Two-Stage Surrogate Workflow

• The paper’s main contribution is a two-stage surrogate workflow that screens candidates using a rapid learner and refines them with a rigorous evaluator.
• The methodology leverages a clear separation between fast approximation and detailed validation, reducing computational cost and enhancing reliability.
• The approach applies across multiple domains—from composite microstructure design to surgery scheduling—demonstrating improved accuracy and efficiency.

A two-stage surrogate workflow is a computational pattern for expensive inverse, design, optimization, or inference tasks in which a first stage performs rapid screening, approximation, or proposal generation, and a second stage applies a stricter evaluation, realization, or validation criterion to the reduced problem. In "Two-Stage Surrogate Modeling for Data-Driven Design Optimization with Application to Composite Microstructure Generation" (Pourkamali-Anaraki et al., 2024), this pattern is instantiated as a “learner” that narrows the search space and an “evaluator” wrapped by conformal inference; related decompositions appear as convex recourse surrogates in surgery scheduling (Almoghrabi et al., 2023), SVM halfspace replacements for network constraints in stochastic unit commitment (Javadi et al., 12 Sep 2025), matrix-space training followed by electromagnetic realization in photonic inverse design (Muda et al., 23 Apr 2026), and Gaussian-process gating between crystal generation and expensive property evaluation (Alvi et al., 26 Jun 2026). This suggests that the expression denotes an architectural principle rather than a single algorithmic template.

1. Core architecture

In the inverse-design formulation of Pourkamali-Anaraki et al., the first stage trains a learner $\hat f_{\mathcal D}$ on data $\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$, computes a screening error

$$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$

sorts this error over the search space $\Omega$, and retains the top $B$ points as a reduced candidate set $\Omega_r$. The second stage trains an evaluator $\hat\mu$, constructs a conformal prediction interval $\hat C_\alpha(x)$ for each $x\in\Omega_r$, and accepts a candidate if and only if $y^{\text{target}}\in \hat C_\alpha(x)$; if multiple candidates are feasible, the one with the smallest $\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$0 is returned, and if none satisfy the rule the method reports “no reliable solution at level $\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$1” (Pourkamali-Anaraki et al., 2024).

Other two-stage surrogates preserve the same separation while changing the semantics of the stages. In elective and emergency surgery planning, the first stage is an offline two-stage stochastic program, and the second-stage recourse cost is replaced by a convex piecewise-linear surrogate computed in preprocessing; this produces a single-stage MIP for the offline plan and is paired with a greedy online policy for actual emergency insertion and cancellation (Almoghrabi et al., 2023). In stochastic unit commitment, the expensive object is not a forward map but a large family of transmission constraints, and the second-stage surrogate is a learned halfspace that replaces the original $\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$2 line-flow inequalities with $\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$3 explicit network constraints (Javadi et al., 12 Sep 2025).

A broader reading appears in surrogate-based Bayesian inference, where Stage 1 is initial surrogate construction from an initial design and Stage 2 is sequential refinement via active learning. There the second stage does not merely score fixed candidates; it repeatedly acquires new simulator evaluations, updates the emulator, and updates the surrogate posterior until a stopping criterion is met (Roberts et al., 13 Mar 2026). The common structure is therefore not tied to a particular model family, but to a deliberate decomposition between inexpensive approximation and more decision-relevant refinement.

2. Mathematical forms

The most explicit canonical formulation is the learner–evaluator scheme. The learner solves empirical risk minimization,

$\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$4

with $\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$5 typically the squared loss, and the evaluator uses cross-validation+ conformal prediction to construct $\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$6-level intervals satisfying

$\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$7

This stage separation is presented as a form of data-driven regularization that reduces reliance on hand-crafted regularizers and adds statistical guarantees on solution reliability (Pourkamali-Anaraki et al., 2024).

A second mathematical pattern replaces a hard-to-solve downstream term by a surrogate that is simpler but still optimization-compatible. In surgery scheduling, the expected second-stage cost for specialty $\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$8 and load $\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$9 is approximated by

$$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$0

and this convex surrogate is embedded directly into a mixed-integer formulation through auxiliary variables and linear inequalities (Almoghrabi et al., 2023). In stochastic unit commitment, the replacement object is not a scalar cost but a polyhedral feasibility region: a learned SVM hyperplane

$$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$1

acts as a surrogate for the original set of DC power-flow transmission limits at each scenario and time (Javadi et al., 12 Sep 2025).

A third pattern uses latent-space reduction before emulation. In high-dimensional inverse uncertainty quantification, outputs $$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$2 are mapped to a reduced space by PCA,

$$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$3

and a multivariate surrogate is trained in the reduced space, specifically a PCE for each latent coordinate, after which predictions are reconstructed in the original output space (Yang et al., 2024). A related decomposition appears in photonic inverse design: the first stage optimizes a passive complex matrix

$$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$4

directly in matrix space, while the second stage fixes the target operator $$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$5 and performs fabrication-aware adjoint realization with the operator-matching loss

$$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$6

The two stages thereby decouple task learning from electromagnetic realization (Muda et al., 23 Apr 2026).

A further variant appears in contextual scenario generation for two-stage stochastic programming. A mapping $$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$7 first predicts a fixed-size set of surrogate scenarios from context, and a second stage either trains this map by a distributional loss such as $$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$8 or $$e(x)=\|\hat f_{\mathcal D}(x)-y^{\text{target}}\|_2^2,$$9, or by a task-based surrogate “loss-net” $\Omega$0 that approximates the downstream decision loss for reduced-scenario 2SPs (Islip et al., 7 Feb 2025). These formulations show that “surrogate” may stand for a screened candidate set, a convex recourse model, a feasibility halfspace, a latent reduced representation, or a differentiable proxy for task loss.

3. Reliability, uncertainty, and validation

A defining feature of many two-stage surrogate workflows is that the second stage is not only more selective, but also uncertainty-aware. In the learner–evaluator framework, conformal inference provides finite-sample, distribution-free guarantees on coverage, and once the residual-based nonconformity scores are computed, evaluating $\Omega$1 for any $\Omega$2 costs only $\Omega$3 quantile computations, independent of $\Omega$4 (Pourkamali-Anaraki et al., 2024). This makes uncertainty auditing compatible with aggressive first-stage screening.

Gaussian-process-based workflows make uncertainty explicit through predictive means and variances. In surrogate-gated materials design, the gate is trained on ORB embeddings reduced by PCA to $\Omega$5, uses an anisotropic RBF kernel, and ranks survivors by Expected Improvement under a fixed oracle budget of $\Omega$6 per cycle after warm-up. The purpose of the second stage is not full validation of all proposals, but triage: only the top-$\Omega$7 survivors are sent to the property oracle (Alvi et al., 26 Jun 2026). In surrogate-based Bayesian inference, emulator uncertainty is propagated into posterior approximations through plug-in, Expected Unnormalized Posterior, or Expected Posterior constructions, and Stage 2 uses acquisition functions such as maximum predictive variance, Expected Integrated Variance, Expected Improvement, or Mutual Information to refine the emulator where posterior uncertainty matters (Roberts et al., 13 Mar 2026).

Uncertainty propagation can itself become a two-stage Bayesian problem. In efficient uncertainty propagation for Bayesian two-step procedures, Stage 1 samples surrogate parameters $\Omega$8 from $\Omega$9, and Stage 2 approximates

$B$0

using a mixture proposal, Pareto smoothed importance sampling, and, when needed, importance weighted moment matching. The workflow is designed to retain both aleatoric uncertainty and epistemic uncertainty while reducing the need to fit a separate second-stage model for every surrogate draw (Jedhoff et al., 15 May 2025).

A semiparametric analogue appears in surrogate-assisted targeted learning for delayed outcomes under administrative censoring. There the “surrogate” is the intermediate measurement $B$1, and the two-stage targeting algorithm first updates the observed-outcome regression and then applies a second targeting update to the bridge function $B$2, explicitly to remove the second-order cross-product remainder $B$3. The resulting estimator is asymptotically linear and doubly robust under the stated conditions (Li, 11 Mar 2026). The common thread is that the second stage is where uncertainty, bias control, or reliability constraints are enforced.

4. Domains and reported empirical behavior

The two-stage surrogate workflow has been reported across inverse design, stochastic optimization, photonics, quantum circuit screening, materials discovery, and simulation exploration. The table summarizes representative instantiations.

Setting	Stage 1	Stage 2 / reported outcome
Inverse design and composites (Pourkamali-Anaraki et al., 2024)	Learner screens top $B$4 candidates	Conformal evaluator; bias reduced to $B$5 of target and variance cut by $B$6–$B$7
Surgery scheduling (Almoghrabi et al., 2023)	Precompute convex piecewise-linear surrogate of recourse cost	MIP + online greedy policy; up to $B$8 lower expected total cost; $B$9 patients in about one minute
Stochastic unit commitment (Javadi et al., 12 Sep 2025)	DCOPF data and SVM training	One halfspace per scenario/time; $\Omega_r$0 and $\Omega_r$1 accuracy; time reductions of $\Omega_r$2 and $\Omega_r$3
Photonic neural networks (Muda et al., 23 Apr 2026)	Matrix-space task training	Adjoint electromagnetic realization; MedMNIST reproduced within $\Omega_r$4 percentage points after $\Omega_r$5 adjoint epochs
GBS screening for GKP states (Khanpour et al., 4 Jun 2026)	Herald-pattern classifier	Fidelity and probability regressors plus exact simulation; $\Omega_r$6 detection accuracy and approximately $\Omega_r$7 lower simulation burden
Bayesian materials design (Alvi et al., 26 Jun 2026)	RL-steered diffusion generation	GP EI gate before oracle; within $\Omega_r$8 of exhaustive oracle spending at roughly one-fifth of the calls
Automated simulation exploration (Saves et al., 19 Oct 2025)	Surrogate modeling	Explainability and consistency diagnostics; exploration in seconds and up to $\Omega_r$9 acceleration

On the Ishigami benchmark, the learner–evaluator workflow was tested with training $\hat\mu$0, test $\hat\mu$1, $\hat\mu$2, and $\hat\mu$3. For scalar targets $\hat\mu$4, the reported means and standard deviations of $\hat\mu$5 were $\hat\mu$6 and $\hat\mu$7 for the single-stage method, $\hat\mu$8 and $\hat\mu$9 for the two-stage method with $\hat C_\alpha(x)$0, and $\hat C_\alpha(x)$1 and $\hat C_\alpha(x)$2 for the two-stage method with $\hat C_\alpha(x)$3 (Pourkamali-Anaraki et al., 2024). In the composite microstructure case study, using only the learner stage often produced outliers and sometimes failed to generate any microstructure near the target, whereas adding the evaluator with a polynomial regressor, $\hat C_\alpha(x)$4, and $\hat C_\alpha(x)$5 eliminated these outliers and consistently yielded at least one microstructure in a $\hat C_\alpha(x)$6 neighborhood of the target (Pourkamali-Anaraki et al., 2024).

In GBS circuit screening for GKP-state creation, the pipeline predicts the optimal heralding pattern first and then predicts fidelity and post-selection probability conditioned on that pattern. On the $\hat C_\alpha(x)$7-sample holdout set, the reported metrics are $\hat C_\alpha(x)$8 GKP-detection accuracy, fidelity regressor $\hat C_\alpha(x)$9, holdout $x\in\Omega_r$0, and log-probability regressor $x\in\Omega_r$1, with only the top-ranked $x\in\Omega_r$2 of circuits forwarded for exact validation (Khanpour et al., 4 Jun 2026). In photonic neural networks, the same decoupling between cheap surrogate learning and expensive physical realization yielded a realized all-optical MedMNIST classifier at $x\in\Omega_r$3 after $x\in\Omega_r$4 adjoint epochs, compared with a surrogate-stage test accuracy of $x\in\Omega_r$5, while RSSCN7 improved by more than $x\in\Omega_r$6 percentage points over a linear readout baseline (Muda et al., 23 Apr 2026).

5. Design parameters and implementation patterns

Although the architecture is generic, the literature identifies recurring control variables. In the learner–evaluator framework, the central choices are the learner and evaluator architectures, the miscoverage level $x\in\Omega_r$7, the number of folds $x\in\Omega_r$8 with typical recommendation $x\in\Omega_r$9, and the candidate budget $y^{\text{target}}\in \hat C_\alpha(x)$0. The reported interpretation is explicit: smaller $y^{\text{target}}\in \hat C_\alpha(x)$1 yields wider intervals and more candidates retained but less precision, while larger $y^{\text{target}}\in \hat C_\alpha(x)$2 reduces conservatism of intervals but increases retraining cost (Pourkamali-Anaraki et al., 2024).

A second recurring pattern is stage-specific feature engineering. In GBS screening, Stage 1 uses raw circuit parameters, configuration metadata, and eleven physics-inspired aggregates; Stage 2 appends eight summary statistics of the predicted herald pattern, bringing the total to $y^{\text{target}}\in \hat C_\alpha(x)$3 features (Khanpour et al., 4 Jun 2026). In Bayesian materials design, pretrained ORB embeddings, optionally concatenated to Magpie features and reduced by PCA to $y^{\text{target}}\in \hat C_\alpha(x)$4, were adopted because cross-factorial benchmarking identified pretrained ORB embeddings with a Gaussian process as the most reliable combination (Alvi et al., 26 Jun 2026). In TSUC, the feature vector for the SVM surrogate includes wind parameters, demand descriptors, and generator outputs, and all features are normalized to zero mean and unit variance over the training set (Javadi et al., 12 Sep 2025).

Some workflows also make stage selection adaptive. AdaE-SAEA defines a bagging ensemble as Stage 1 for robustness during exploration and a boosting ensemble as Stage 2 for bias reduction and high-precision fitting during exploitation. Rather than using a fixed switch, a DQN with a $y^{\text{target}}\in \hat C_\alpha(x)$5 architecture selects both the surrogate module and the infill criterion from a 15-dimensional ELA state and an action space of size $y^{\text{target}}\in \hat C_\alpha(x)$6 (Jin et al., 30 May 2026). This suggests that, in some formulations, the two-stage workflow is not a one-time pipeline but a state-dependent control policy over alternating surrogate regimes.

Offline/online separation is another implementation motif. Contextual scenario generation emphasizes that offline training may take hours, but online deployment reduces to one forward pass through the context-to-scenarios map and a small reduced-scenario 2SP solve in seconds (Islip et al., 7 Feb 2025). Surgery scheduling follows a similar logic: the convex surrogate of recourse cost is computed in preprocessing, embedded into an offline MIP, and complemented by an online greedy insertion and cancellation policy at run time (Almoghrabi et al., 2023). Across domains, the practical objective is consistent: expensive computation is concentrated where it changes decisions most.

6. Limitations, misconceptions, and open directions

A frequent misconception is that a two-stage surrogate workflow simply means “use two models.” The cited literature shows a broader picture. The second stage may be a conformal auditor, a convex reformulation, an exact quantum simulation reserved for surrogate-selected candidates, an adjoint realization problem, an explainability-and-consistency layer, or a targeted statistical update. This suggests that the defining property is staged allocation of computation and reliability, not the mere presence of two regressors.

The workflows also retain substantive failure modes. Inverse design may legitimately return “none” or report “no solution,” and in the composite study the benefit of the evaluator was precisely that it signaled infeasibility or unreliability instead of outputting spurious candidates (Pourkamali-Anaraki et al., 2024). In contextual scenario generation, offline training requires solving many reduced 2SP instances, the quality of the final policy depends on the representational power of the loss-net $y^{\text{target}}\in \hat C_\alpha(x)$7 and the distributional regularizer, gradients through $y^{\text{target}}\in \hat C_\alpha(x)$8 are approximate, and stability can degrade if $y^{\text{target}}\in \hat C_\alpha(x)$9 wanders outside the training-data manifold (Islip et al., 7 Feb 2025). In high-dimensional inverse UQ, the total error is the sum of dimensionality-reduction reconstruction error and surrogate regression error; moreover, PCA is linear and high-dimensional inputs can still strain PCE (Yang et al., 2024).

The literature on surrogate modeling plus explainability adds a further caution: low NDCG, volatile SHAP or PDP curves, poor PVA, or high RMSE and low MCC are not merely descriptive defects, but indicators that the surrogate may be inadequate for explanation or decision support. The proposed response is targeted resampling, model-family changes, or recalibration rather than blind trust in the two-stage pipeline (Saves et al., 19 Oct 2025). Likewise, the review of surrogate-based Bayesian inference emphasizes that robust workflows require explicit uncertainty propagation and active learning; surrogate use does not exempt the analyst from model checking, stopping rules, or posterior diagnostics (Roberts et al., 13 Mar 2026).

Open directions are stated explicitly in several of the cited works. These include extending the learner–evaluator framework to high-dimensional problems, extreme/regional coverage analysis, and multi-agent surrogate interactions (Pourkamali-Anaraki et al., 2024); expanding GBS screening to larger circuits, six-mode and higher-$\mathcal D=\{(x^{(i)},y^{(i)})\}_{i=1}^n$00 configurations, active learning for uncertainty hotspots, and additional targets such as Wigner logarithmic negativity (Khanpour et al., 4 Jun 2026); and extending contextual scenario generation to sequential or multi-stage settings (Islip et al., 7 Feb 2025). Taken together, these directions indicate that the two-stage surrogate workflow is evolving from a computational shortcut into a general design principle for coupling approximation, uncertainty control, and task-aware validation in expensive scientific computation.