Neural Surrogates: Deterministic vs Probabilistic

• Deterministic and probabilistic neural surrogates are deep learning models that approximate complex systems, with deterministic surrogates providing fixed predictions and probabilistic ones incorporating uncertainty.
• They leverage methodologies such as latent variable models, diffusion models, and hybrid frameworks to balance physical constraints with robust uncertainty quantification.
• Applications span computational science, neuroscience, engineering design, and energy modeling, significantly reducing computational cost while enhancing prediction fidelity.

Deterministic and probabilistic neural surrogates are mathematical models, principally based on deep learning architectures, constructed to approximate the behavior of complex systems or simulators. Deterministic surrogates provide point predictions for a given input, while probabilistic surrogates return distributions or uncertainty quantification alongside the prediction. These approaches are pivotal in computational science, neuroscience, engineering design, spatiotemporal forecasting, and building energy modeling, where the high computational cost or stochastic nature of the underlying system necessitates efficient, flexible, and interpretable approximations. The development of these surrogates has spurred methodologies that bridge strict dynamical modeling with stochastic or information-theoretic descriptions, often exploiting latent variables, hybrid frameworks, or architectural constraints for improved fidelity, uncertainty representation, and generalization.

1. Fundamental Principles and Distinctions

The defining distinction between deterministic and probabilistic neural surrogates lies in how they characterize the output for a given input:

• Deterministic surrogates establish a mapping $x \mapsto y$ via a trained neural network, producing a unique output (e.g., predicted state, field, or decision variable) for each input. Classical frameworks include regression neural networks, operator-learning architectures (e.g., Fourier Neural Operators), and deterministic function-approximation surrogates for PDEs, optimal control, or simulator outputs (Ilievski et al., 2016, Longo et al., 2020, Holzschuh et al., 12 Sep 2025).
• Probabilistic surrogates model not the mapping $x \mapsto y$, but the conditional distribution $p(y|x)$. These networks predict either the parameters of prescribed distributions (such as $\mathcal{N}(\mu(x), \sigma(x))$), mixtures (e.g., Gaussian mixture models), or implicitly define sampleable distributions for $y$ given $x$ (as in diffusion models or latent variable generative models) (Yang et al., 2019, Maulik et al., 2020, Fukami et al., 2020, Peršak et al., 2023, Sheng et al., 16 Feb 2025, Holzschuh et al., 12 Sep 2025). They allow the calculation of confidence intervals, quantiles, or even full generative samples.

The distinction extends beyond outputs: probabilistic surrogates integrate uncertainty (aleatoric and/or epistemic) into the modeling pipeline, often crucial for risk-averse decision-making, uncertainty propagation, or robust optimization.

2. Theoretical Foundations and Model Construction

The theoretical underpinnings of both approaches draw from disparate traditions:

• Mechanistic (deterministic) surrogates are typically built to directly interpolate or map high-dimensional data: for example, using autoencoders for dimensionality reduction with a subsequent regression mapping from inputs to a low-dimensional latent code, followed by decoding (Deshpande et al., 15 Jul 2024). In optimal control, local quadratic expansions and dynamic programming techniques yield deterministic updates (e.g., DDP gain matrices or closed-form control policies) (Filabadi et al., 18 Jul 2024).
• Probabilistic surrogates involve statistical and information-theoretic frameworks:
  • Latent Variable Models: Conditional deep surrogates introduce latent variables $z$ with $p(y|x) = \int p(y|x, z) p(z|x) dz$, often trained with variational inference (VAE-style) or adversarial approaches for implicit posteriors (Yang et al., 2019, Rixner et al., 2020).
  • Gaussian Process Surrogates: By compressing high-dimensional outputs via neural network autoencoders, GP regression in a latent space provides scalable Bayesian uncertainty quantification (Deshpande et al., 15 Jul 2024).
  • Diffusion Models and Flow Matching: Deep generative surrogates for fields or images use forward and reverse stochastic processes to sample from complex conditional distributions in high dimensions (Sheng et al., 16 Feb 2025, Holzschuh et al., 12 Sep 2025).
  • Surrogates for Stochastic Simulators: Models such as Probabilistic Surrogate Networks match the structure and stochastic control flow of nontrivial simulators (e.g., with unbounded or dynamically evolving sets of latent variables) (Munk et al., 2019).
  • Quantile Regression and Conformal Prediction: In cases where explicit distributions are intractable or multi-modal, quantile predictions—extended with conformalization for calibration—are used for coverage guarantees (Krannichfeldt et al., 23 Jul 2025).

Probabilistic surrogates are closely linked to the theory of neural coding in neuroscience, where the stochastic spiking statistics of deterministic neurons are mapped to a probabilistic code via conditional models as in adaptive gain control theory (Famulare et al., 2011).

3. Methodological Advances and Hybrid Approaches

Modern research increasingly blurs the boundary between deterministic and probabilistic surrogates:

• Hybrid Deterministic–Probabilistic Decompositions: Some frameworks decompose the output as $$x^{(\mathrm{ta})} = \mathbb{E}[x^{(\mathrm{ta})}|x^{(\mathrm{co})}] + r$$, where a deterministic neural network estimates the conditional mean and a probabilistic model (e.g., diffusion or residual learning network) models the stochastic residual (Sheng et al., 16 Feb 2025). Scale-aware mechanisms (e.g., spatially-varying priors $Q$) are included for region-specific uncertainty.
• Regularization and Robustness: Surrogates for robust optimization problems employ deterministic surrogates for the nominal (or "quick-solve") problem, with an embedded "worst-case" evaluation layer that regularizes the surrogate through adversarial or robustness penalties (Peršak et al., 2023). This approach yields neural surrogates that produce robust solutions without full-blown robust optimization at inference.
• Physics-Based and Data-Driven Integration: In building energy modeling, deterministic outputs of physics simulators inform or are combined with probabilistic neural quantile regression models, with residual learning providing both correction and intuitive out-of-distribution behavior (Krannichfeldt et al., 23 Jul 2025).
• Constraint-Informed Surrogates: For physical sciences, geometric (symmetry) and physical (conservation law) constraints in the architecture (input/output layers) augment deterministic surrogates so that they respect invariances and conservation properties. This approach improves generalization and reduces long-term error accumulation (Huang et al., 5 Jun 2025).

4. Uncertainty Quantification and Calibration

Uncertainty modeling is central to probabilistic neural surrogates:

• Aleatoric and Epistemic Uncertainty: Epistemic uncertainty, originating from model limitations and inflicted by insufficient training data, can be modeled via Bayesian inference over neural network weights (Variational Bayes) (Deshpande et al., 2021). Aleatoric uncertainty arising from intrinsic system stochasticity is represented via output variance parameters or sampled latent variables (Maulik et al., 2020, Deshpande et al., 15 Jul 2024).
• Calibration Strategies: Accurate uncertainty quantification requires calibrated predictions. Metrics such as the continuous ranked probability score (CRPS), reliability score (RS), and accuracy–reliability (AR) cost functions are used in model selection and neural network training (Camporeale et al., 2018). Conformalized quantile prediction is used to correct the coverage of prediction intervals in quantile-regression surrogates (Krannichfeldt et al., 23 Jul 2025).
• Function-Space and Distributional Matching: In high-resolution 3D surrogates, probabilistic sampling via diffusion models or flow matching is used to generate plausible samples whose statistics match ground-truth simulations (e.g., turbulent profiles, higher moments across Reynolds numbers) (Holzschuh et al., 12 Sep 2025).

5. Practical Applications and Performance Benchmarks

Empirical studies across domains uniformly indicate the following:

• Computational Efficiency: Deterministic surrogates, especially those built on RBF interpolation (Ilievski et al., 2016) or trained with quasi-Monte Carlo designs (Longo et al., 2020), achieve rapid and dimension-robust performance, particularly for optimization or design parameter sweeps. Pretraining on small 3D volumes with patch fusion enables scaling to unprecedented resolutions (Holzschuh et al., 12 Sep 2025).
• Predictive Accuracy and Adaptability: Probabilistic surrogates (mixture density networks, Gaussian process latent mappings, flow-matched diffusion) consistently attain superior accuracy and more reliable uncertainty quantification than point predictors, especially in stochastic, multi-modal, or under-sampled settings (Maulik et al., 2020, Fukami et al., 2020, Yang et al., 2019).
• Generalization and Stability: Hard-coded physical or symmetry constraints in architecture, and well-calibrated uncertainty estimates, yield surrogates that generalize reliably across initial conditions, lower error accumulation in autoregressive rollouts, and better handle out-of-distribution scenarios (Huang et al., 5 Jun 2025, Zhou et al., 17 Dec 2024).

Representative performance outcomes, as explicitly reported:

• The HORD deterministic surrogate requires only 27% of the evaluations consumed by state-of-the-art probabilistic GP-EI or SMAC in 19-dimensional DNN hyperparameter tuning; it achieves up to 6× speedup in function evaluations (Ilievski et al., 2016).
• The CoST hybrid deterministic–probabilistic surrogate achieves 20–25% relative improvement over best-in-class baselines for spatiotemporal forecasting while accelerating training and inference (Sheng et al., 16 Feb 2025).
• Constrained neural PDE surrogates outperform strong baselines, with doubly-constrained (symmetry + conservation) models demonstrating the lowest nRMSE and best spectral accuracy in both synthetic and real-world data (Huang et al., 5 Jun 2025).

6. Impact, Limitations, and Future Directions

• Resource Efficiency and Scalability: Deterministic RBF surrogates and QMC-trained DNNs are highly scalable in dimensionality but cannot quantify predictive uncertainty unless extended via post-processing or hybridization (Ilievski et al., 2016, Longo et al., 2020). Probabilistic surrogates based on diffusion or latent-variable methods are now viable for large-scale 3D domains and high-dimensional outputs, with adaptation to nanoscale turbulent flows and multi-physics problems (Holzschuh et al., 12 Sep 2025, Deshpande et al., 15 Jul 2024).
• Integration of Constraints and Hybrid Models: Incorporating symmetries, physics, and robustness within the architecture, along with mean-residual decompositions or residual learning, marks a maturing in surrogate modeling where deterministic and probabilistic paradigms are mutually reinforcing (Huang et al., 5 Jun 2025, Peršak et al., 2023, Sheng et al., 16 Feb 2025).
• Challenges: Probabilistic surrogates incur greater computational cost at training, sensitivity to misspecified uncertainty models, and rely on regularization and calibration for reliable deployment. Deterministic surrogates may generalize poorly without intrinsic bias pursuit or cannot reflect outcome variability.
• Open Research Spaces: Future work includes: unifying constraint enforcement with probabilistic quantification, expanding efficient surrogate modeling to real-time adaptive control or risk-sensitive planning (Filabadi et al., 18 Jul 2024), and extending hybrid and probabilistic surrogates (such as diffusion models) to even higher-dimensional and more multi-modal settings.

7. Representative Mathematical Formulations

Surrogate Type	Core Mathematical Structure	Key Reference(s)
Deterministic RBF	$$S_n(x) = \sum_{i=1}^n \lambda_i \phi(\|x-x_i\|) + p(x)$$	(Ilievski et al., 2016)
Probabilistic Gaussian	$p(y|x) = \mathcal{N}(\mu(x), \sigma(x))$	(Maulik et al., 2020)
Latent Variable Model	$p(y|x) = \int p(y|x, z)p(z|x)dz$	(Yang et al., 2019)
Hybrid Mean-Residual	$$x^{(\mathrm{ta})} = \mathbb{E}[x^{(\mathrm{ta})}|x^{(\mathrm{co})}] + r$$	(Sheng et al., 16 Feb 2025)
Conformal Quantile	$$s(x, y) = \max\{f_{\alpha/2}(x) - y, y - f_{1-\alpha/2}(x)\}$$	(Krannichfeldt et al., 23 Jul 2025)
Flow Matching Diffusion	$$x_t = t \cdot u_{\text{out}} + [1-(1-\sigma_{\min}) t] \cdot \epsilon$$	(Holzschuh et al., 12 Sep 2025)

This mathematical summary captures the essential predictive and uncertainty structures that distinguish deterministic and probabilistic neural surrogates across the surveyed literature.