Predictive Surrogates: Fast, Accurate Models

• Predictive Surrogates are learned models that approximate expensive target procedures while adhering to application-specific error bounds.
• They encompass diverse methods including deterministic emulation, probabilistic forecasting, active learning, and hybrid physics-guided approaches.
• Their efficacy relies on task-specific fidelity, robust uncertainty quantification, and incorporation of structural priors to support reliable decision-making.

Searching arXiv for recent and relevant papers on predictive surrogates. Predictive surrogates are learned models used in place of more expensive, slower, or less accessible target procedures while preserving the predictive quantity of interest to a degree specified by an application-dependent error criterion. Across scientific computing, stochastic partial differential equations, hybrid physical modeling, robotics, quantum hardware, causal inference, and clinical trial methodology, the term refers to a common pattern: a map $f$ or an effect quantity that is costly to obtain directly is approximated by a surrogate $\hat f$, and the surrogate is deployed only to the extent that its predictive fidelity, uncertainty behavior, or inferential validity is adequate for the downstream task (Fink et al., 2024). In some settings the target is a simulator or operator $f:X\to Y$; in others it is a conditional distribution, a treatment effect, a long-term endpoint, or the mean-value behavior of a quantum processor (Srikishan et al., 30 Jun 2026, Ji et al., 16 Jan 2025, Liao et al., 23 Jul 2025). The modern literature therefore treats predictive surrogates not as a single model class but as a broad methodological category spanning deterministic emulation, probabilistic forecasting, active-learning surrogates, hybrid residual models, reduced-order approximations, and surrogate endpoints.

1. Conceptual scope and formal definitions

In computational physics and scientific computing, a predictive surrogate model is a data-driven approximation $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$ of a computationally expensive simulator $F^*(x)$, trained from pairs $\{x_i,y_i=F^*(x_i)\}$ so that $\hat y=\mathcal{F}(x)\approx F^*(x)$ (Anirudh et al., 2019). HPAC-ML formulates the same structure in operator form: scientific applications often rely on expensive operators $f:X\to Y$, and a learned surrogate $\hat f:X\to Y$ replaces or “surrogates” selected regions of code when the approximation error $\epsilon(x)=\|f(x)-\hat f(x)\|$ is acceptable under a user-specified error budget $\hat f$0 (Fink et al., 2024).

This operator view extends naturally to uncertainty-aware settings. For stochastic PDEs, deterministic prediction of a single trajectory is insufficient; a useful surrogate must produce samples $\hat f$1 whose ensemble statistics match those of the high-fidelity solver and whose spread reflects epistemic and aleatoric uncertainty (Srikishan et al., 30 Jun 2026). In that literature, the object being approximated is not merely a pointwise map but a forecast distribution over future states. TRIE makes this distinction explicit by evaluating whether a surrogate reproduces invariant measures, delivers calibrated predictive uncertainty, and supports efficient probabilistic generation (Srikishan et al., 30 Jun 2026).

A second major formulation arises in hybrid models. BITS for GAPS assumes a known physics map $\hat f$2 and represents the discrepancy by a latent function $\hat f$3, yielding

$\hat f$4

The predictive surrogate is then the posterior mean or predictive distribution of the latent residual, learned nonparametrically by a hierarchical Gaussian process and embedded back into the governing physical equation (Jones et al., 20 Nov 2025). This suggests a general distinction between full surrogates, which emulate the entire map, and residual surrogates, which emulate only the part left unexplained by known structure.

A third formulation appears in statistics and causal inference, where the surrogate is itself an outcome proxy. “Predictions as Surrogates” shows that pre-trained model predictions $\hat f$5 can be treated as cost-effective surrogates for expensive outcomes within the surrogate-outcome framework, and plug-in surrogate endpoints are defined as functions $\hat f$6 of post-treatment variables substituted directly for the primary outcome in randomized experiments (Ji et al., 16 Jan 2025, Margueritte et al., 12 May 2026). In these settings, predictive adequacy is judged not by approximation of a simulator, but by effect-predictiveness or inferential efficiency.

The quantum-computing literature introduces a narrower and more formal usage. A predictive surrogate is defined there as a purely classical function $\hat f$7, where $\hat f$8 is the mean-value behavior of a noisy quantum processor under parameterized circuit $\hat f$9, initial state $f:X\to Y$0, and observable $f:X\to Y$1 (Liao et al., 23 Jul 2025). The paper further requires computational efficiency in training and evaluation time polynomial in $f:X\to Y$2, $f:X\to Y$3, and $f:X\to Y$4, using a polynomial number of quantum measurements (Liao et al., 23 Jul 2025).

Across these formulations, a common core remains: the surrogate stands in for a target quantity that is expensive, latent, delayed, or experimentally constrained, and its usefulness depends on matching the target under task-specific notions of fidelity.

2. Deterministic operator surrogates in scientific computing

A central use of predictive surrogates is acceleration of large scientific codes by replacing computational hotspots with learned inference. HPAC-ML provides a directive-based programming model for this purpose (Fink et al., 2024). It extends HPAC’s #pragma approx with tensor functors, tensor maps, and an ML clause. Tensor functors declare how to materialize applicative tensor entries from strided application memory; tensor maps bind those functors to concrete array slices; and the ML clause wraps a structured block so that the runtime either collects training data or performs inference (Fink et al., 2024).

The HPAC-ML runtime has two coordinated layers. The data bridge lowers functors and maps into lightweight descriptors and, at runtime, resolves slice extents, wraps application pointers as Torch tensor views, concatenates feature slices into input tensors, and maps output tensors back to application arrays (Fink et al., 2024). Execution control then decides between collection and inference. In “collect” mode, the original operator is run and $f:X\to Y$5 tensors are appended to HDF5 for offline training; in “infer” mode, a TorchScript model is loaded and used to produce surrogate outputs; in “predicated” mode, the runtime switches between the two paths based on a Boolean condition such as iteration count, residual norm, or a user-exposed flag (Fink et al., 2024).

The paper formalizes the resulting performance trade-off through per-invocation speedup $f:X\to Y$6 and end-to-end application speedup

$f:X\to Y$7

where $f:X\to Y$8 is the fraction of runtime spent in the surrogated operator (Fink et al., 2024). Accuracy is constrained by an RMSE budget $f:X\to Y$9, and model selection is performed by Bayesian-optimization-driven search over the Pareto frontier between speedup and error (Fink et al., 2024).

The reported results illustrate the practical range of this deterministic surrogate pattern. Across five GPU benchmarks and over 5000 ML models, HPAC-ML reports up to $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$0 speed improvements with minimal accuracy loss as low as $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$1 RMSE (Fink et al., 2024). The benchmark summary includes $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$2 maximum speedup for MiniBUDE with MAPE $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$3, $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$4 for Binomial Options with RMSE $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$5, $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$6 for Bonds with RMSE $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$7, $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$8 for MiniWeather with RMSE $\mathcal{F}:\mathcal{X}\to\mathcal{Y}$9, and $F^*(x)$0 for ParticleFilter with RMSE $F^*(x)$1 (Fink et al., 2024). The MiniWeather case also shows a characteristic limitation of iterative surrogates: auto-regressive error amplification can produce catastrophic drift, and interleaving inference with accurate steps, such as using the surrogate in one out of two time steps, can reduce RMSE by $F^*(x)$2 while preserving $F^*(x)$3 speedup (Fink et al., 2024).

Reduced-order surrogate modeling provides a different deterministic strategy. Arby constructs $F^*(x)$4 for parameterized solution families $F^*(x)$5 by combining a Reduced Basis greedy algorithm, the Empirical Interpolation Method, and a regression stage that learns $F^*(x)$6 at empirical nodes (Villanueva et al., 2021). The final surrogate has the form

$F^*(x)$7

with offline cost $F^*(x)$8 and online evaluation $F^*(x)$9 (Villanueva et al., 2021). In the CMB power-spectrum example, a set of $\{x_i,y_i=F^*(x_i)\}$0 solutions with $\{x_i,y_i=F^*(x_i)\}$1 multipole indices is described, at a given tolerance error, by a subset of $\{x_i,y_i=F^*(x_i)\}$2 solutions (Villanueva et al., 2021). This suggests the importance of structural compression when the target map is smooth over parameter space.

Online surrogate maintenance has also been studied for expensive simulations of complex systems. The optimizer-driven workflow of “Efficient Learning of Accurate Surrogates for Simulations of Complex Systems” uses local optimization to ensure turning points on the response surface enter the training set, tests surrogate validity by graphical-distance scores, and retrains whenever the score drops below threshold (Diaw et al., 2022). Thin-plate radial-basis-function interpolation is used as the surrogate form, and the method is demonstrated on benchmark functions and on a nuclear matter equation-of-state application where $\{x_i,y_i=F^*(x_i)\}$3 expensive evaluations yield a surrogate with approximately $\{x_i,y_i=F^*(x_i)\}$4 speed-up per evaluation (Diaw et al., 2022). Here the emphasis is not only approximation quality but automatic regeneration of a valid surrogate as new information arrives.

3. Probabilistic, uncertainty-aware, and active-learning surrogates

For stochastic or noisy systems, a point predictor can be misleading even when short-horizon errors appear small. TRIE evaluates stochastic PDE surrogates under three complementary criteria: invariant-measure reproduction, predictive-uncertainty calibration, and efficiency of probabilistic generation (Srikishan et al., 30 Jun 2026). Invariant-measure fidelity is assessed by Wasserstein distances between empirical long-time measures; uncertainty calibration is scored by the Continuous Ranked Probability Score

$\{x_i,y_i=F^*(x_i)\}$5

and efficiency is measured by end-to-end inference time per sample (Srikishan et al., 30 Jun 2026).

The framework is demonstrated on stochastic Kuramoto–Sivashinsky and stochastic Kolmogorov flow across $\{x_i,y_i=F^*(x_i)\}$6 parameter settings (Srikishan et al., 30 Jun 2026). Standard pointwise-trained neural surrogates such as ResNet, UNet, and PDE-Transformer may produce plausible short rollouts yet fail to match long-time statistical structure (Srikishan et al., 30 Jun 2026). Approximate uncertainty methods, including Monte Carlo dropout and heteroscedastic Gaussian likelihoods, generate stochastic forecasts but are often miscalibrated and overconfident under temporal and spatial diagnostics (Srikishan et al., 30 Jun 2026). Across all reported probabilistic settings, generative models achieve the lowest CRPS, and the Stochastic Interpolant achieves the lowest invariant-measure Wasserstein distances; for example, in Kuramoto–Sivashinsky with $\{x_i,y_i=F^*(x_i)\}$7 and $\{x_i,y_i=F^*(x_i)\}$8, SI achieves approximately $\{x_i,y_i=F^*(x_i)\}$9 versus approximately $\hat y=\mathcal{F}(x)\approx F^*(x)$0 for ResNet and approximately $\hat y=\mathcal{F}(x)\approx F^*(x)$1 for UNet (Srikishan et al., 30 Jun 2026). A latent SI variant preserves invariant-measure fidelity within approximately $\hat y=\mathcal{F}(x)\approx F^*(x)$2 of full SI while reducing per-sample inference time on Kolmogorov flow from $\hat y=\mathcal{F}(x)\approx F^*(x)$3 s to $\hat y=\mathcal{F}(x)\approx F^*(x)$4 s, approximately a $\hat y=\mathcal{F}(x)\approx F^*(x)$5 speedup (Srikishan et al., 30 Jun 2026).

Deep Gaussian processes provide another uncertainty-aware surrogate family for nonstationary response surfaces. “Active Learning for Deep Gaussian Process Surrogates” uses nested Gaussian-process layers to induce automatic input warping and nonstationary covariance, performs Bayesian inference with elliptical slice sampling, and drives sequential design by IMSE or ALC criteria averaged over posterior draws (Sauer et al., 2020). The two-layer DGP outperforms one-layer GP baselines in several nonstationary settings, concentrating runs in “interesting” regimes rather than at large Euclidean distances (Sauer et al., 2020). The same paper emphasizes the cubic cost of DGP inference and therefore the role of careful acquisition in keeping $\hat y=\mathcal{F}(x)\approx F^*(x)$6 small (Sauer et al., 2020). This illustrates a recurrent theme: predictive surrogates are often inseparable from adaptive sampling policies that determine where expensive truth evaluations should be spent.

Generative Bayesian Computation via Implicit Quantile Networks extends this logic to large-$\hat y=\mathcal{F}(x)\approx F^*(x)$7, non-Gaussian emulation. Instead of a GP posterior, GBC learns the full conditional quantile function $\hat y=\mathcal{F}(x)\approx F^*(x)$8 and uses forward passes at sampled $\hat y=\mathcal{F}(x)\approx F^*(x)$9 values to produce draws from $f:X\to Y$0 (Polson et al., 24 Feb 2026). Across fourteen benchmarks, GBC improves CRPS by $f:X\to Y$1–$f:X\to Y$2 on piecewise jump-process benchmarks, by $f:X\to Y$3 on a ten-dimensional Friedman function, and scales linearly to $f:X\to Y$4 training points where dense-covariance GPs are infeasible (Polson et al., 24 Feb 2026). In active learning, a randomized-prior IQN ensemble attains nearly three times lower RMSE than deep GP active learning on Rocket LGBB (Polson et al., 24 Feb 2026). The paper also states that GPs retain an edge on smooth surfaces where their smoothness prior is effective regularization (Polson et al., 24 Feb 2026). That qualification is important: predictive surrogates are not uniformly interchangeable, and the appropriate class depends on whether the target surface is smooth, discontinuous, heteroskedastic, or distributional.

Uncertainty-aware surrogates are also prominent in engineering and biomedicine. In reconstructive surgery design, Gaussian-process surrogates are trained on PCA-compressed finite-element strain fields over a five-dimensional material-parameter space, then used for Sobol sensitivity analysis and optimization under uncertainty (Stowers et al., 2020). The workflow relies on surrogate predictive distributions to estimate expected clinical costs under uncertain material parameters and optimize flap orientation, the only controllable material parameter clinically (Stowers et al., 2020). This is a canonical example of a predictive surrogate serving not merely as an emulator but as an enabling layer for sensitivity analysis, risk-neutral design, and risk-averse optimization.

4. Hybrid, physics-guided, and structure-preserving surrogates

A major trend in predictive surrogates is the incorporation of domain structure to improve sample efficiency, robustness, and extrapolative behavior. BITS for GAPS exemplifies serial hybrid modeling by combining known physics $f:X\to Y$5 with a latent residual GP $f:X\to Y$6, equipped with hierarchical priors on length scales, output scale, and noise to encode physically meaningful structure in the predictive posterior (Jones et al., 20 Nov 2025). The sequential sampling policy is entropy-based, using differential entropy of the predictive posterior and an information-gain proxy $f:X\to Y$7 under the one-step GP update (Jones et al., 20 Nov 2025). In the vapor–liquid equilibrium application, embedding the surrogate activity coefficient into extended Raoult’s law yields accurate predictions across composition–temperature space, quantified uncertainty, and reported sample efficiency of more than $f:X\to Y$8 fewer high-fidelity runs than space-filling designs for the same RMS error in $f:X\to Y$9 (Jones et al., 20 Nov 2025).

Manifold and cycle consistencies are a different kind of structural prior. In inertial confinement fusion, the MaCC framework first trains an autoencoder so that high-dimensional outputs lie near a learned physical manifold, then trains the forward surrogate in latent space while coupling it to an inverse model through cycle consistency (Anirudh et al., 2019). The cycle-consistency penalty enforces $\hat f:X\to Y$0 and $\hat f:X\to Y$1, thereby discouraging unphysical latent codes and implausible outputs (Anirudh et al., 2019). On a $\hat f:X\to Y$2D semi-analytic ICF simulator with $\hat f:X\to Y$3 training samples, the MaCC surrogate improves image-band MSEs relative to a baseline and improves small-data performance by up to approximately $\hat f:X\to Y$4 when training on as few as $\hat f:X\to Y$5 of samples (Anirudh et al., 2019). The paper also reports improved robustness to perturbations and better self-consistency under forward–inverse scans (Anirudh et al., 2019). This suggests that predictive surrogates benefit from constraints operating in output geometry, not only in input–output regression loss.

In legged robotics, smoothness and derivative quality are the critical structural properties. Smooth Neural Surrogates learn discrete-time dynamics

$\hat f:X\to Y$6

using an SNS-MLP architecture with tunable first- or second-order smoothness bounds and a heavy-tailed Cauchy likelihood to handle impulsive residuals (Moore et al., 17 Jan 2026). The explicit motivation is that standard neural dynamics models can inherit stiff transitions from contact events, develop non-physical local nonsmoothness, and exhibit non-Gaussian model errors (Moore et al., 17 Jan 2026). By penalizing Lipschitz and Jacobian-Lipschitz surrogates, the method yields bounded, informative derivatives for trajectory optimization through contact (Moore et al., 17 Jan 2026). In zero-shot locomotion tasks, the authors report cost reductions typically $\hat f:X\to Y$7–$\hat f:X\to Y$8 on well-conditioned tasks, and on difficult tasks standard neural models often fail while smoothing enables reliable execution from $\hat f:X\to Y$9 to $\epsilon(x)=\|f(x)-\hat f(x)\|$0 success and about $\epsilon(x)=\|f(x)-\hat f(x)\|$1–$\epsilon(x)=\|f(x)-\hat f(x)\|$2 lower cumulative cost (Moore et al., 17 Jan 2026). The surrogate is therefore judged less by one-step prediction error than by whether its derivatives are sufficiently well-behaved to support MPC.

Physics-guided operator surrogates for multiphase flow show another decomposition principle. Hybrid DeepONet surrogates split spatial and temporal learning between branch and trunk networks, combining FNO, MLP, and KAN components to reduce memory requirements while preserving predictive performance on 2D Darcy flow and 2D/3D SPE10 reservoir problems (Santos et al., 4 Nov 2025). The surrogate operator is represented as

$\epsilon(x)=\|f(x)-\hat f(x)\|$3

with spatial inputs routed to the branch and temporal or spatio-temporal queries routed to the trunk (Santos et al., 4 Nov 2025). FNO-branch hybrids achieve lower errors than MLP- or KAN-only variants in Darcy and SPE10 cases, and on the largest 3D benchmark only FNO-branch hybrids fit the largest model size on a single NVIDIA H100 $\epsilon(x)=\|f(x)-\hat f(x)\|$4 GB GPU (Santos et al., 4 Nov 2025). The paper states near real-time inference, yielding approximately $\epsilon(x)=\|f(x)-\hat f(x)\|$5–$\epsilon(x)=\|f(x)-\hat f(x)\|$6 speedups over OPM Flow solves (Santos et al., 4 Nov 2025).

Structure-preserving surrogates also appear in kinetic plasma UQ. “Micro-Macro Tensor Neural Surrogates for Uncertainty Quantification in Collisional Plasma” replaces repeated evaluations of the Landau collision term by neural surrogates based on VPFP and Euler–Poisson equations, using an anisotropic micro–macro decomposition and tensorized SPINN factorization (Chen et al., 30 Dec 2025). The AP loss is scaled to recover the EP limit as $\epsilon(x)=\|f(x)-\hat f(x)\|$7 and the Vlasov–Poisson regime as $\epsilon(x)=\|f(x)-\hat f(x)\|$8 (Chen et al., 30 Dec 2025). In UQ experiments, the VPFP-based control-variate surrogate reduces Monte Carlo $\epsilon(x)=\|f(x)-\hat f(x)\|$9 error by roughly two orders of magnitude relative to plain Monte Carlo, often from approximately $\hat f$00 to approximately $\hat f$01, while surrogate inference is approximately $\hat f$02–$\hat f$03 faster than deterministic VPFP or VPL sampling once many samples are required (Chen et al., 30 Dec 2025). Here, predictive surrogates are embedded inside a variance-reduction estimator rather than used directly as stand-alone predictors.

5. Predictive surrogates in inference, endpoints, and meta-analysis

In biostatistics and causal inference, the term “surrogate” historically refers to an endpoint that substitutes for a long-term or clinically relevant outcome. Recent work connects that tradition to predictive modeling more explicitly. “Predictions as Surrogates” formalizes pre-trained model outputs as surrogate outcomes and develops recalibrated prediction-powered inference (Ji et al., 16 Jan 2025). The target parameter $\hat f$04 is defined by a loss or estimating equation involving the expensive outcome $\hat f$05, while the surrogate $\hat f$06 is available widely. The optimal imputed loss uses

$\hat f$07

and RePPI learns an approximation to this conditional expectation by flexible machine learning during a recalibration step (Ji et al., 16 Jan 2025). The key guarantee is that even if the learned calibration is inconsistent, the resulting estimator always improves upon the estimator using only labeled data with true outcomes, and if the estimate is consistent it attains the smallest asymptotic variance among PPI estimators (Ji et al., 16 Jan 2025). Applications using XGBoost and GPT-4o mini predictions report effective sample-size gains, including approximately $\hat f$08 fewer labels than PPI++ in the US Census example, $\hat f$09–$\hat f$10 in politeness-score regression, and $\hat f$11–$\hat f$12 in wine-review analysis (Ji et al., 16 Jan 2025).

A related but distinct line addresses plug-in surrogate endpoints in randomized experiments. “Learning plug-in surrogate endpoints for randomized experiments” defines a composite surrogate $\hat f$13 of post-treatment measurements and selects it by minimizing mean-squared CATE error

$\hat f$14

under causal assumptions including ignorability, Prentice surrogacy, and transportability (Margueritte et al., 12 May 2026). Two methods are proposed: Surrogate-Sampling, which uses a generative model for $\hat f$15, and Bound-Regression, which optimizes a weighted regression upper bound (Margueritte et al., 12 May 2026). In synthetic and IHDP experiments, Surrogate-Sampling gives the smallest ATE-MAE and highest $\hat f$16 among plug-in-only methods, while Bound-Regression is especially helpful in linear cases (Margueritte et al., 12 May 2026). This literature treats predictive surrogates as learned substitutes that retain effect heterogeneity information without requiring pre-treatment covariates at trial time.

Meta-analytic surrogate evaluation extends the same logic across studies. RISE-Meta evaluates trial-level surrogate markers in high-dimensional settings by first estimating study-level surrogacy metrics nonparametrically via U-statistics, then combining them through random-effects meta-analysis and equivalence testing (Hughes et al., 5 May 2026). Trial-level bias is $\hat f$17, with $\hat f$18 and $\hat f$19 defined as probabilities that a treated unit exceeds a control unit on primary and surrogate outcomes, respectively (Hughes et al., 5 May 2026). Random-effects REML and Hartung–Knapp variance estimation produce pooled effects $\hat f$20, prediction intervals, and TOST-based validity assessments (Hughes et al., 5 May 2026). In influenza-vaccination gene-expression data, after screening $\hat f$21 Blood Transcription Modules, $\hat f$22 modules are significant after FDR correction, and the composite signature yields pooled $\hat f$23, TOST $\hat f$24, $\hat f$25, and CCC $\hat f$26 (Hughes et al., 5 May 2026). The paper thereby frames surrogate validity as generalizable prediction of trial-level treatment effects across studies.

Older meta-analytic methods similarly distinguish settings where surrogate relationships vary across treatment classes or treatment contrasts. The Bayesian hierarchical model of treatment-class-specific surrogate relationships borrows information across classes via fully or partially exchangeable priors on intercepts and slopes, improving precision in sparse classes and avoiding over-shrinkage when classes differ (Papanikos et al., 2019). Bivariate network meta-analysis further allows contrast-specific surrogate relationships and across-treatment predictions for new contrasts (Bujkiewicz et al., 2018). These works suggest a broader point: predictive surrogates in clinical research are evaluated not only by accuracy within observed studies but by whether they remain predictive under heterogeneity across treatment classes, trial populations, and contrasts.

The term “surrogate-assisted performance prediction” broadens the scope beyond medicine. Funkner et al. use random-forest and MLP surrogates to predict both clustering quality metrics and execution time of an evolutionary clinical-pathway discovery algorithm from algorithm hyperparameters and dataset descriptors (Funkner et al., 2020). For $\hat f$27 sequence sets and $\hat f$28 target-algorithm runs per set, the general RF model achieves test MAPE $\hat f$29 for execution time, $\hat f$30 for Calinski–Harabasz, $\hat f$31 for Davies–Bouldin, $\hat f$32 for number of clusters, and $\hat f$33 for non-clustered sequences (Funkner et al., 2020). In this case the surrogate predicts the behavior of an algorithm rather than a physical process, yet the logic is the same: use a cheap predictive layer to guide tuning, early stopping, or Pareto selection under budget constraints.

6. Emerging domains, comparative issues, and recurring limitations

Predictive surrogates now extend into domains where direct access to the target system is physically constrained. In quantum computing, the surrogate need not reproduce full output distributions; it may target only the mean-value behavior relevant to optimization or phase identification (Liao et al., 23 Jul 2025). Two constructions are proposed: a classical-shadow kernel predictor $\hat f$34 and a truncated-feature ridge-regression surrogate $\hat f$35, both with polynomial efficiency guarantees under Pauli noise (Liao et al., 23 Jul 2025). On a $\hat f$36-qubit superconducting processor, the surrogates support efficient pre-training of VQEs for transverse-field Ising models and identification of Floquet symmetry-protected topological phases (Liao et al., 23 Jul 2025). For $\hat f$37, $\hat f$38, $\hat f$39 achieves normalized deviation approximately $\hat f$40 versus original VQE approximately $\hat f$41, using only $\hat f$42 of the $\hat f$43 million shots required by full VQE; with $\hat f$44 shadows, MAE is approximately $\hat f$45, $\hat f$46 approximately $\hat f$47, and correlation $\hat f$48 approximately $\hat f$49 (Liao et al., 23 Jul 2025). The paper explicitly characterizes predictive surrogates as a practical pathway for broadening access to rare, fabrication-expensive quantum processors (Liao et al., 23 Jul 2025).

Not every use of surrogate models is benign or constructive. In adversarial machine learning, QueryNet uses multiple surrogate networks to approximate a victim model’s gradients and predictions, exploiting both Gradient Similarity and Prediction Similarity to reduce black-box query counts (Chen et al., 2021). Although this is an attack setting rather than a scientific one, it demonstrates that predictive surrogate quality can be operationalized in terms of transferability rather than pointwise regression. QueryNet reports average query reductions of roughly an order of magnitude, including from $\hat f$50 to $\hat f$51 queries on CIFAR-10 WRN28 and from $\hat f$52 to $\hat f$53 on MNIST WRN10 under specified attack budgets (Chen et al., 2021). A plausible implication is that “predictive surrogate” is now a sufficiently general concept to apply wherever expensive or rate-limited access to a target model can be substituted by learned approximation.

Across domains, several recurrent limitations appear.

First, approximation error is often not the correct deployment criterion. HPAC-ML shows that low RMSE can still be unsafe in iterative, auto-regressive settings because error amplification causes drift (Fink et al., 2024). TRIE similarly shows that plausible short rollouts can coexist with poor invariant-measure fidelity and miscalibrated uncertainty (Srikishan et al., 30 Jun 2026). Smooth Neural Surrogates show that slightly worse zeroth-order prediction can nonetheless yield drastically better control performance if derivatives are smoother and more informative (Moore et al., 17 Jan 2026). These results undermine the common misconception that one-step prediction error alone determines surrogate quality.

Second, uncertainty estimates from approximate methods can be unreliable. TRIE finds Monte Carlo dropout and heteroscedastic Gaussian likelihood surrogates often overconfident or underconfident depending on spatial region and diagnostic (Srikishan et al., 30 Jun 2026). GBC notes that calibration can deviate from nominal in low-data or high-noise regimes, although conformal post-processing can restore finite-sample coverage (Polson et al., 24 Feb 2026). GP-based methods supply principled uncertainty, but their computational cost can be prohibitive at scale (Sauer et al., 2020, Polson et al., 24 Feb 2026).

Third, surrogate validity is highly task-specific. In surgery planning, what matters is whether the surrogate supports sensitivity analysis and expected-cost optimization under uncertainty (Stowers et al., 2020). In randomized experiments, the question is whether the surrogate endpoint predicts the treatment effect on the primary outcome under identifiable assumptions (Margueritte et al., 12 May 2026). In quantum emulation, computational efficiency and mean-value fidelity under noise are the defining criteria (Liao et al., 23 Jul 2025). In SPDE forecasting, long-time measure preservation and CRPS dominate (Srikishan et al., 30 Jun 2026). This suggests that “predictive surrogate” should be understood operationally: the surrogate is adequate only relative to the downstream task and evaluation protocol.

Finally, structural priors and hybridization repeatedly improve practical performance. Residual physics-plus-surrogate decompositions (Jones et al., 20 Nov 2025), manifold and cycle consistency (Anirudh et al., 2019), smoothness control for dynamics derivatives (Moore et al., 17 Jan 2026), branch–trunk factorization in neural operators (Santos et al., 4 Nov 2025), and asymptotic-preserving tensorization (Chen et al., 30 Dec 2025) all show that unconstrained universal approximation is rarely the best engineering choice. The literature consistently favors surrogates that preserve the target system’s manifold, invariants, asymptotic limits, latent structure, or deployment-time requirements.

Taken together, predictive surrogates are best understood as a family of approximation strategies that mediate between expensive truth and practical decision-making. Their contemporary development spans directive-based integration into legacy codes, distributional forecasting for stochastic systems, uncertainty-aware active learning, hybrid physics residuals, structure-preserving neural operators, inference with surrogate outcomes, and classical emulation of quantum devices. The unifying question is not merely whether a model predicts well, but whether it predicts the right object, with the right uncertainty, at the right cost, for the right downstream use.