Adaptive surrogate modeling for high-dimensional spatio-temporal output

Structured Neural Chaos: An Adaptive Surrogate Modeling Framework for Functional Uncertainty Quantification and Global Sensitivity Analysis

Variance-based global sensitivity analysis (GSA) plays a key role in uncertainty quantification by identifying the contributions of uncertain inputs to the variability of the model response. The repeated model evaluations required for these tasks are often prohibitively expensive; surrogate models provide an efficient alternative by constructing inexpensive approximations of the underlying system response. Constructing surrogate models that combine scalability and interpretability for systems with high-dimensional stochastic inputs and functional responses remains challenging, particularly when sensitivity estimates are required across spatial or temporal domains. Polynomial chaos expansion (PCE) provides an effective framework for uncertainty propagation and sensitivity analysis due to its orthogonal structure and direct relationship with variance-based sensitivity measures. However, PCE suffers from the curse of dimensionality, whose computational burden is amplified for problems with functional responses. In this work, we introduce the Structured Neural Chaos (sNC) expansion as a surrogate modeling framework for variance-based GSA, inspired by the interpretability and orthogonal structure of PCE. The proposed framework retains the interpretability of structured decompositions while leveraging the expressive power of neural networks. The sNC expansion mirrors a truncated functional ANOVA decomposition, where each interaction component admits a separable low-rank approximation whose basis functions and coefficients are parameterized by neural networks. The expansion is constructed sequentially, adaptively identifying the dominant modes within each ANOVA subspace and determining the effective complexity of the representation. The resulting structure enables the extraction of statistical and sensitivity quantities directly from the coefficients of the sNC expansion at negligible cost.

Enhancing computational efficiency in multiscale systems using deep learning of coordinates and flow maps

Multiscale systems are expensive to simulate because fast dynamics require small time-steps, while slow dynamics require long prediction horizons. We propose latent hierarchical time-stepping (L-HiTS), which combines nonlinear coordinate discovery with multiscale flow-map learning. A deep autoencoder first compresses the high-dimensional PDE state into a validated low-dimensional latent space. Residual neural network time-steppers are then trained and coupled directly in this reduced space using validation-based hierarchy selection and vectorized prediction. Unlike multiscale HiTS, L-HiTS performs recursive forecasting in latent coordinates and reconstructs the full state only after prediction. The method is validated on the FitzHugh–Nagumo model, the chaotic Kuramoto–Sivashinsky equation, and a two-dimensional Burgers’ system. L-HiTS achieves comparable prediction accuracy to multiscale HiTS while substantially reducing training and prediction costs, with near order-of-magnitude prediction-time savings in the reported cases.

Machine-learning surrogate models for nonlinear energetic-particle transport predictions in ITER

Fast and accurate prediction of energetic-particle transport driven by Alfv\'en eigenmode (AE) instabilities is essential for integrated modeling workflows used in the design and optimization of burning plasma fusion reactors. In this work, we develop machine-learning-based surrogate models for rapid prediction of energetic beam and alpha-particle transport fluxes, together with predictive uncertainty estimates, for an ITER steady-state scenario. Two complementary surrogate methodologies, Gaussian process (GP) regression and hierarchical neural networks (NNs), are trained using nonlinear FAR3d gyrofluid simulations of energetic-particle transport. A flux-variability analysis demonstrates that the selected plasma-state representation provides a sufficiently unique parameterization of the nonlinear transport response over most of the sampled feature space, thereby justifying the surrogate formulation. Both surrogate models reproduce the nonlinear transport fluxes with high predictive accuracy while reducing the computational cost of transport evaluation by approximately five to six orders of magnitude relative to direct nonlinear FAR3d simulations. Although the two approaches achieve comparable predictive accuracy, they exhibit distinct uncertainty characteristics: the GP provides more consistent global uncertainty estimates, whereas the NN more clearly distinguishes between different transport regimes. This work establishes a proof of concept for developing machine-learning surrogate models of energetic-particle transport that are sufficiently accurate and computationally efficient to be incorporated into future integrated modeling workflows.

Efficient Bayesian calibration of many-parameter system models

Computer models of complex engineering systems rely on proper tuning of their model parameters to ensure accurate predictions of the system behavior. The challenge of effectively calibrating many-parameter models is the difficulty of sampling in high-dimensional spaces and the computational expense of generating a large number of samples to characterize the calibrated parameter distributions. The method of active subspaces has been shown to be effective at constructing low-dimensional latent spaces for Bayesian inverse problems when the misfit function (i.e., negative log-likelihood) is treated as the function of interest. On the other hand, works that implement surrogate modeling for inference often focus on approximating the predictive model itself. In this work, an integrated dimension reduction and surrogate modeling framework for efficient and robust model calibration based on the Kennedy O'Hagan framework is proposed, with the following key components. First, an active subspace of the misfit function is identified. Then, a surrogate model for the misfit is constructed in this low-dimensional latent space. Care is taken to ensure that the assumed probabilistic structure of the misfit surrogate is compatible with the structure imposed on the misfit by the observation noise and computer model discrepancy. Further, a generalized likelihood function is defined that can account for the misfit surrogate uncertainty along with the other usual sources of uncertainty, e.g., experimental noise, model inadequacy, etc. This general formulation is shown to be valid for surrogates of any deterministic bijective function of the original likelihood, not just the misfit. Finally, a strategy for incorporating the uncertainty in identifying the active subspace is included.

Deep Spatially Varying Coefficient Model for Interpolation of Non‐Stationary Meteorological Data

In spatial statistics, the spatially varying coefficient model (SVCM) is widely applied in the analysis and interpolation of non‐stationary spatial data. By incorporating spatially varying coefficients, the model can capture spatial heterogeneity and provide an attractive interpretation of response‐covariate associations. However, intensive matrix operations are inevitable in the inference of SVCM, which limits its scalability to massive spatial datasets. In contrast, deep learning has demonstrated considerable potential in various regression and classification tasks with large‐scale data in terms of accuracy and computational efficiency. Nevertheless, many deep learning models fail to capture the spatial variability when directly applied to non‐stationary spatial data and often lack interpretability. To address these challenges, we propose a deep spatially varying coefficient model (DSVCM) that combines the strengths of SVCM and deep learning. The proposed model aims to provide a computationally efficient interpolation method for non‐stationary spatial data, while also facilitating the interpretation of the effects of input covariates. In the proposed model, a deep neural network (DNN) is utilized to estimate spatially varying coefficients and responses, where spatial basis functions are served as input to capture spatial variability. By leveraging spatial basis functions, we establish the connection between DSVCM and SVCM, and theoretically prove the superiority of DSVCM in prediction accuracy. The effectiveness of the model is further demonstrated through extensive simulation studies and experiments on Singapore air temperature data. The results show that the proposed DSVCM outperforms several baseline models for spatial interpolation in both prediction accuracy and computational efficiency.

Prediction of physical fields under linear equality constraints