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

math.OCOct 20, 2026

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

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

Organizations: Department of Electrical, Computer & Energy Engineering, University of Colorado Boulder · Davidson School of Chemical Engineering, Purdue University · ExxonMobil Technology and Engineering Company

Abstract

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.

Figures & tables

Figure 1: Geometric intuition of tuning low fidelity surrogate to better approximate a high-fidelity model.

Figure 2: Cost and feasibility evaluation for the OPF benchmark instances. The top panels show cost correlation, and the bottom panels show feasibility across the test samples.

Panel of Figure 3 — Figure 3: Relative error across systems under transductive and inductive learning. Filled and unfilled violins indicate transductive and inductive results, respectively.

Figure 4: Cost and feasibility evaluation for the OGF benchmark instances. The top panels show cost correlation, and the bottom panels show feasibility across the test samples.

Figure 5: Relative objective error for the inductive OGF experiment on GasLib-135-F. Violin plots show the distributions for the self-supervised, semi-supervised, and untuned models.

Figure 6: Cost and feasibility evaluation for the pooling benchmark instances. The top panels show cost correlation, and the bottom panels show feasibility across the test samples.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Graph Representation
Stage	Layer	Input	Output
Node features	Input	–	$6\times\|N\|$
Edge features	Input	–	$5\times\|E\|$
GNN Encoder
Message passing	GATv2Conv 1	$(6,5)$	$1024\times\|N\|$
	GATv2Conv 2	$(1024,5)$	$512\times\|N\|$

Appendix

Table 1: GNN architecture and optimization-aware pipeline for OPF.

Graph Representation
Stage	Layer	Input	Output
Node features	Input	–	$4\times\|J\|$
Edge features	Input	–	$3\times\|P\|$
GNN Encoder (5 Attention Heads)
Message passing	GATv2Conv 1	$(4,3)$	$512\times 5=2560$
	GATv2Conv 2	$(2560,3)$	$256\times 5=1280$

Appendix

Table 2: GNN architecture and optimization-aware pipeline for the gas-flow problem.

Graph representation
Component	Operation	Input	Output
Node features	Input	–	$4\times\|V\|$
Edge features	Input	–	$5\times\|E\|$
Input embedding
Node embedding	Dense	$4$	$32$
GNN encoder

Appendix

Table 3: GNN architecture and optimization-aware pipeline for the pooling problem.

Index	Description
$i,j$	Buses
$g$	Generators

Appendix

Table 10

Set	Description
$B$	Set of buses
$G$	Set of generators
$E$	Set of branches represented by bus pairs $(i,j)$
$R$	Set of reference buses
$G_{i}$	Set of generators connected to bus $i$
$E_{i}$	Set of branches connected to bus $i$

Appendix

Table 11

Parameter	Description
$P_{i}^{D}$	Active-power demand at bus $i$
$Q_{i}^{D}$	Reactive-power demand at bus $i$
$V_{i}^{L}$	Lower voltage-magnitude bound at bus $i$
$V_{i}^{U}$	Upper voltage-magnitude bound at bus $i$
$P_{g}^{L}$	Lower active-power bound of generator $g$
$P_{g}^{U}$	Upper active-power bound of generator $g$

Appendix

Table 12

Variable	Description
$p_{g}$	Active-power output of generator $g$
$q_{g}$	Reactive-power output of generator $g$
$v_{i}^{r}$	Real component of the complex voltage at bus $i$
$v_{i}^{i}$	Imaginary component of the complex voltage at bus $i$
$p_{ij}$	Active-power flow from bus $i$ to bus $j$
$q_{ij}$	Reactive-power flow from bus $i$ to bus $j$

Appendix

Table 13

Index	Description
$i,j$	Junctions
$k$	Pipes
$c$	Compressors
$r$	Receipts (gas supplies)
$d$	Deliveries (gas withdrawals)
$v$	Vertices in the polyhedral pipe relaxation

Appendix

Table 14

Set	Description
$J$	Set of junctions
$P$	Set of pipes
$C$	Set of compressors
$R$	Set of dispatchable receipts
$D$	Set of nondispatchable deliveries
$R_{i}$	Set of receipts connected to junction $i$

Appendix

Table 15

Parameter	Description
$F_{k}$	From-junction of pipe $k$
$T_{k}$	To-junction of pipe $k$
$F_{c}$	From-junction of compressor $c$
$T_{c}$	To-junction of compressor $c$
$P_{i}^{L}$	Lower pressure bound at junction $i$
$P_{i}^{U}$	Upper pressure bound at junction $i$

Appendix

Table 16

Variable	Description
$g_{r}$	Injection at receipt $r$
$f_{k}$	Signed mass flow through pipe $k$
$h_{c}$	Signed mass flow through compressor $c$
$\pi_{i}$	Squared pressure at junction $i$
$\rho_{c}$	Squared pressure-ratio variable of compressor $c$
$q_{k}$	Low-fidelity auxiliary for $f_{k}\|f_{k}\|$

Appendix

Table 17

Index	Description
$i$	Raw materials
$j$	Products
$k$	Qualities
$l$	Pools

Appendix

Table 18

Set	Description
$I$	Set of raw materials
$J$	Set of products
$K$	Set of qualities
$L$	Set of pools
$T_{X}$	Set of input-to-pool connections $(i,l)$ (pool feeds)
$T_{Y}$	Set of pool-to-product connections $(l,j)$

Appendix

Table 19

Parameter	Description
$F_{ij}$	Per-unit flow weight for direct arc $(i,j)$
$F_{il}$	Per-unit flow weight for input-to-pool arc $(i,l)$
$F_{lj}$	Per-unit flow weight for pool-to-product arc $(l,j)$
$A_{i}^{U}$	Upper bound on the availability of raw material $i$
$P_{i,k}$	Value of quality $k$ for raw material $i$
$D_{j}^{U}$	Upper bound on demand for product $j$

Appendix

Table 20

Variable	Description
$y_{l,j}$	Flow from pool $l$ to product $j$
$z_{i,j}$	Direct flow from raw material $i$ to product $j$
$p_{l,k}$	Level of quality $k$ in pool $l$
$q_{i,l}$	Proportion of flow from raw material $i$ to pool $l$
$v_{i,l,j}$	Flow from raw material $i$ to product $j$ through pool $l$

Appendix

Table 21

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