturbed by up to ±30% and reactive demand from nominal to 15% lower.Evaluation uses 1,000 unseen operating conditions from the same distribution; inductive tests apply a GNN trained only on the 500-bus system to larger networks without retraining.Source: §B.3 · pp. 27–28

B.4. Graph Representation and GNN Architecture

  • The GNN represents buses as graph nodes and active transmission branches as edges, using three edge-aware GATv2 layers and a decoder to output a positive coefficient per bus.The convolutional hidden dimensions are 1024, 512, and 256; the decoder dimensions are 256 →64 →1, with a Softplus output.Source: §B.4 · p. 28

B.5. Training Procedure

  • After 100-epoch coefficient initialization to βᵢ=1 at every bus (the original untuned DC-OPF), self-supervision uses the complete LF–HF pipeline and HF evaluation objective, while semi-supervision omits the HF evaluation layer.HF evaluation-value sensitivities to DC-OPF dispatch are obtained from dual multipliers of generator-linking constraints.Source: §B.5 · pp. 28–29

B.6. Large-Scale 30,000-Bus Implementation

  • For the 30,000-bus system, the HF AC-OPF correction model runs on a GPU, and dual multipliers provide gradients without differentiating through solver iterations.This retains the self-supervised training principle used for smaller systems while using GPU acceleration for the HF model.Source: §B.6 · p. 29

B.7. Training Time

The appendix reports matched training-time budgets for self-supervised and semi-supervised OPF training, then describes the gas-flow models and how the LRWP objective is parameterized. The gas parameterization reshapes the tractable low-fidelity objective rather than reconstructing the nonlinear gas equations.

B.7. Training Time

  • Self-supervised and semi-supervised OPF training use the same time budget per system, although semi-supervised training generally completes more epochs because it omits the HF model during training.Table 7 summarizes computational cost, but the supplied text gives no specific timing values.Source: §B.7 · p. 29, Table 7 · p. 29

C.1. HF and LF Models

  • The nonlinear WP gas model captures pressure-flow relationships and operating features, while the easier LRWP low-fidelity model uses polyhedral and McCormick relaxations that can yield suboptimal or nonlinear-incompatible receipt injections.The receipt injections are the shared decisions, and slack variables quantify adjustments needed for compatibility with the nonlinear high-fidelity model.Source: §C.1 · p. 30

C.2. Parameterized LRWP Model

  • For gas optimization, the GNN modifies the LRWP objective with learned positive coefficients, reshaping its optimizer’s receipt decisions to perform well under nonlinear WP evaluation.The learned terms do not explicitly reconstruct the nonlinear gas equations.Source: §C.2 · p. 30

C.3. Benchmark Systems and Instance Generation

The gas-flow experiments use modified GasLib networks to assess generalization, with a graph neural network producing positive coefficients for the LRWP model.

C.3. Benchmark Systems and Instance Generation

  • Each GasLib network has 1,000 realizations; one trains the model, while the remaining operating conditions assess generalization.The GasLib-40-E-trained model is also applied directly to GasLib-135-F in an inductive experiment.Source: §C.3 · pp. 30–31
  • Selected compressor ratios and pipe diameters are changed and receipt locations added to increase the relaxed–nonlinear mismatch without changing the learning framework.The benchmarks are GasLib-40-E and GasLib-135-F; Table 8 reports their problem sizes.Source: §C.3 · pp. 30–31, Table 8 · p. 30

C.4. Graph Representation and GNN Architecture

  • Junctions are graph nodes and physical pipes are edges oriented according to their stored from-junction and to-junction direction.Pipe diameter, length, and friction factor are identified as edge attributes.Source: §C.4 · p. 31
  • Three edge-aware GATv2 layers with five attention heads feed junction- and pipe-level decoders that produce positive β_i and α_k coefficients.The layers have 512, 256, and 128 features per head, yielding concatenated dimensions of 2560, 1280, and 640; both decoder outputs use Softplus.Source: §C.4 · p. 31

C.5. Training Procedure

The gas-flow training procedure initializes the GNN to reproduce the nominal LRWP objective, then trains it using either HF feedback or HF reference receipt injections.

C.5. Training Procedure

  • Initialization trains the GNN for 100 epochs to output coefficients equal to one, reproducing the nominal LRWP objective before either training approach.AdamW uses a learning rate of 10^-4 for this initialization.Source: §C.5 · p. 31
  • Self-supervised training uses GNN-predicted α and β to parameterize LRWP, with DiffOpt.jl differentiation and HF receipt-linking dual multipliers providing sensitivity to the LF receipt vector.Source: §C.5 · p. 31
  • Semi-supervised training omits the HF evaluation layer and compares the predicted receipt vector with HF reference receipt injections.Both approaches use AdamW at 10^-5, mini-batches of one, and random seed 20.Source: §C.5 · p. 31

C.6. Training Time

The pooling pipeline evaluates solutions from a parameterized low-fidelity relaxation in the original high-fidelity model; gas-network training takes a few minutes under equal training-time budgets.

C.6. Training Time

  • Gas-network models are substantially smaller than OPF and pooling instances, so training finishes in a few minutes, with equal training time for both approaches on each network.Table 10 summarizes OGF training times in minutes.Source: §C.6 · p. 32, Table 10 · p. 32

D.1. HF and LF Models

  • The pooling HF model is the original nonconvex QCP, while its LF counterpart uses a McCormick relaxation with small quadratic terms to discourage undesirable extreme-point solutions.The paper calls this LF formulation the Quadratic McCormick relaxation; after solving LF, its q-variables are fixed in HF while remaining variables are optimized under the original bilinear relationships, with slack variables where needed.Source: §D.1 · p. 32

D.2. Parameterized Quadratic McCormick Model

  • The GNN predicts four families of node-, edge-, and path-level objective coefficients, changing which point the tractable LF feasible region selects without changing its constraints.For self-supervised learning, the resulting LF solution is evaluated using the HF pooling model.Source: §D.2 · p. 33

D.3. Benchmark Systems and Instance Generation

The pooling experiments evaluate generalization across perturbed synthetic instances and use a heterogeneous-graph GNN with staged HF-feedback training.

D.3. Benchmark Systems and Instance Generation

  • Five synthetic pooling instances increase in size, and test realizations perturb nominal supply and demand by ±10%.Training starts on the nominal instance, then fine-tunes on it plus 24 target-distribution realizations; evaluation uses unseen realizations from that distribution.Source: §D.3 · p. 33, Table 11 · p. 33

D.4. Graph Representation and GNN Architecture

  • The heterogeneous graph represents sources, pools, and terminals as node types, with three edge relations and separate message-passing operators aggregated by summation.Features include node type and capacity or demand, and edge transportation cost, flow bound, and relation; the three GATv2 layers use four attention heads and residual connections.Source: §D.4 · pp. 33–34
  • Edge decoders map endpoint embeddings and edge features to η, β, and σ, while triplet decoders use source–pool–terminal embeddings to produce additional coefficients.Table 12 summarizes the GNN architecture and optimization-aware pipeline.Source: §D.4 · p. 34, Table 12 · p. 34

D.5. Training Procedure

  • Self-supervised training first uses a conservative slack-augmented HF objective for 100 epochs, then an aggressive HF evaluation model for 100 more, before fine-tuning on 25 instances.The fine-tuning set comprises the nominal instance and 24 additional target-distribution samples; batches contain five instances.Source: §D.3 · p. 33, §D.5 · p. 35
  • Semi-supervised training instead solves each training instance's full-space QCP under a 500-second limit to generate a target q, then trains and fine-tunes by MSE.Its fine-tuning uses the same nominal and 24 additional samples as self-supervised training, with batches of five.Source: §D.5 · p. 35

D.6. Training Time

Pooling experiments require substantially more computation than gas experiments, and the two training approaches receive equal time budgets per instance.

D.6. Training Time

  • Pooling experiments are substantially more computationally demanding than gas experiments because both LF and HF models grow rapidly in size.Source: §D.6 · p. 35
  • Self-supervised and semi-supervised approaches receive the same total training-time budget for each benchmark instance.Source: §D.6 · p. 35
  • Table 13 summarizes training times for the pooling experiments in minutes.Source: Table 13 · p. 35

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