Conditional Hybrid Neural Operator (CHNO)

• CHNO is a composite machine learning framework that integrates resolution-agnostic neural operators with conditional diffusion models for stochastic closure in PDEs.
• It fuses a deterministic operator stage with a generative corrector stage to capture non-local dependencies and high-frequency details.
• The framework improves spectral fidelity, enhances uncertainty quantification, and accelerates runtime across multiscale dynamical systems.

A Conditional Hybrid Neural Operator (CHNO) is a composite machine learning framework that integrates neural operator architectures—with resolution-agnostic operator learning—and conditional score-based generative models, particularly diffusion-based methods, to address stochastic, non-local closure modeling and multi-scale surrogate prediction tasks governed by partial differential equations (PDEs). CHNOs enable both deterministic representations of large-scale structure and stochastic, high-frequency corrections, facilitating accurate modeling where classical approaches lack generalization, spectral fidelity, or uncertainty quantification.

1. Formal Definition and Core Principles

CHNOs combine neural operators (e.g., Fourier Neural Operators (FNOs), Physics-Informed Neural Operators (PINOs), DeepONet) and score-based conditional diffusion models. This fusion enables modeling the conditional distribution of fine-scale unknowns or residuals given coarse-scale predictions or partial measurements. The generic structure is two-stage:

1. Deterministic Operator Stage An operator neural network maps initial or resolved states to a prediction of the field evolution:

$$u(x) = G_\theta(a(x)) = Q \circ (L_T \circ \ldots \circ L_1) \circ P$$

where $L_t$ denotes Fourier or spectral convolution layers.

1. Conditional Generative Corrector Stage A score-based diffusion model is conditioned explicitly on the output of the operator network to stochastically correct the residuals or reconstruct fine-scale details:

$$dx = \left[ f(x, t) - g(t)^2 \nabla_x \log p_t(x_t \mid y) \right]dt + g(t)d\bar{w}$$

with $y$ the operator prediction and $\nabla_x \log p_t(x_t \mid y)$ the conditional score learned from data.

Key principles of CHNO include:

• Stochastic modeling: Directly learning distributions of unresolved closure effects or residuals.
• Non-locality: Capturing spatial and temporal dependencies beyond local neighborhoods.
• Resolution invariance: Neural operator architectures enable mesh-independent field modeling.
• Conditionality: Conditioning generative models on operator outputs or observed data for context-aware sampling.

2. Mathematical and Algorithmic Frameworks

a. Closure Modeling via CHNO

Consider a reduced system state $V = \mathcal{K}(v)$, with the governing equation augmented by a closure term $U$: $$\frac{\partial V}{\partial t} = \widetilde{M}(V) + U$$ The stochastic closure $U$ is modeled via the conditional distribution $p(U \mid \mathbf{y})$, where $\mathbf{y}$ includes resolved states, measurements, or model-based estimates.

Conditional Diffusion Model

The generative model learns the score (gradient of log-density) via denoising score matching: $$\theta^* = \arg\min_\theta \mathbb{E}_{\tau} \mathbb{E}_{U_\tau} \mathbb{E}_{U_0, \mathbf{y}} \left\|\nabla_{U_\tau} \log p(U_\tau|U_0) - s_\theta(\tau, U_\tau, \mathbf{y})\right\|^2_2$$ Sampling employs the reverse SDE: $$U_{\tau_{i+1}} = U_{\tau_i} + \sigma^{2\tau_i} s_\theta(\tau_i, U_{\tau_i}, \mathbf{y}) \Delta\tau + \sigma^{\tau_i} \sqrt{\Delta\tau} z_i$$

Hybridization

The operator output (e.g., FNO or PINO solution) is passed to the diffusion model as conditioning information—either as sequence (for full temporal correction in MHD (Kacmaz et al., 2 Jul 2025)) or as multimodal field data (as in stochastic closure (Dong et al., 6 Aug 2024)).

b. Operator Learning Surrogates

Operator learning is employed for surrogate modeling in reliability analysis (Li et al., 2023):

• DeepONet architecture: Inputs are encoded functionally; outputs are learned as operator evaluations, ensuring generalization across domain or discretization.

3. Architectural Features and Conditional Fusion Strategies

CHNOs are defined by their multimodal, resolution-agnostic, and conditionally fused design:

• Neural Operator Backbone:
  • FNOs utilize Fourier convolutions for non-local field mapping.
  • PINOs embed PDE residuals directly in the loss, assuring physical consistency.
• Generative Corrector:
  • Diffusion models (Elucidated Diffusion Model, UNet-based) are employed.
  • Conditioning strategy: operator output is concatenated or embedded as input channels, providing prior for generative refinement.
• Fusion Mechanisms:
  • Temporal sequence conditioning (hybrid PINO-Diffusion (Kacmaz et al., 2 Jul 2025)).
  • Multimodal input fusion (score network with FNO pipelines for field, measurements, and noise level (Dong et al., 6 Aug 2024)).
  • Measurement and model estimate upsampling for improved conditional closure generation.

4. Applications Across Multiscale Dynamical Systems

CHNO frameworks have been applied in:

• Turbulent Closure Modeling (Navier-Stokes-α, 2D MHD):
  • Generation of stochastic, non-local closure terms for climate and fluid simulations (Dong et al., 6 Aug 2024, Kacmaz et al., 2 Jul 2025).
  • Recovery of energy spectra and non-Gaussian structures in high-Reynolds-number turbulence.
  • Efficient sampling for online integration with PDE solvers (up to 100x acceleration (Dong et al., 6 Aug 2024)).
• Reliability Analysis and Failure Probability Estimation:
  • Operator hybrid approaches with DeepONet surrogates enable accurate, scalable failure probability estimation in engineering systems (Li et al., 2023).
  • Hybrid MC algorithms employ operator-based surrogates and selective full-model recalculation for boundary points.

5. Empirical Results and Performance Characterization

Aspect	Operator Only	CHNO Hybrid
Large-scale, low-frequency	Excellent	Excellent
High-frequency, small-scale	Poor (bias)	Excellent
Temporal coherence	Deterministic, strong	Preserved
Spectral accuracy (turbulent)	Sharp drop at high-k	DNS-level recovery
Conditional generalization	Limited	Robust (multimodal)
Efficiency (runtime)	Fast inference; limited fidelity	Fast with accelerated sampling, high fidelity

Quantitative results (Kacmaz et al., 2 Jul 2025):

• At $Re=1000$: PINO-only error 25.5%, hybrid error 10.3%.
• High-wavenumber recovery for magnetic fields at $Re=10^4$: first surrogate model to achieve this.

Resolution invariance (Dong et al., 6 Aug 2024):

• Training at $64 \times 64$, consistent results up to $256 \times 256$.

Hybrid reliability estimation (Li et al., 2023):

• In high-dimensional settings ($n=50$), NOH achieves 0.81% relative error with only 150 evaluation calls.

6. Scientific Advances and Limitations

Advances

• Stochastic, non-local modeling beyond deterministic, local closures.
• Operator-learning paradigm: Mesh-independent field representations and flexible input modalities.
• Conditional generative modeling: Incorporating measurements, estimates, and historical context for improved accuracy.
• Accelerated sampling: Order-of-magnitude runtime improvements via adaptive time stepping and conditional prior selection.

Limitations

• Performance of generative corrector depends critically on the fidelity of operator prior; incomplete large-scale predictions are not fully corrected.
• Current generative stage may lack strict physics-informed enforcement, introducing residual physical inconsistencies at small scales.
• High computational resource demands during training, though inference remains significantly faster than direct simulation.

7. Outlook and Generalizability

The CHNO paradigm is extensible to a broad class of PDE-driven and multiscale systems, including climate modeling, plasma kinetics, and high-dimensional uncertainty quantification. A modular architecture allows for updates and improvement as new operator or generative models emerge. Potential future directions include physics-informed generative modeling, more robust uncertainty quantification, and advanced adaptive sampling techniques for reliability analysis.

CHNOs represent a principled fusion of operator learning and generative modeling, enabling accurate, efficient, and scalable surrogate models for stochastic, multi-physics, and high-dimensional systems, with demonstrated state-of-the-art performance in turbulent closure modeling and reliability quantification (Dong et al., 6 Aug 2024, Kacmaz et al., 2 Jul 2025, Li et al., 2023).