Ensemble Surrogate Model Overview

• Ensemble surrogate models are frameworks that aggregate diverse predictive surrogates to enhance accuracy and provide robust uncertainty quantification.
• They integrate methods like polynomial sparse grids, Gaussian processes, decision trees, and deep neural networks through weighted aggregation or architectural grouping.
• Applications span uncertainty quantification, black-box optimization, and data assimilation in fields such as fluid dynamics, environmental modeling, and adversarial robustness.

An ensemble surrogate model is a computational framework in which multiple surrogate models—predictive approximations of expensive or complex simulations—are strategically combined to optimize accuracy, uncertainty estimation, or computational efficiency. Surrogate ensemble approaches appear in diverse settings, including uncertainty quantification for PDEs, scientific data assimilation, black-box optimization, environmental forecasting, adversarial robustness, and parameter sensitivity analysis. Methodological advances in this area exploit ensemble strategies for balancing error, maximizing transferability, calibrating predictive intervals, and enabling robust, scalable deployment on modern architectures.

1. Fundamental Principles and Construction

Ensemble surrogate models can be constructed as weighted aggregations of diverse surrogate models or as specialized architectures embedding multiple predictors. Individual surrogates may consist of polynomial sparse grids, Gaussian processes, decision tree ensembles (forests), deep neural networks, or reduced-order physical models. The common principle is to exploit the diversity or complementary strengths among member surrogates to improve accuracy, provide robust point estimates, and capture epistemic (model-based) uncertainty.

Mathematically, the output $\hat{y}(x)$ of an ensemble of $M$ surrogates $\{f_m(x; \theta_m)\}_{m=1}^M$ is typically defined via

$\hat{y}(x) = \sum_{m=1}^{M} w_m f_m(x; \theta_m)$

with weights $w_m$ potentially set by cross-validated error, regression tuning, or learned adaptively during the surrogate's operation (Audet et al., 2021, Kim et al., 2022, Kalaydjian et al., 2023, Sun et al., 22 Aug 2025). Some recent works embed several specialized branches within a single predictive architecture and average outputs, an approach exemplified in packed-ensemble surrogate models for fluid flows (Kalaydjian et al., 2023). SurroFlow (Shen et al., 16 Jul 2024) introduces a conditional normalizing flow combined with an autoencoder, where bidirectional mapping and latent variable sampling offer ensemble-like capability for both prediction and uncertainty quantification.

Key member surrogates may be:

• Polynomial sparse grid surrogates in adaptive collocation (D'Elia et al., 2017)
• Physics-informed autoencoders for nonlinear model reduction (Popov et al., 2021)
• Tree-based models with bagging/oversampling (e.g., BwO forest) (Kim et al., 2022)
• Ensembles of GPs or randomized surrogates for Bayesian optimization (Lu et al., 2022, Audet et al., 2021)
• Lightweight neural ensembles for spatiotemporal forecasting (Borisova et al., 2023)
• Ensembles of neural operators tuned via hyperparameter search (Sun et al., 22 Aug 2025)

2. Ensemble Strategies and Uncertainty Quantification

Ensemble surrogates enable rigorous quantification of both predictive mean and epistemic uncertainty—critical for uncertainty-aware optimization and scientific inference. Two principal methodologies are employed:

• Variance/Spread-Based Quantification: For a query $x$, uncertainty is evaluated using the weighted variance among surrogate predictions: $$\sigma^2(x) = \sum_{m=1}^M w_m \left(f_m(x) - \hat{y}(x)\right)^2$$ This variance reflects the epistemic uncertainty: if surrogates agree, uncertainty is low; if they disagree, uncertainty is high. Such measures drive exploration in Bayesian optimization and inform reliability bounds (Audet et al., 2021, Kim et al., 2022, Sun et al., 22 Aug 2025, Borisova et al., 2023).
• Distributional and Conformal Approaches: Some frameworks directly output full predictive distributions for $y$ conditioned on $x$, with higher-order priors yielding robust and calibrated intervals. ConfEviSurrogate (Duan et al., 3 Apr 2025) employs deep evidential regression with conformal prediction-based calibration to provide guaranteed coverage for predictive intervals, separating aleatoric and epistemic contributions using hierarchical priors and Student-t marginals.
• Ensembles of Generative Models: For probabilistic surrogate modeling (e.g., industrial control), ensembles of conditional GANs (cGANs) produce both mean forecasts and variance, with measures like Hellinger distance used to penalize overconfident optimization in out-of-distribution settings (Feng, 2022).

3. Computational Efficiency and Architectural Considerations

Ensemble surrogates are engineered to be compatible with contemporary computing architectures:

• Vectorized and Parallel Processing: Embedded ensemble propagation strategies enable batch processing where data structures and computation graphs are shared among ensemble members, optimized for vectorized and GPU-parallel operations (D'Elia et al., 2017).
• Parameter Sharing and Grouping: Packed-ensemble architectures exploit grouped convolutions and sub-networks with tunable capacity and sparsity to reduce memory and speed up training while retaining smooth ensemble properties appropriate for physical simulations (Kalaydjian et al., 2023).
• Hyperparameter and Model Selection: Construction often follows an initial phase of large-scale hyperparameter search to identify a pool of diverse, high-performing surrogates, after which ensemble aggregation is performed via bagging, stacking, or regression weighting to maximize robustness (Sun et al., 22 Aug 2025).
• Adaptive Model Selection: In Bayesian optimization, kernel selection and surrogate weighting are adaptively updated based on model fit to data, with the ensemble posterior formed as a mixture of surrogate posteriors (Lu et al., 2022).

4. Applications in Scientific Computing and Optimization

Ensemble surrogate modeling has demonstrable advantages across multiple domains:

• Uncertainty Quantification: Adaptive surrogate-based grouping strategies in the context of PDE uncertainty quantification minimize ensemble divergence by grouping samples with predicted similar computational cost, thus exploiting vectorization and communication-reduction opportunities on modern HPC architectures (D'Elia et al., 2017).
• Bayesian Inverse Problems and Data Assimilation: Multi-fidelity EnKF frameworks combine full-order and reduced-order surrogate dynamics (with surrogates constructed using autoencoder or polynomial chaos expansions) for sequential data assimilation, enabling efficient, robust Bayesian inference even in non-Gaussian and hierarchical models (Ba et al., 2018, Popov et al., 2021).
• Black-box and Bayesian Optimization: Ensembles of GPs, forests, or arbitrary surrogates serve as stochastic models that enable acquisition functions (e.g., Expected Improvement) to exploit local surrogate disagreement as a guide for exploration, leading to improved convergence over deterministic models in both continuous and mixed-variable settings (Audet et al., 2021, Lu et al., 2022, Kim et al., 2022).
• Adversarial Robustness and Transferability: In adversarial attacks, ensembles of surrogate models expand the transferable subspace; theoretical analyses decompose transferability error into vulnerability and diversity, motivating the explicit design of diverse, lower-complexity ensembles to maximize cross-model adversarial effectiveness and minimize transfer error (Cai et al., 2022, Chen et al., 2023, Yao et al., 9 Oct 2024).
• Earth System and Environmental Modeling: Applications include surrogate ensemble models for spatially resolved sea-ice concentration forecasts (Borisova et al., 2023), wave forecasting with historical data-driven aggregation (O'donncha et al., 2018), and large-scale operator surrogates with uncertainty-aware parametric sensitivity for ocean modeling (Sun et al., 22 Aug 2025).
• Interactive Simulation Exploration and Visualization: SurroFlow (Shen et al., 16 Jul 2024) integrates normalizing flow-based surrogates and a genetic algorithm into a visual interface, enabling scientists to both recommend optimal simulation parameters and assess prediction reliability via built-in uncertainty.

5. Comparative Evaluation and Practical Guidelines

Empirical results repeatedly demonstrate that ensemble surrogates outperform single-model surrogates in both predictive accuracy and reliability of uncertainty estimates. For example:

• In ocean modeling, ensemble FNOs yield lower RMSEs and more robust parametric sensitivities than any individual surrogate, with ensemble variance providing a usable epistemic uncertainty estimate (Sun et al., 22 Aug 2025).
• Surrogate-based ensemble grouping in PDE UQ applications maintains near-minimal extra computational work across a wide range of solver costs and preserves the acceleration benefit when sample-to-sample costs vary widely (D'Elia et al., 2017).
• In adversarial transfer scenarios, incorporating more surrogate models, increasing their diversity, and carefully managing model complexity each demonstrably reduce transferability error and increase black-box attack success rates (Yao et al., 9 Oct 2024, Chen et al., 2023).

Three practical guidelines for ensemble surrogate design—supported by both theoretical and experimental evidence—are:

1. Increase Ensemble Size: Larger ensembles narrow the generalization gap and constrict error bounds on transferability and uncertainty.
2. Promote Diversity: Ensembles should include heterogeneous architectures, training example shuffling, or distinct hyperparameters to maximize the spread among predictions.
3. Control Complexity: Overly complex (overfit) surrogates inflate ensemble Rademacher complexity, leading to poorer generalization and unreliable uncertainty estimates; regularization and capacity modulation can mitigate this.

A summary of typical ensemble strategies and uncertainty quantification methods is presented in the following table:

Ensemble Strategy	Uncertainty Quantification	Typical Domain/Benefit
Weighted aggregation (bagging, stacking)	Prediction variance or spread, distributional modeling	Black-box optimization, UQ, sensitivity
Embedded architectural grouping (packed-ensemble)	Aggregate output variance, Spearman’s correlation	Fluid dynamics, CFD design
Adaptive kernel/weight update (EGP, meta-learning)	Posterior mixture variance, Bayesian regret	Bayesian optimization
Deep evidential regression + conformal calibration	Student-t predictive intervals, explicit epistemic/aleatoric decomposition	Scientific simulations, robust UQ

6. Limitations, Challenges, and Future Directions

Despite their strengths, ensemble surrogate models present notable challenges:

• Computational overhead can be nontrivial if ensemble members are large, but this is mitigated by architectural choices (e.g., packed-ensembles) and vectorized hardware deployment (Kalaydjian et al., 2023).
• Quality of uncertainty quantification rests critically on ensemble diversity; highly correlated surrogates yield overconfident, unreliable intervals.
• Proper calibration of prediction intervals may require post-hoc methods such as conformal prediction to guarantee finite-sample or distribution-free coverage (Duan et al., 3 Apr 2025).
• In high-dimensional or data-limited regimes, the ensemble methodology may need to be coupled with advanced dimensionality reduction (e.g., autoencoders (Shen et al., 16 Jul 2024, Popov et al., 2021)) or specialized architectures for tractability.

Recent trends point to further integration of ensemble surrogates with genetic algorithms and interactive visualization for parameter exploration (Shen et al., 16 Jul 2024), deeper theoretical exploration of transferability and ensemble complexity (Yao et al., 9 Oct 2024), and automated model selection via large-scale hyperparameter search and meta-learning (Sun et al., 22 Aug 2025). AutoML approaches and domain-adaptive ensembling offer promising directions for rapid, robust surrogate model deployment in scientific and engineering domains.