Direct Interval Propagation Methods using Neural-Network Surrogates for Uncertainty Quantification in Physical Systems Surrogate Model

Neural network surrogate models have emerged as a promising approach to model solution fields for a wide variety of boundary value problems encountered in physical modeling. Stochastic problems represent an area of particularly high interest because of the potential to significantly reduce the repeated evaluation of expensive forward models via traditional numerical solvers when conducting parametric analysis. However, many studies found in the literature primarily focus on the ability of neural network surrogate models to represent deterministic samples or mean field solutions and largely overlook surrogate model performance at the tails of the distribution. The present study examines in detail the ability of neural network surrogate models to capture the full distribution of solution fields over the entire probability space, while emphasis is placed at the tails of the distribution. Serving as a canonical problem is the heat conduction equation with a highly stochastic source term, inducing extremely large variation in the thermal solution field. Comparisons are made between a classic feed-forward fully connected network and a Deep Operator Network architecture, using both data-driven and physics-informed loss functions. Results show that the worst-case prediction errors are an order of magnitude larger than the mean field error, highlighting the importance of the outlier samples. The large errors associated with extreme samples result from the networks having to extrapolate beyond the bounds of the training data. A method for identifying these samples is presented along with a discussion of potential approaches to account of their errors. Among the models considered, the fully connected neural network trained using a weak form residual loss performs best in handling these extrapolated inputs, achieving the highest prediction accuracy for the numerically produced datasets.

Neural surrogates can substantially accelerate computer-aided engineering (CAE) workflows, but their use in design requires uncertainty estimates that remain meaningful across varying geometries, spatial prediction fields, and engineering quantities of interest. We investigate how established uncertainty quantification (UQ) approaches behave when adapted to geometry-conditioned neural surrogates. We compare one closed-form and two sampling-based approaches-a Gaussian process (GP)-based method, concrete Monte Carlo (MC) dropout, and deep ensembles-and evaluate them on three large, industry-relevant CAE datasets for external aerodynamics and crash dynamics. We examine whether predicted uncertainties have credible magnitudes, identify locations with larger prediction errors, respond to unfamiliar inputs, and remain informative for derived engineering quantities. On the DrivAerStar dataset, where all three methods are compared, each generally assigns higher uncertainty to locations with larger prediction errors, and validation-based rescaling brings interval coverage close to nominal on a disjoint in-distribution test set. Results on AirFRANS and automotive crash also show useful error ranking and interval estimates, but the relative performance of the methods changes with the dataset and evaluation criterion. UQ methods and evaluation metrics should therefore be selected based on the intended downstream CAE decision.

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.