CoNBONet: Conformalized Neuroscience-inspired Bayesian Operator Network for Reliability Analysis

Neural operators such as the Fourier Neural Operator (FNO) have emerged as powerful surrogates for solving partial differential equations (PDEs), achieving speedups of several orders of magnitude over traditional numerical solvers. However, deploying these models in safety-critical engineering applications -- such as thermal management of electronic components and battery systems -- requires not only accurate point predictions but also rigorous uncertainty guarantees. Existing uncertainty quantification (UQ) methods for neural operators, including Monte Carlo Dropout and Deep Ensembles, provide only relative uncertainty estimates without formal coverage guarantees. In this work, we propose the first application of split conformal prediction to neural operator-based physics simulation, providing distribution-free prediction intervals with finite-sample coverage guarantees. We further introduce a normalized conformal prediction scheme that leverages MC Dropout uncertainty to produce adaptive-width intervals, yielding tighter intervals in regions of low uncertainty and wider intervals where the model is less certain. Full-scale experiments (33.7M parameters, 800 training samples, 5 ensemble members, NVIDIA V100) on steady-state heat conduction benchmarks demonstrate that our method achieves 89.1% empirical coverage at the target level of alpha=0.1, while producing spatially adaptive prediction intervals that reflect the underlying physical uncertainty structure. We also provide an uncertainty decomposition framework that separates epistemic uncertainty (68% of total) from aleatoric uncertainty (32% of total), offering actionable guidance for data collection and model improvement. Our method is implemented in an open-source platform with REST API endpoints and interactive 3D visualization.

Neural operators (NOs) provide fast, resolution-invariant surrogates for mapping input fields to PDE solution fields, but their predictions can exhibit significant epistemic uncertainty due to finite data, imperfect optimization, and distribution shift. For practical deployment in scientific computing, uncertainty quantification (UQ) must be both computationally efficient and spatially faithful, i.e., uncertainty bands should align with the localized residual structures that matter for downstream risk management. We propose a structure-aware epistemic UQ scheme that exploits the modular anatomy common to modern NOs (lifting-propagation-recovering). Instead of applying unstructured weight perturbations (e.g., naive dropout) across the entire network, we restrict Monte Carlo sampling to a module-aligned subspace by injecting stochasticity only into the lifting module, and treat the learned solver dynamics (propagation and recovery) as deterministic. We instantiate this principle with two lightweight lifting-level perturbations, including channel-wise multiplicative feature dropout and a Gaussian feature perturbation with matched variance, followed by standard calibration to construct uncertainty bands. Experiments on challenging PDE benchmarks (including discontinuous-coefficient Darcy flow and geometry-shifted 3D car CFD surrogates) demonstrate that the proposed structure-aware design yields more reliable coverage, tighter bands, and improved residual-uncertainty alignment compared with common baselines, while remaining practical in runtime.

Bayesian neural network predictions are commonly approximated using a fixed number of Monte Carlo samples per input, without controlling the resulting error that comes from this finite sample. We propose the use of confidence sequences to dynamically determine how many samples are needed while maintaining statistical guarantees. We consider several ways in which predictive probabilities are used, including identifying the most likely class, approximating the full predictive distribution, and resolving probability-threshold decisions. Sampling stops once the corresponding decision can be made with the desired guarantee. Experiments show that the method allocates the computational budget efficiently, assigning more samples to ambiguous inputs than to easy inputs while preserving reliable decisions and reducing overall latency relative to a fixed Monte Carlo budget.