Surrogate Models

Finite Element Analysis (FEA) is widely used for transient mechanical simulations, but its high computational cost limits real-time and high-resolution applications. Deep learning surrogate models can reduce this cost; however, many existing approaches are restricted to steady-state prediction or cannot jointly predict Node- and Element-based Outputs (NEO) over time. The state-of-the-art DeepFEA framework has addressed these issues but remains limited to structured finite element (FE) meshes. To overcome this limitation, this study proposes DeepFEAv2, a deep learning surrogate framework that enables prediction of transient FEA simulations across different FE mesh topologies and element types. The main contributions of DeepFEAv2 are: (a) a module that uses the FE connectivity matrix to organize input features by element and arrange them into an input sequence guided by the mesh topology; (b) a novel neural network architecture designed to process the input sequence and jointly predict NEO over time; and (c) a FEA-informed optimization strategy for regularizing these NEO predictions. DeepFEAv2 was evaluated on structured and unstructured 3D linear elastic datasets, as well as on a pressure-driven aortic valve dataset. DeepFEAv2 achieved R^2 values up to 0.99 and normalized errors as low as 0.38%. Compared with DeepFEA, it achieved up to 38.0% relative increase in R^2 and up to 87.1% reduction in normalized error. DeepFEAv2 also performed inference up to three orders of magnitude faster than traditional FEA. These results demonstrate that DeepFEAv2 can efficiently model transient FEA simulations across increasingly complex FE settings, providing a scalable surrogate framework for transient FEA.

Surrogate-assisted evolutionary algorithms (SAEAs) are effective methods for solving expensive optimization problems (EOPs), where surrogate models replace most expensive evaluations and critically influence the final optimization results. In recent years, tabular foundation models have advanced rapidly, and the Tabular Prior-data Fitted Network (TabPFN) has been adopted as a surrogate model for EOPs due to its strong predictive capability, demonstrating promising performance. Motivated by its potential as a surrogate model in SAEAs, this work conducts a comprehensive study that combines extensive experiments with in-depth theoretical analysis to investigate the effectiveness of TabPFN. Specifically, we perform experiments across both offline and online SAEA settings, covering diverse problem scenarios such as single-objective, multi-objective, constrained, combinatorial, mixed-variable, and engineering optimization problems. In addition, we further analyze the advantages and limitations of TabPFN within SAEAs and provide practical guidelines for its application in different optimization settings. Results show that the effectiveness of TabPFN is highly problem dependent, and it cannot replace conventional surrogates universally. Overall, TabPFN should be adopted selectively according to data availability, landscape complexity, search space characteristics, and its role within the algorithm. Customized model management strategies and role-specific algorithm design are necessary to fully exploit its advantages and avoid its pitfalls.

Discrete black-box optimization is often addressed using approaches such as Sequential Model-Based Optimization (SMBO), which aims to improve sample efficiency by fitting surrogate models that approximate a costly objective function over a discrete search space. In many real-world problems, the set of feasible inputs is often given by logical constraints known in advance. However, existing surrogate modeling techniques generally fail to capture the symbolic rules governing feasibility in discrete input spaces. In this paper, we propose a surrogate modeling approach based on tensor decomposition that captures the structure of discrete search spaces while directly integrating feasibility information. To implement this approach, we formulate surrogate model training as a constrained polynomial optimization problem and solve a relaxed formulation using a differentiable penalty term derived from T-norms. Our experiments on both synthetic and real-world benchmarks, including a pressure vessel design task, demonstrate that the proposed method improves sample efficiency by effectively guiding the search away from infeasible regions.

We study the local errors of classical machine-learning surrogate models, which approximate the time evolution of the one-dimensional viscous Burgers equation. Four models are compared on the same prediction task, using the spatial grid values directly: radial basis function (RBF) kernel ridge regression (KRR), linear Ridge, ExtraTrees, and Random Forests. Across all four models, the one-step residual, defined here as the true value minus the predicted value at each grid point, forms clear curved branches near predicted maxima and minima. A more detailed analysis of KRR shows that these errors are much more strongly related to the second spatial derivative, which measures local curvature, than to the first spatial derivative. Near a smooth extremum, predicted value and curvature form a local two-branch fold. Under our local curvature-based model of the residual, this fold predicts a leading-order near-parabolic relation between predicted value and residual. This geometric result motivates a direct test of the Burgers advection (transport) and diffusion (smoothing) terms. For KRR and Ridge, regression tests on held-out trajectories, a control that breaks the spatial alignment of the diffusion term, and a spectral test of high-frequency content are consistent with insufficient viscous smoothing at moderate and high viscosity. In this case, the surrogate retains more small-scale structure than the true future state. The same physical explanation is much weaker for the tree models. Finally, a correction that uses only predicted quantities reduces both one-step error and error during recursive rollout, where each prediction is used as the next input.

Reliable transport models are essential when modelling and optimising many chemical engineering processes, yet, most models assume hand-picked constitutive laws which may not reflect reality, and often assume initial conditions are known exactly. Both restrictions can significantly bias model predictions and lead to systematic error when used in predictive and control settings. Black-box neural surrogate alternatives for modelling can better match real example data, but are confined to the task they were trained on and cannot be interrogated for physical consistency. Here we introduce a general-purpose differentiable hybrid modelling framework for transport processes, specifically for the case of population balance equations. Our framework integrates a JAX finite volume population balance solver with learnable neural network components which are trained to both discover constitutive laws and fit initial conditions from real experimental data, allowing us to better model real experimental transport systems. Furthermore, we use our framework for process optimisation, using its differentiability to allow us to direct optimising experimental settings for quantities of interest. This work highlights the huge potential of such differentiable hybrid modelling frameworks for learning and optimising any given chemical separation which involves mass, energy, and/or momentum transport.

Predicting storm surge induced by landfalling tropical cyclones is crucial for flood mitigation and coastal risk management. Traditionally, physics-based numerical models simulate storm surge by solving the Navier--Stokes equations using numerical methods, but these simulations are computationally expensive. Generative models are promising for storm surge emulation because they can generate diverse realizations rather than producing a single deterministic prediction. However, their use for storm surge emulation remains largely unexplored. In this paper, we leverage diffusion models for storm surge surrogate modeling, combining a baseline prediction stage with conditional generation to provide a more interpretable modeling framework. We develop SurgeGen, a two-stage generative framework for generating storm surge scenarios conditioned on hypothetical storms with parameters defined in a continuous space. First, a baseline model produces a coarse estimate of the storm surge height. This estimate then conditions a diffusion model, which generates refined storm surge scenarios that better capture spatial patterns and variability. We demonstrate that our approach can generate realistic and diverse storm surge scenarios under conditions both within and outside the training distribution.

Observations increasingly reveal the coupled radiative, chemical, and dynamical processes that shape exoplanet atmospheres. Interpreting these atmospheres requires models that can capture this complexity. However, multidimensional models remain fundamentally limited by computational cost, and answering key questions requires simulating the governing physical mechanisms at speeds classical methods cannot achieve. As a result, models often rely on simplifying approximations, such as equilibrium chemistry, even when those assumptions miss important effects. There is a pressing need for fast and accurate chemical kinetics solvers to model planetary atmospheres. Here we present a machine learning local-box chemical kinetics solver for exoplanet atmospheres using a residual flow-map architecture. We demonstrate that this surrogate model is several orders of magnitude faster than a classical solver, achieving microsecond-scale inference while retaining percent-level accuracy. The surrogate model covers a parameter space that spans $T=300$-$3000$ K, $P=10^{-6}$-$10^{4}$ bar, $Δt=10^{-3}$-$10^{8}$ s, and compositions ranging from $10^{-2}$ to $10^{3}$ times solar in both C/O ratio and metallicity. Our model outperforms several commonly used machine learning architectures and performs robustly under the extreme stiffness characteristic of atmospheric chemistry. The machine learning framework presented here is a flexible and efficient approach to emulating state-to-state flow-map problems that commonly arise in numerical simulations.

Classical numerical solvers for partial differential equations (PDEs) are computationally expensive to solve repeatedly across varying initial conditions, motivating the need for learned surrogates. In this paper, we propose a trainable Neural Cellular Automata (NCA) based surrogate model for learning long time PDE dynamics. Rather than mapping an entire initial field to a full trajectory in one shot, our proposed model learns a small, local, homogeneous update rule that is applied identically and repeatedly at every grid cell, mirroring the locality of differential operators. We benchmark this framework against three baselines: PDE - Net, a modified physics-informed neural network (PINN), and a Fourier Neural Operator (FNO), on five canonical PDEs (heat, advection, Burgers, Allen - Cahn, and Fisher - KPP), evaluated at temporal domain two times beyond the training temporal domain. The proposed model achieves the lowest long-horizon relative errors on the majority of the experiments.

Streaming-potential-mediated transport of viscoelastic fluids has attracted research attention owing to its applications in electrokinetic energy conversion and microfluidic transport. Existing analytical and semi-analytical models in published literature provide valuable physical insights, but require repeated numerical evaluations for exploring large design spaces and identifying the optimal operating conditions. In this work, a surrogate-assisted framework is developed for rapid design optimization of pressure-driven electrokinetic transport of simplified Phan-Thien-Tanner fluids in a slit microchannel. A high-fidelity numerical database is generated over a broad range of governing dimensionless parameters, which includes the zeta potential, the Debye parameter, the Dukhin number, and the viscoelastic parameter. A Machine Learning surrogate model is subsequently trained to accurately approximate the nonlinear relationship between the governing parameters and the streaming potential, while the volumetric flow rate and hydroelectric energy conversion efficiency were calculated from closed form equation by using the streaming potential predicted by the surrogate. This is coupled with a multi-objective optimization strategy to identify operating conditions that simultaneously maximize energy conversion efficiency and volumetric flow rate. The proposed methodology can significantly accelerate parametric exploration compared with repeated numerical simulations across different parameters and provides practical design guidelines for electrokinetic microfluidic devices. The study demonstrates the potential of combining computational fluid mechanics with data-driven surrogate modeling for efficient engineering design and optimization.

We present a natively multiscale operator learning method for the surrogate modeling of (numerical solvers for) multiscale partial differential equations (PDEs). The primary novelty of our method lies in a novel multiscale kernel frame function approximation technique. Leveraging this new kernel frame technique, we cast the operator learning problem as one of learning frame coefficients of output functions as a function of frame coefficients of input functions. The generalization step then automatically allows for a multiscale decomposition of the output functions. Our method is applicable to both tensor-product grids and point clouds. We present interpolation proofs, error estimates, and numerical convergence rates for our frame approximation. We the demonstrate the applicability of our method for the surrogate modeling of inherently multiscale PDEs. The new multiscale frame kernel method is significantly more accurate than popular neural operators on challenging problems from the literature, while simultaneously admitting an a posteriori multiscale decomposition upon generalization.

Forward and inverse modeling of parametric dynamical systems requires surrogate models that are not only accurate for state prediction, but also informative for parameter calibration. However, a systematic end-to-end differentiable formulation for coupling deep-learning-based reduced-order surrogates with variational parameter estimation remains underdeveloped. In this work, we introduce a physics-aware neural-network-based latent-space framework for reduced-order forward modeling and variational parameter estimation. The proposed autoencoder-based approach yields a differentiable surrogate that maps physical parameters to predicted flow fields through a latent representation. The observable supervision is used during offline training to encourage the latent variables to retain information correlated with system parameters, while the online inverse problem is solved in the parameter space through the surrogate-induced observation operator. The method is evaluated on two computational-fluid-dynamics benchmarks. The results show that reconstruction accuracy alone is insufficient for inverse modeling, owing to the lack of end-to-end differentiability or physics awareness for variational parameter calibration. Quantitative latent-space analysis further shows that observable supervision improves case-level separability and temporal organization of latent representations. Experiments with realistic measurement settings, including noisy, low-resolution, randomly masked, and block-wise partial observations, demonstrate the robustness of the proposed framework and show that it generally reduces calibration error and variability compared with the standard surrogate models.

Advanced air mobility operations hold the potential to enhance and expand regional transportation of both people and goods in populated areas. However, hazardous flight conditions arising from interactions between wind and the built environment remain a significant challenge for aerial vehicles in urban settings. This work proposes a novel framework towards safe flight planning of aerial vehicles in windy urban environments. A learning-based surrogate model is trained to rapidly predict flow fields from readily available information such as building geometry and incident wind. This surrogate prediction is used to calculate a volumetric flight challenge scalar field based on critical flow parameters and proximity to structures. A safe, flow-informed flight trajectory is then identified through a cost-minimizing pathfinder. The complete system is demonstrated experimentally through flight tests of a micro aerial vehicle through a model urban geometry placed in a large fan-array wind tunnel. Comparing this approach to trajectories generated without knowledge of the wind field, we find the flow-informed approach reduces undesired vehicle displacement and improves flight stability. This work is among the first practical demonstrations of safe, wind-aware methodologies for advanced air mobility in urban environments.

Supervised training of finite-element (FE) surrogate models requires reference solutions, and each reference solution is obtained by solving the system that the surrogate is intended to replace. The assembled discrete potential energy provides a training signal that requires no reference solution. This note records, with proofs, the identities that make this signal exact for linear elastostatics: the difference between the energy of a prediction and the energy of the reference solution equals one half of the squared stiffness-norm error, and the gradient of the energy equals the stiffness-weighted error. Label-free discrete-energy minimisation and supervised regression in the stiffness norm therefore have the same unique minimiser and identical gradients at every point. Around this central result, the note states a conditioning lemma that bounds the displacement error by the energy gap, a modewise contraction identity that explains why the Euclidean displacement error is an unsuitable primary metric, the Chebyshev bound that governs conjugate-gradient post-processing of surrogate predictions, and a conditional latent-separation proposition for joint-embedding predictive architecture (JEPA) pretraining on a shared stiffness operator, with an explicit numerical counterexample that delimits its scope. Every claim with numeric content is implemented as an executable falsification check; the checks were executed twice, on synthetic test problems and on a probe set of 16 instances from the validation split of a pre-registered experimental run, and every inequality holds, with the measured tightness reported. A closing section explains why the construction does not extend to elastodynamics through direct minimisation of the action functional, and which time-discrete formulation restores exactness.

An ensemble of surrogate models helps improve the prediction quality and robustness of surrogate models, and in turn, the search performance of surrogate-assisted evolutionary algorithms (SAEAs). Although different degrees of smoothness of the approximated fitness landscapes need to be carefully designed for an effective ensemble, little attention has been paid to the explicit tuning of the degree of smoothness derived by surrogate models. This study proposes an adaptive ensemble SAEA, which automatically constructs plausible ensemble models by optimizing their parameter settings. Unlike existing adaptive/ensemble SAEAs, which consider prediction accuracy alone, the proposed algorithm optimizes the structure of radial basis function networks (RBFNs) by solving bi-objective minimization problems of approximation error and model complexity, resulting in robust ensemble models of accurate surrogate models with different degrees of smoothness of the approximated fitness landscapes. As a result, the over/under-fittings are reduced. Additionally, an infill criterion is designed so that surrogate models with different degrees of smoothness can contribute to the solution prescreening. The experimental results demonstrated the statistical superiority of our algorithm over state-of-the-art SAEAs on a single-objective benchmark and real-world problem sets under an expensive optimization scenario. The source code of the proposed algorithm is available at https://github.com/haranychan/EPOS

Understanding and controlling complex dynamical systems often requires executing thousands of numerical simulations across vast parametric landscapes, which is time-consuming. Machine learning surrogates significantly accelerate simulation by predicting state trajectories across different initializations and parameter values. However, surrogate models are specialized to one simulation model. Modifying the underlying differential equations - e.g., adding a physiological state or altering an epidemiological contact network - renders trained models obsolete and forces computationally expensive retraining from scratch. We introduce HyperODE, a surrogate capable of operating across an entire class of approximately mass-conserving compartmental models without retraining. By mapping the structure of ordinary differential equations (ODEs) into directed hypergraphs, HyperODE decouples the functional form of system interactions from the neural network architecture. HyperODE takes a compartmental model in the form of an ODE with an arbitrary parameter distribution defined through quantiles and transforms it into a hypergraph. It outputs the distribution of the trajectories for all the states in the original ODE in the form of quantiles. We then use this surrogate to build an encoder that takes a noisy trajectory and outputs a distribution over the parameters of the original ODE, thus calibrating the model in a single pass. On families and system sizes never seen in training, HyperODE produces calibrated quantile bands in a single forward pass, with weighted-interval score and coverage on par with specialized surrogates for each structure. For inverse inference, HyperODE produces calibration from noisy state trajectories in a few milliseconds with a single shared encoder, competitive with existing methods. HyperODE extends zero-shot to ODEs that break mass conservation and to external forcing.

Bayesian Optimization (BO) is widely adopted for data-efficient optimization in scientific and engineering applications, yet its computational cost is rarely evaluated alongside optimization performance. Here we present a systematic, compute-aware study of BO that evaluates surrogate models along two axes: optimization quality and computational frugality. Across eight benchmark functions and nine real-world datasets spanning materials science, mechanics, robotics, chemistry, and machine learning, we benchmark four surrogate models: Gaussian Processes, Random Forests, NGBoost, and Bayesian Adaptive Spline Surfaces. We show that Gaussian Process-based BO consistently incurs the highest time and memory overhead without delivering superior optimization or sample efficiency. In contrast, scalable alternatives achieve equal or better performance at a fraction of the computational cost. Motivated by these findings, we introduce a surrogate-recommendation framework that predicts the most suitable BO surrogate from inexpensive dataset characteristics. Together, these results establish FruBO as a reproducible, compute-aware baseline for Bayesian Optimization and provide practical guidance for surrogate selection under limited computational and experimental budgets.

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.

Phase-field modeling provides a powerful approach for predicting microstructure evolution but becomes computationally prohibitive for multicomponent and multiphase systems over large spatial and temporal scales. This work presents an AE-GCN-LSTM surrogate framework for long-horizon forecasting of microstructure evolution in the multicomponent AlCrFeNi high-entropy alloy system containing coexisting BCC and FCC phases. A multi-head autoencoder compresses the four elemental concentration fields and phase-field order parameter into latent representations, which are formulated as graphs for learning their spatial and temporal evolution. The framework accurately forecasts microstructure evolution over horizons extending to 3,000,000 simulation timesteps. Its robustness is systematically evaluated under previously unseen conditions without retraining, fine-tuning, or parameter adaptation. These evaluations include variations in FCC precipitate size and initial position, microstructures containing one, two, and five FCC precipitates, and complex phase interactions involving precipitate merging and splitting. Although trained only on 100 x 100 computational domains containing a single nominal alloy composition, the framework is successfully transferred to larger 256 x 256 and 512 x 512 systems and to previously unseen AlCrFeNi compositions. Across the evaluated configurations, the model preserves the dominant phase morphology and compositional evolution while providing computational speedups ranging from approximately 7200 to 62300 relative to conventional phase-field simulations. These results demonstrate that latent graph-based AE-GCN-LSTM forecasting provides a scalable and computationally efficient surrogate for long-horizon simulation of multicomponent, multiphase microstructures and offers a promising foundation for high-throughput alloy design.

We present a novel approach to regression tasks using classification which is motivated by the mechanism used by fruitflies to sense their environment. Specifically, we formulate a general framework for learning nonlinear input-output relationships by replacing complex global surrogate models with a finite library of representative local patterns. Since scientific data often occupy limited and recurring regions of the input space, we generate predictions by measuring similarities between a query and stored patterns, then combining their associated responses through weighted reconstruction. We apply this approach to nonlinear dynamical systems, data-driven regression, and physics-informed learning using suitable embeddings and similarity measures. For dynamical systems, our offline-online workflow extracts patterns from data or governing equations during the offline phase, while online prediction requires only similarity evaluation and response aggregation. This structure helps us reduce computational and memory demands while providing explicit control over the trade-off among accuracy, storage, and inference cost.

Ensembles are a standard way to improve the performance and robustness of deep neural networks, but their effectiveness crucially depends on both the quality and the diversity of individual models. Most neural architecture search (NAS) methods are computationally expensive. Extending them to neural ensemble search (NES), which requires joint optimization of individual architectures and their ensemble composition, leads to an exponential growth of the search space and makes the problem computationally intractable. To address this, we introduce a dual-objective surrogate-guided ensemble search: candidate architectures are represented as directed acyclic graphs, and two surrogate models are trained independently to estimate predictive accuracy and diversity potential. Their combined estimates guide an NES framework that efficiently identifies architectures that are both individually strong and collectively diverse. Our final ensemble achieves competitive or superior performance compared to standard baselines such as Deep Ensembles and Random Search on FashionMNIST, CIFAR-10, and CIFAR-100.

Transformer-based surrogate models are increasingly used to replace expensive first-principles simulation in engineering design. But conventional transformer architectures are often over parameterized for the small, low-dimensional datasets typical of engineering design spaces, where large simulation data is expensive to generate. Under these conditions, excess parameter capacity leads to overfitting rather than improved accuracy, while also incurring unnecessary memory and compute overhead. This motivates a shift towards architectures that focus on additional compute rather than additional learnable parameters. This paper presents a hardware-aware evaluation of three recursive transformer paradigms for surrogate thermo-mechanical analysis of advanced packages: a)Tiny Recursive Model, b) our proposed Depth Recursive transformer, c) and a simple recursive transformer. We systematically compare their predictive performance (Recall, Mean Reciprocal Rank), parameter count, computational complexity (FLOPs), providing practical design guidelines for selecting recursive transformer architectures under resource-constrained scenarios. We validate this principle on two low-dimensional engineering prediction tasks: 1) thermo-mechanical reliability analysis of advanced semiconductor packages, where stress and warpage from thermal cycling must be evaluated repeatedly across a design-of-experiments sweep under costly finite element analysis (FEA). 2) Laplace PDE iterative numerical solver for capacitance field. Overall, recursive weight-sharing transformers provide an effective and generalizable trade-off between prediction accuracy, parameter efficiency, and computational cost for small data engineering surrogate modeling.

In this work, we present a simulation-based parameter estimation framework for a model defined by a computational simulation of a physical system. We specifically outline an estimation framework consisting of two closely-integrated steps that facilitate an overall end-to-end parameter estimation scheme. The first step involves utilizing an embedded normalizing flow which is used to transform the unknown complex distribution of the residual information into a simple base distribution corresponding to the transformed residual information. In the second step, an empirical-likelihood estimator, under moment restrictions, is utilized for imposing an indirect constrain on the base distribution, where such an instantiated task reasonably allows us to treat the transformed residual information as random variables arising from discretely distribution population with each transformed data point as a single-cell from a set of finite-cell contingencies. Moreover, we use first-order gradient methods for updating the estimated parameter values of the model defined by the computational simulation and the corresponding parametrized embedded normalizing flow, that call for all gradient-related information by leveraging implicitly differentiations of the empirical-likelihood function, which is constructed from the implied empirical probabilities under moment restrictions. Here, it is worth mentioning that the problem formulation presented in this work, which highlights an information-theoretic interpretation, allows to present a computational framework for algorithmic implementations. Finally, as a-by-product, the inverse of the parametrized embedded normalizing flow, w.r.t. the estimated parameter values, serves as a surrogate model for the computational simulation model, which provides useful information for quantifying model discrepancies and sensitivity analysis.

The use of variable well perforation and injection strategies can improve the efficiency of geological carbon storage operations. We develop a new multimodal auto-regressive transformer surrogate to model these operations under geological uncertainty. A modified SEAM CO2 geomodel, which involves a faulted system with three stacked aquifers, is considered. The two injection wells are perforated in stages, from bottom to top, with the stage durations and individual well injection rates treated as control variables. The surrogate model processes three input modalities - the 3D geomodel, scalar parameters characterizing relative permeability functions, and control variables - through separate encoders. These are fused via self-attention in a transformer encoder, and a temporal decoder generates predictions auto-regressively through encoder-decoder cross-attention. The surrogate is trained, using 4000 GEOS flow simulations, to predict saturation and pressure at monitoring locations, total injected and mobile CO2 mass, and saturation footprints. For a new test set, involving randomly sampled geomodels and control variables, the surrogate achieves a median saturation MAE of 0.028 and median relative errors of 0.2-5% for the other quantities of interest. Importantly, it captures the switch from rate to bottom-hole-pressure control. The surrogate model is used within a hierarchical Markov chain Monte Carlo data assimilation procedure for a synthetic true model under three operational strategies. Substantial uncertainty reduction is achieved for key metaparameters, particularly the fault permeabilities. Posterior predictions for saturation footprints and total injected and mobile CO2 mass are also shown to be generally consistent with true model results.

Gene regulatory network modeling often requires balancing predictive accuracy and mechanistic interpretability. In this work, we compare continuous surrogate models and a discrete mechanistic model on the same \textit{Arabidopsis thaliana} induced systemic resistance (ISR) dataset, using both the raw continuous gene-expression measurements and their sign-binarized representation. The study considers eight defense-related genes measured over nine time points and evaluates two continuous predictors, Random Forest (RF) regression and a Multi-Layer Perceptron (MLP), against a threshold Boolean network (TBN). The models are assessed using rolling-origin one-step prediction, recursive multi-step rollout, and interpretability analysis. RF achieved the best average one-step numerical performance in the continuous domain, with an MAE of 1.910 and an RMSE of 2.836, compared with 2.089 and 3.106 for the MLP. In the binary domain, the TBN obtained the best average one-step qualitative performance, with a binary accuracy of 0.550 and a Hamming distance of 3.600, compared with 0.500 and 4.000 for RF, and 0.495 and 4.040 for the MLP. In recursive rollout, the TBN exactly reproduced the observed binarized trajectory, while the MLP also showed near-perfect fidelity, with a trajectory binary accuracy of 0.986, and RF accumulated substantially larger deviation, with a trajectory binary accuracy of 0.708. These results highlight that local numerical accuracy and global qualitative dynamical fidelity are not necessarily aligned, and suggest that continuous surrogates and threshold Boolean networks should be viewed as complementary tools for modeling biological regulation.

Bayesian optimization (BO) is an optimization method that sequentially proposes the next candidate explainable variables for optimizing target variables by balancing exploration and exploitation. BO is often used under a limited evaluation budget, such as hyperparameter tuning of deep learning. Despite its effectiveness, conventional BO may have poor convergence in practical experimental science where each evaluation is often costly and time-consuming. Recently, BO methods have been proposed that accelerate optimization by using pseudo-experimental data that simulate experimental data. However, when only a limited number of experimental data are available, the generated pseudo-experimental data may be of insufficient quality. In this study, we developed PolyBO to improve optimization time by generating high-quality pseudo-experimental data even when the number of trials is limited. PolyBO performs BO efficiently by generating pseudo-experimental data with an adaptively updated versatile parametric model. This low-capacity polynomial regression model is intended to enable efficient BO even with limited experimental data. PolyBO updates the BO surrogate model with a combined dataset consisting of experimental data and pseudo-experimental data and then performs optimization. Using synthetic benchmark functions with diverse landscapes, we found that PolyBO reduced the optimization time by a median of 42%. For a real-world material composition optimization problem, PolyBO reduced the optimization time by a median of 96% compared with conventional methods. Overall, PolyBO achieves efficient optimization in settings where each experiment requires a long time.

Digital Twins rely on surrogate models to mirror physical systems in real time, yet these models can degrade as operating conditions evolve, a phenomenon known as concept drift. Maintaining surrogate fidelity under drift, particularly when models must also capture aleatoric uncertainty, remains an open challenge. Existing adaptive frameworks lack principled mechanisms for detecting when updates are needed, for efficiently adapting models from limited streaming data, and for certifying that updates genuinely improve predictive performance. Here we present an adaptive Digital Twin framework that integrates a Fisher score--based multivariate drift detector, Low-Rank Adaptation (LoRA) for parameter-efficient continual learning, and a Mann--Whitney $U$ test for online statistical validation. The framework monitors surrogate-model confidence via Fisher score vectors, triggers targeted fine-tuning of fewer than 1% of model parameters upon drift detection, and statistically certifies predictive improvement before deploying the updated surrogate. Applied to a stochastic linear system and a directed energy deposition additive manufacturing process as case studies, the framework successfully detects distributional shifts with short delays and restores both predictive accuracy and uncertainty quantification under abrupt and incremental drift. These results establish a statistically rigorous and computationally tractable pathway for sustaining the trustworthiness of neural-network--based Digital Twins throughout their operational life cycle.

Nuclear fusion has made significant progress in recent years and is expected to become one of the most important pathways to addressing global energy challenges. This paper focuses on observing plasma using visible-light cameras, analyzing its spatio-temporal motion cues, and predicting the two-dimensional spatial distribution of light intensity, aiming to provide a foundational basis for future scientific experiments using deep neural networks. Specifically, we propose Delta-InvFormer, a novel backbone network centered on a differential Transformer. The key insight is that by taking consecutive video frames as input, we can better capture the dynamics of the plasma. Moreover, spatial and temporal differential self-attention effectively mitigates interference from noisy signals, ensuring high-quality feature extraction. These features are then fused into a compact and informative representation, which is fed into a decoder network to predict the distribution. Based on real experimental data collected from the Experimental Advanced Superconducting Tokamak (EAST) large-scale scientific facility, our results demonstrate that the proposed model not only significantly accelerates traditional methods for distribution prediction but also achieves competitive reconstruction accuracy. The source code of this paper will be released on https://github.com/Event-AHU/OpenFusion

Many engineering problems involve phenomena whose governing equations are poorly characterized or only partially known. Surrogate modeling techniques such as neural networks can capture the behavior of these systems, but they typically demand large training datasets that are difficult to obtain in engineering contexts and yield models with limited physical interpretability. The Sparse Identification of Nonlinear Dynamics (SINDy) method addresses both limitations by performing sparse regression over libraries of candidate nonlinear terms, recovering interpretable governing equations from comparatively small datasets. Although SINDy has been demonstrated extensively on canonical benchmark systems, its application to practical engineering problems is less widely documented. This tutorial introduces the SINDy method and progressively builds toward its main extensions, from noise-robust weak-form and ensembling-based variants to constrained and parametrizable formulations. The paper and the accompanying tutorial (available at https://github.com/paullililili/SINDy4Engineers) is organized in three parts: the first introduces the standard SINDy algorithm and progressively extends it, inviting readers without prior knowledge to follow each step and adapt the methods to their own problems; the remaining two parts present detailed case studies on (1) the system identification of an unmanned aerial vehicle and (2) a chaotic thermosyphon heat exchanger. Through these examples, we aim to demonstrate that SINDy is simple to implement yet flexible enough to serve as a valuable identification tool for advanced engineering applications.

Incorporating hysteresis and eddy currents into finite element simulations of laminated-core electrical machines is computationally challenging. Resolving the fields inside the laminations at each integration point and at every nonlinear iteration leads to computational costs several orders of magnitude higher than anhysteretic simulations, making such approaches impractical for design applications. Conversely, simplified models accounting only for magnetic saturation are becoming increasingly inadequate as electrical machine topologies and operating conditions grow in complexity. In this context, machine learning surrogate modeling has emerged as a promising alternative, offering efficient and accurate approximations of complex electromagnetic behaviors. In this paper, a recurrent neural network is trained as a surrogate of a laminated-core material model for an isotropic laminated core, and is integrated into realistic two-dimensional magnetodynamic finite element simulations based on a magnetic vector potential formulation. The proposed approach achieves excellent agreement with the reference laminated-core model while limiting the computational cost to about twice that of an anhysteretic simulation. By training the recurrent neural network on a sufficiently diverse set of artificially generated magnetic field sequences designed to mimic those encountered in electrical machine simulations, the proposed approach can be readily applied across a wide range of finite element simulations. Furthermore, the trained surrogate model is provided as a standalone component that can be easily incorporated into existing computational frameworks. It is publicly available at https://gitlab.onelab.info/getdp/lamnet.

Emerging sustainable materials increasingly rely on engineered hierarchy and microstructure to achieve control of their properties and mechanical behavior. Optimizing these materials with controllable microstructures requires efficient multiscale simulations. Data-driven surrogate models for the microscale can accelerate multiscale simulations, but require large amounts of data even for a fixed microstructure. When a range of microstructures is considered, as is the case in multiscale optimization, even more data is needed to train a surrogate. To overcome this challenge, we condition a hybrid physics-data surrogate on microstructural variables using a hypernetwork. This approach enables accurate predictions of multiscale mechanical behavior for a mycelium-woodchip composite material, even when trained on small datasets. The conditioned surrogate makes multiscale simulations of functionally graded structures tractable, and we validate it against a full FE^2 simulation. We optimize a graded multiscale disk, and reduce the peak stress by 42% compared to one with a random microstructure. Then, we go one step further, conditioning the network directly on manufacturing variables that can have a complex influence on the microstructure. This is a practical route to engineer the microscale for desired macroscale behavior. This contribution highlights the benefits of microarchitectured structures and demonstrates how conditioned surrogate models enable their multiscale optimization, which will accelerate the development and design of future sustainable materials and structures.

Multiscale problems are notoriously difficult to tackle using traditional numerical methods, as accurately resolving fine-scale features often requires prohibitively fine discretizations. This challenge is particularly pronounced in applications such as materials science, fluid dynamics, climate systems, chemical processes, and complex networks. Recent neural operator models provide a promising data-driven alternative, but frequently struggle to achieve sufficient accuracy in the presence of strongly heterogeneous or oscillatory coefficients. In this work, we focus on the solution of elliptic PDEs with rough and high-contrast inputs. The Localized Orthogonal Decomposition (LOD) method is a well-established numerical approach for such problems, but it comes, however, at a substantial computational cost. We investigate the performance of popular neural operator architectures on these challenging multiscale problems and identify key limitations in their ability to resolve fine-scale structure. To overcome these challenges, we introduce LOD-MSNO (LOD-Multiscale Neural Operator), a hybrid approach that leverages the LOD method as a strong multiscale prior by building on its representation of the solution as a linear combination of problem-adapted basis functions, while addressing its main computational bottlenecks through data-driven operator learning. We further provide theoretical error estimates for the proposed coefficient-learning framework. Lastly, we demonstrate the potential of our proposed method to outperform current neural operator baselines in terms of accuracy for challenging multiscale inputs, while mainly retaining the computational efficiency of neural operator models.

Long-horizon physics-based simulations of battery degradation provide mechanistic insight but remain computationally expensive, limiting their use for dense exploration of operating conditions over extended cycle life. Here, we propose a hybrid physics-probabilistic learning framework for surrogate modeling of lithium-ion battery degradation trajectories at unseen charging rates. Cycle-resolved degradation data generated with a DFN/P2D electrochemical model in PyBaMM are first transformed into capacity-aligned voltage and derivative features and encoded using a Variational Autoencoder (VAE). The resulting two-dimensional latent space organizes degradation trajectories according to both cycle progression and charging protocol. A sparse multitask Gaussian process (GP) is then trained in this latent space using cycle number and C-rate as input variables, providing continuous interpolation of latent degradation dynamics together with posterior uncertainty estimates. Under protocol-level holdout evaluation, the latent-space GP accurately recovers unseen C-rate trajectories and exhibits uncertainty behavior consistent with the support of the training data. When queried at unseen interior C-rates, the model generates latent trajectories that remain coherently positioned between neighboring simulated protocols. Decoding the GP-predicted latent states through the frozen VAE decoder yields smooth voltage-capacity evolution, while Monte Carlo propagation of the GP latent posterior through an auxiliary latent to State of Health (SOH) predictor provides uncertainty-aware SOH estimates. The proposed BattVAE-GP framework therefore offers a computationally efficient and uncertainty-aware surrogate for long-horizon degradation modeling, providing a structured basis for extending battery health prediction toward richer operating conditions and future simulation-experiment fusion.

This work aims to develop a fast and physically consistent surrogate model for real-time structural health monitoring of fractured elastic domains. We propose a physics-informed DeepONet framework that predicts displacement fields from both boundary conditions and fracture geometry, using a dedicated encoding strategy for the latter and without relying on finite-element-generated training data. The traction-free condition on the fracture boundary is imposed weakly through a localized penalty term. The presented numerical example focuses on one representative fracture geometry, demonstrating the feasibility of the formulation and laying the groundwork for extensions to surrogate modeling across diverse fracture geometries.

In agentic scientific machine learning (SciML), large language model (LLM) agents can discover surrogate models and select one by an automated score, typically an error metric. A low error, however, does not establish that the predicted fields satisfy the physics that matter for mechanics, such as boundary conditions, superposition, stiffness scaling, or causality. We introduce Physics-Audited Agentic SciML (PA-SciML), a verification-first workflow for agentic SciML discovery. The workflow fixes a scoring evaluator before search, derives reviewable machine-checkable physics requirements, checks each trained candidate on its outputs, and separately searches prescribed input ranges or measured load-history spans for high-violation cases without reference solution fields. A surrogate is reported as verified only under the stated checks. When enabled, the workflow also adds advisory numerical probes before training and tests one modeling change at a time to record which isolated edits are associated with score gains before reuse. In the reported computational-solid-mechanics numerical examples, the static elasticity run selects a surrogate with lower validation error than the error-only baseline while both selected models pass the common linear-elastic checks. In the transient elastodynamics run, an error-only baseline with similar mean error fails a stricter causality check by responding to future parts of the loading history, while the selected surrogate passes the stated checks. The main distinction is per-candidate physics evidence on predicted fields, not a richer aggregate score.

Foundation Models for Electronic Health Records (FEMRs) are pretrained on large-scale structured patient data, enabling them to convert longitudinal patient trajectories into generalizable representations for diverse clinical prediction tasks. Despite their effectiveness, FEMRs remain black-box models, raising concerns about bias, interpretability, and clinical trust. To address this, we propose the first token-level explainability approach for FEMRs. We train a Transformer-based surrogate model on input-output pairs from the FEMR across two prediction tasks, approximating its behavior while preserving temporal dynamics. We identify the most influential tokens, providing insights into how FEMRs leverage different aspects of patient history for predictions. To evaluate clinical relevance, we introduce a novel clinical alignment metric that quantifies the correspondence between the surrogate model's key tokens and clinically validated features. Our results demonstrate that the surrogate closely approximates FEMR predictions and that token-level explanations align well with clinical knowledge, offering a practical framework for interpretable and trustworthy clinical AI.

Grain growth is governed by the reduction in grain boundary energy and exhibits well-established statistical scaling laws. Developing data-driven surrogates that preserve these physical invariants while remaining computationally scalable remains challenging, especially in 3D. We present 3D-PRIMME (Physics-Regulated Interpretable Machine Learning for Microstructure Evolution) for learning three-dimensional grain growth dynamics. The model is trained using only two consecutive time steps yet accurately reproduces the linear coarsening law and preserves topological statistics over extended time scales. Despite being trained on a $100^3$ grid points with 512 grains, the learned evolution operator is applied to domains up to $1024^3$ grid points with 550000 grains without retraining, maintaining consistent kinetics and grain topology across orders-of-magnitude increases in system size. These results demonstrate that 3D-PRIMME learns a scale-independent and temporally stable local evolution rule, enabling efficient and robust large-scale surrogate prediction of 3D microstructure evolution.

Integral codes like the Accident Source Term Evaluation Code (ASTEC) are powerful tools to study the physics of Severe Accidents (SAs) in nuclear reactors. Real time SA simulators can also be helpful in training operators of nuclear plants to react correctly to malfunctions. However, SA simulators can take up to several days per simulation, making their use infeasible for real time applications. In this work we show how to speed up a SA simulator with a fast, Deep Learning based (DL), surrogate model (SM). The SM is built as a combination of a dimensionality reduction stage, via an AutoEncoder, and a time-stepping stage, via a Neural Ordinary Differential Equation. The data on which the SM is trained are obtained from the ASTEC simulator, by sampling a set of operator actions for station blackout (SBO) and loss-of-coolant accidents (LOCA). The objective of the developed SM is to approximate multiple spatio-temporal fields for the thermal-hydraulic physics, core degradation, and fission product release modules in ASTEC's vessel domain. The SM predicts simultaneously around $80$ different physical variables (both scalar and fields), maintaining a stable autoregressive rollout up to $50$ thousand time steps. In addition, the AutoEncoder achieves a dimensionality reduction by a factor of over $300$, which allows the SM to predict up to $40$ hours of simulation in under a minute, both on CPU and GPU. This work is the first study of the capabilities and limits of DL based surrogate modeling in approximating the challenging, highly non-linear physics of ASTEC.

The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Recently proposed data-driven techniques, like extended dynamic mode decomposition (EDMD), its kernelized variant, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.

Many scientific systems exhibit uncertainty from stochastic forcing, unresolved degrees of freedom, or imperfect observations, making reliable surrogate forecasting fundamentally distributional rather than pointwise. For such systems, deterministic neural surrogates fail to capture statistical measures and forecast uncertainty. We introduce TRIE, an evaluation framework for stochastic PDE surrogates that asks whether models reproduce invariant measures, provide trustworthy predictive uncertainty, and scale to efficient probabilistic generation. We demonstrate TRIE on two stationary chaotic spatially extended SPDEs, stochastic Kuramoto--Sivashinsky and stochastic Kolmogorov flow, across 11 parameter values. Our evaluation shows that standard pointwise-trained neural surrogates can produce plausible short rollouts while failing to match long-time statistical structure. Approximate uncertainty methods such as Monte Carlo dropout and heteroscedastic Gaussian likelihoods produce stochastic forecasts, but are often miscalibrated and overconfident under temporal and spatial uncertainty diagnostics. Across these criteria, generative models provide the most consistent performance, accurately capturing invariant measure statistics and achieving the lowest CRPS in all reported probabilistic settings. Finally, we show that latent generative models with automatic dimension discovery retain much of this statistical fidelity while reducing Kolmogorov inference time by roughly $12\times$. We release our code and data at https://github.com/scailab/TRIE-SPDE-Bench to support reproducible evaluation of stochastic PDE forecasting models.

Short-fiber thermoplastic (SFT) composites are increasingly employed in lightweight aerospace and automotive structures owing to their favorable strength-to-weight ratio, high production rates, and recyclability. Unlike continuous-fiber systems, the mechanical response of SFTs is governed by mesoscale interactions among fiber orientation, spatial clustering, and manufacturing-induced porosity. These features exhibit significant spatial variability in manufactured components and influence stiffness, damage initiation, and nonlinear deformation. Although mesoscale finite element (FE) models can resolve such heterogeneity, their application to realistic three-dimensional microstructures remains computationally intractable. A data-driven surrogate framework is proposed to predict the mechanical behavior of additively manufactured, compression-molded (AM-CM) SFTs. Microstructures reconstructed from micro-computed tomography data were discretized into Voronoi-based cells representing distinct fiber-interaction neighborhoods. Each cell was homogenized via nonlinear FE simulations incorporating matrix damage, and the resulting stress-strain responses trained a hybrid Graph Neural Network-Long Short-Term Memory (GNN-LSTM) architecture encoding microstructural topology and history-dependent mechanical evolution. The surrogate accurately predicts stiffness and stress-strain behavior of unseen microstructures, achieving $R^2\approx 0.98$ relative to high-fidelity FE simulations with over two orders-of-magnitude reduction in computational cost. Coupling the framework with experimentally calibrated damage laws demonstrates that fiber orientation, clustering, and porosity collectively govern local effective stiffness. The approach provides a physics-informed, data-efficient pathway to identify mechanically weak microstructural cells and accelerate digital-twin development for SFT components.

We introduce a physics-constrained neural network for the rapid prediction of rigorous coupled-wave analysis outputs in the form of Jones matrices. Starting from energy conservation in lossless layered periodic structures, we use the fact that the scattering outputs lie on a Stiefel manifold. This energy constraint is enforced as a hard condition by projecting onto the manifold using differentiable symmetric orthogonalization. The resulting surrogate enforces energy conservation by construction while preserving differentiability for gradient-based inverse design. The performance and generality of the proposed approach are demonstrated through the inverse design of a diffractive waveguide combiner for augmented reality glasses.

Physics-informed surrogate models can accelerate computational fluid dynamics simulations. However, many existing methods reproduce global flow patterns more reliably than localized multiscale structures. This study presents a physics-informed Fourier-wavelet transformer for next-step velocity-field reconstruction in real-world flow benchmarks. The proposed formulation combines hybrid Fourier-wavelet spectral encoding with physics-biased self-attention based on partial differential equation residual diagnostics. It also uses self-supervised pretraining through Masked Physics Prediction and Equation Consistency Prediction. The experiments are conducted on two real benchmark cases: cylinder-wake flow and fluid-structure interaction. All approaches are evaluated under a shared local protocol and compared with spectral, transformer-based, operator-learning, and physics-informed neural-network baselines. On the cylinder-wake benchmark, the proposed model achieves the best aggregate accuracy, with an all-channel normalized mean-squared error of 0.05875 and an all-channel Pearson correlation coefficient of 0.97019. On the fluid-structure-interaction benchmark, it gives the lowest all-channel normalized mean-squared error of $2.70 \times 10^{-4}$, compared with $4.02 \times 10^{-4}$ for the strongest baseline. Component-wise field comparisons and scale-separated diagnostics further show stronger recovery of localized wake structures, including near-body, wake-core, and far-wake features. The results demonstrate improved real-world flow reconstruction while maintaining a practical accuracy-cost tradeoff.

Quantum machine learning methods are increasingly explored for modeling complex environmental systems, including groundwater heat plume dynamics. In this work, we explore a Quantum Convolutional Neural Network (QCNN) as a surrogate model for predicting temperature variations in groundwater induced by geothermal heat pumps in the city of Munich. To comply with the scalability constraints of current quantum hardware, the original high-dimensional simulation output is reduced to a compact set of representative parameters that serve as training targets for the surrogate. The proposed QCNN architecture consists of a quantum convolutional layer, a quantum pooling layer, and a fully connected quantum readout stage. Convolution and pooling operations are realized via parameterized quantum circuits based on rotational gates and measurement-driven decoding, while a Hamiltonian-inspired feature-encoding scheme is used to prepare informative input states on the quantum device. We evaluate the QCNN across multiple execution backends, including an ideal statevector simulator, a noisy simulator, IBM's 127-qubit Kyiv quantum processor, and the same hardware augmented with advanced error-mitigation techniques. Realistic noise models are employed to approximate device behavior and to assess the impact of mitigation strategies. Model performance is benchmarked using mean squared error (MSE) on both training and testing sets. The results show that, although classical neural networks still achieve the highest predictive accuracy, the QCNN attains competitive and consistent performance on simulators and exhibits noticeable improvement under error-mitigated hardware conditions. These findings indicate that quantum-enhanced surrogate modeling is a promising direction for future groundwater temperature prediction as quantum hardware and error-mitigation techniques continue to mature.

Black-box optimization (BBO) deals with problems where objective functions lack explicit analytical forms and are expensive to evaluate. Factorization machine with quadratic-optimization annealing (FMQA) constructs a surrogate model using a factorization machine (FM) and optimizes it with an Ising machine. Conventional FMQA applies a single integer-binary encoding throughout the optimization process, although the encoding best suited to surrogate learning may differ from the one best suited to Ising-machine solution search. We propose a stage-dependent FMQA framework and derive conversion formulas between one-hot and domain-wall QUBO matrices that preserve the surrogate objective over feasible integer states up to an additive constant. We evaluate the OhDw variant, which employs one-hot encoding for learning and domain-wall encoding for search, on the Rastrigin function with input dimensions N = 2 and 5 and discretization levels q = 61 and 301. Across all conditions, the dominant factor governing optimization performance is the encoding used in the learning stage, with one-hot encoding consistently yielding lower residual errors than domain-wall or binary encoding. The additional benefit of switching to domain-wall encoding for solution search is condition-dependent. For N = 5 and q = 301, OhDw achieves a lower residual error and solutions closer to the global optimum than one-hot-only FMQA, whereas for N = 5 and q = 61 the latter achieves a lower residual error. These results indicate that one-hot encoding in the learning stage is the primary performance driver and that stage-dependent encoding can provide further improvement under finer discretization.

High-resolution turbulence modeling is essential for scientific computing, but remains constrained by the cost of direct numerical simulation and the scarcity of full-resolution data. Existing scientific compressors reduce storage but typically operate on per-frame representations, whereas learned compressors yield compact latents that are often resolution-dependent and weakly aligned with the physics of turbulence. This raises the need for a compression framework that reduces data size, preserves physical diagnostics, and transfers from low-resolution training fields to high-resolution test fields without retraining. In this paper, we propose Physics-Preserving Latent Compression (PPLC), a patch-local latent compressor for three-dimensional turbulence. Motivated by inertial-range scale similarity, PPLC treats fixed-size patches as transferable units and applies a shared variational autoencoder independently of the global grid size. It combines exact mean preservation, zero-mean fluctuation encoding, an invertible Haar wavelet front-end, shift-consistency regularization, and overlap-aware reconstruction. Instantiated on forced isotropic turbulence, PPLC is trained only on stride-downsampled 256^3 fields and transfers zero-shot to 1024^3 fields. Experiments show that PPLC improves the balance between reconstruction accuracy and physical fidelity over classical and learned baselines, keeping diagnostics such as dissipation, enstrophy, energy spectra, and incompressibility closer to the ground truth. Beyond turbulence compression, PPLC offers a general strategy for physics-preserving latent representations that support data-efficient scientific surrogate modeling.

Wound rotor synchronous motors have emerged as a strong alternative that eliminates dependence on REEs. However, WRSM design requires the simultaneous optimization of numerous geometric and electromagnetic parameters, and the high computational cost of conventional finite element analysis severely limits the rapid exploration of the large parameter space. Although there are machine-learning-based surrogate modeling studies in the literature, they generally compare only a limited number of models, exclude deep learning architectures, and do not provide a comprehensive benchmark specific to WRSM. In this study, the performance of a total of eight machine learning and deep learning models from four different algorithmic families was systematically compared for the prediction of WRSM torque and motor efficiency. On a dataset of 3351 samples generated using Latin Hypercube Sampling in the Motor-CAD simulation environment, each model was trained with 10 different random seed values and tuned via Optuna hyperparameter optimization. Different from the existing literature, this study jointly offers a broad model spectrum including recent deep learning architectures such as FT Transformer, a multi-seed reproducibility protocol, and a Pareto analysis of the computational cost-accuracy trade-off. The results revealed that neural-network-based models systematically outperform tree-based models. The FT-Transformer model achieved the highest single-model accuracy with R^2 = 0.9928, producing predictions in 0.33 milliseconds and thus obtaining several orders of magnitude speedup compared to FEA. Model performances were evaluated in a multidimensional manner using R^2, MAE, RMSE, and MAPE metrics.

We introduce BASIL, a user-friendly desktop application for process optimization. BASIL employs a Bayesian approach, incorporating special acquisition functions that can be used to solve both single and multi-objective optimization problems. It provides a graphical interface that enables users to input their experimental parameters, optimization objectives, and legacy data. This is then used to build surrogate models, which are coupled with acquisition functions to guide and optimize a process towards a desired objective. To facilitate model building, BASIL provides a variety of predefined surrogate model templates. BASIL can be used to optimize any arbitrary experiment or process with known, user-defined input variables, optimization objectives, and defined output.

This chapter reviews recent advances in Scientific Machine Learning (SciML) for modeling coupled fluid flow and transport phenomena governed by the incompressible Navier-Stokes and scalar transport equations. Such systems, found in applications like turbidity currents and thermal convection, feature strong nonlinear coupling and multiscale behavior that make high-fidelity simulations computationally expensive. To address this, the chapter surveys state-of-the-art SciML methods for building efficient surrogate models, including linear reduced-order techniques based on Singular Value Decomposition (such as Dynamic Mode Decomposition) and nonlinear neural network approaches like Physics-Informed Neural Networks (PINNs) and $β$-Variational Autoencoders ($β$-VAEs). It first covers the authors' work combining these models with High Performance Computing strategies, including Adaptive Mesh Refinement/Coarsening (AMR/C) and scientific floating-point data compression. It then presents two new contributions: surrogate modeling of turbidity currents via PINNs, and the extraction of disentangled nonlinear modes from thermal flows using $β$-VAEs. Governing equations and representative benchmarks, including lock-exchange flows and Rayleigh-Bénard convection, illustrate these methodologies. The chapter is intentionally long, covering both the mathematical and physical foundations of coupled fluid flow and the computational aspects of state-of-the-art modeling. Overall, it demonstrates how SciML enables fast, accurate approximations of complex coupled systems within the specific data regimes and modeling assumptions considered, while substantially reducing computational cost relative to full-order simulations. Broader capabilities such as real-time prediction and uncertainty quantification remain active research directions whose feasibility depends strongly on the problem at hand.

As deep learning for physical systems continues to grow in popularity, efforts to improve generalizability have primarily focused on designing architectures that embed physical constraints. However, for machine-learning surrogate climate models (emulators), we show that the low structural diversity in existing scenarios commonly used to generate training data places a ceiling on predictive skill. Here, we examine whether training datasets themselves can be optimized to improve generalization. We introduce a method to create datasets that produce emulators capable of generalizing to new, structurally different scenarios absent from the training data. We use a differentiable Simple Climate Model (SCM) to calculate the sensitivity of emulator loss to perturbations in the training data, iteratively updating the training data to maximize emulator skill. For an SCM, training on one scenario optimized in this fashion outperforms an emulator trained on six standard ScenarioMIP pathways. We achieve this higher predictive skill despite training on a smaller dataset, finding that our emulator successfully isolates distinct physical behaviors of different climate forcing agents (e.g., greenhouse gases vs. aerosols) without single-forcing runs. We then demonstrate that scenarios optimized using an SCM, when used to drive an intermediate-complexity climate model, produce a training dataset that yields a more skillful emulator than training on ScenarioMIP outputs. Our results suggest that, in the compute-constrained environment of running full-scale climate models, generating a small number of dynamically rich scenarios provides greater marginal value for emulation and characterizing system responses than expanding the suite of traditional emissions pathways.

This work presents an extension to Pareto Front Guided Sampling (PFGS), a Human-in-the-Loop (HitL) Bayesian Optimization (BO) framework in which Gaussian process (GP) surrogate-derived quantities are reformulated as objectives of a multi-objective optimization problem, and the resulting Pareto front is exposed to a domain expert for interactive candidate selection rather than returning a single automated recommendation. The framework is extended in two directions: constrained optimization is addressed by incorporating the posterior probability of satisfying output specification limits as an explicit Pareto objective, computed analytically from the GP posterior distribution; robust optimization is addressed by a Monte Carlo sampling strategy that estimates expected lower-confidence performance over a user-defined variability of input perturbations, capturing performance degradation under likely implementation deviations. The resulting multi-dimensional Pareto representation renders trade-offs between predicted performance, model uncertainty, probabilistic constraint satisfaction, and input robustness simultaneously visible through pairwise two-dimensional projections on an interactive dashboard, enabling selection criteria to be iteratively refined as the surrogate model improves and development objectives evolve. The framework is showcased on an eight-dimensional fed-batch Chinese Hamster Ovary (CHO) cell culture simulator demonstrating systematic identification of high-performing, feasibility-compliant, and perturbation-resilient operating conditions, and illustrating how expert-defined requirements provide a principled stopping criterion and support informed allocation of experimental resources.

Operator learning is an emerging field at the intersection of machine learning and scientific computing. By learning mappings between function spaces, neural operators provide data-driven surrogate models for families of partial differential equations (PDEs). Once trained, these models can evaluate solution operators efficiently, making them suitable for many-query applications such as real-time prediction and parameter sweeps. However, maintaining high approximation accuracy and stable long-term predictions remains challenging for complex forward and inverse problems. To address these challenges, we propose the Starter-Iterator Neural Operator (SINO), which incorporates the initialization and residual-correction structures of classical iterative solvers into neural operator learning. The frequency-domain Starter captures dominant global spectral features and provides an informed initial approximation, while the latent-space Iterator applies successive residual-based corrections to refine local and multiscale solution structures. Experiments on representative time-dependent PDEs, including the Navier-Stokes and acoustic wave equations, together with applications to image super-resolution and weather forecasting, show that SINO achieves competitive accuracy and stable performance across the benchmarks considered in this work.

AI-driven engineering workflows face particular challenges in crash safety design: unlike aerodynamics, crash events involve highly nonlinear contact dynamics, material nonlinearity, and discrete state transitions that are difficult to capture with data-driven surrogate models. To the best of our knowledge, we present the first foundation model--orchestrated workflow for crash safety design that enables surrogate-assisted exploration for pedestrian protection, reducing evaluation time from hours per CAE simulation to seconds. The workflow integrates four components: (1) a surrogate trained on CAE crash simulations to predict pedestrian leg injury metrics from design parameters, achieving an average $R^2=0.87$ and providing distribution-free conformal prediction intervals; (2) multiobjective evolutionary search (NSGA-II) to discover diverse feasible parameter sets under user-specified constraints; (3) a morphing-based geometry generator that maps parameters to topology-preserving 3D shapes; and (4) a natural-language interface in which an LLM orchestrates the workflow and a vision--language model supports semantic comparison of generated designs. In an automotive front-bumper case study, the workflow produces 35 distinct safety-compliant alternatives from a single exploration, a process that would require weeks with conventional CAE iteration. These results suggest that foundation models can serve as integration layers between ML surrogates and physics-based simulation, helping bring AI capabilities to safety-critical engineering domains.

Neural operators are widely used as surrogate solution maps for partial differential equations (PDEs), but full-size models can be costly to store, deploy, and evaluate in many-query scientific workflows. This work introduces Operator Boosting, a stagewise residual-learning framework for constructing compact neural-operator surrogates directly, rather than training a large model and compressing it afterward. Starting from the empirical mean predictor in normalized output coordinates, the method trains a sequence of tiny same-family neural operators on residual fields and incorporates each correction through validation-selected shrinkage. We instantiate the framework with Fourier neural operators (FNOs), DeepONets, and convolutional neural operators (CNOs), and compare boosted tiny stacks against full-size monolithic baselines across one-, two-, and three-dimensional PDE benchmarks from PDEBench, APEBench, and The Well. Across 30 dataset-architecture pairs, 21 show positive mean accuracy gains and 17 have positive confidence intervals, while all boosted stacks reduce trainable parameter count by approximately 72-95%. Best-model comparisons show empirical Pareto improvements on 7 of 10 completed PDE benchmarks, including two-dimensional Navier-Stokes, shallow-water dynamics, Darcy flow, one-dimensional transport and reaction systems, and three-dimensional compressible Navier-Stokes. These results show that Operator Boosting often improves the empirical accuracy-parameter Pareto frontier of neural PDE surrogates, while also exposing PDE- and architecture-dependent regimes where residual boosting fails to offset compression.

In many engineering applications, a single high-fidelity model produces multiple quantities of interest (QoIs) under the same input parameters, e.g. finite element models of complex physical systems. To alleviate the high computational cost of direct model evaluations, surrogate models are widely used to construct efficient approximations of model responses. Naturally, the accuracy of surrogates strongly depends on the quality of the experimental design (ED). However, a single ED may not provide an adequate representation for all outputs simultaneously, especially when different outputs exhibit varying sensitivities to the input variables. A straightforward solution is to perform separate sampling for each output, but this results in increased sampling complexity and computational cost. From a statistical perspective, such an approach also ignores potential correlations among all outputs and may compromise data consistency. To address this issue, an adaptive sequential sampling method for constructing polynomial chaos expansion surrogate models is generalized for vector valued QoIs. The method sequentially selects new samples from a candidate pool based on their local contribution to the output variance, while balancing distance-based exploration of the input space and exploitation of aggregated variance information across all outputs. Its performance is compared with non-sequential Latin Hypercube Sampling through several numerical examples from engineering problems. Numerical results demonstrate that the proposed strategy improves both surrogate accuracy and stability, and provides a more reliable estimation of second-order statistics.

This chapter discusses how a data-driven machine learning approach can reproduce key aspects of the physical behavior of multiphase flows in complex geological formations. We propose an end-to-end graph neural surrogate tailored to CO$_2$ plume migration forecasting in geological storage. The method is evaluated on the SPE11A benchmark, a well-known industry test case designed to assess CO$_2$ storage scenarios and characterized by sharp gas-water interfaces, strong advective transport, and rapid convective mixing with fingering development. The benchmark is reformulated as a graph in which nodes represent computational cells and edges encode transmissibility-based interactions enriched with geometric attributes. Directional transport arising from grid geometry, permeability contrasts, and geological heterogeneity is captured through an anisotropic message-passing mechanism, where interaction weights are computed via geometry-conditioned edge embeddings, biasing message aggregation toward physically relevant transport directions. Temporal evolution is modeled in latent space using an autoregressive residual formulation trained with multi-step supervision. The proposed model produces competitive forecasts of gas saturation and liquid-phase density, which are key indicators for CO$_2$ storage monitoring, with cumulative errors that remain moderate over extended forecasting horizons.

Evaluating neural operators for 3D turbulent flow requires validated datasets with physical benchmarks. We present a reproducible pipeline generating training data for 3D channel flows around generated geometries at Re=1,000-10,000. Our lattice Boltzmann solver with cumulant collision operators is rigorously verified against experimental measurements (Strouhal number, drag coefficients, turbulent fluctuations) with comprehensive grid convergence studies at resolution 1024x512x512. Building upon an established framework, this validated pipeline enables standardized surrogate model comparison. We outline planned systematic evaluation of Fourier Neural Operator and U-Net variants on forecasting, super-resolution, and error correction tasks, using physics-informed metrics to assess turbulent energy cascade representation. Future work will compare computational efficiency between numerical solvers and neural surrogates, exploring practical application. We seek community feedback on our validation approach, planned benchmark methodology, and evaluation priorities for neural operators in turbulent flows.

Digital twin modeling, including control and data assimilation under model uncertainty, often faces an open-ended fidelity problem: adding variables, data streams, and time scales can indefinitely increase model complexity, ultimately producing systems that are difficult to maintain, validate, interpret, and use for stress or safety testing. As an alternative, one can seek parsimonious stochastic surrogate models built only on the variables needed to describe the relevant quantities of interest. We introduce a framework for discovering such variables from observational data by identifying which candidate inputs influence the full conditional law of a target quantity, rather than only its conditional mean. This distinction is essential in stochastic, coarse-grained, or partially observed systems, where dependencies may appear through changes in variability, tail behavior, multimodality, or uncertainty rather than through deterministic functional relationships. The framework couples conditional generative modeling, which learns the conditional distribution of the target given candidate inputs, with Gaussian-process-based analysis of variance (through kernel mode decomposition), which enables iterative pruning of non-influential inputs and interpretable structure discovery. In control settings, the resulting surrogate can be interpreted as a learned Markov decision process: the method identifies not only a transition model, but also the state, action, and memory variables needed to make the learned dynamics effectively Markovian. Across examples involving stochastic dynamical systems, missing variables, PDE control, reinforcement learning, and economic data, the discovered structures yield interpretable stochastic surrogates whose downstream performance is comparable to models trained on the full variable set.

Modern particles physics experiments have demonstrated an increasing need for fast, high-fidelity detector simulation as detector components have improved and subsequent computational requirements approach the limits of available resources. Recently, deep generative models have emerged as a promising alternative to traditional Monte-Carlo methods, with recent works drawing inspiration from large language models (LLMs) and self-supervised next-token prediction methods. In this work, we present an application of a GPT-style autoregressive transformer as a fast surrogate model for the calorimeter inside the CLAS12 experiment at the Thomas Jefferson National Accelerator Facility. The model is conditioned on incident momentum and generates realistic detector hits autoregressively across all nine calorimeter layers as sequences of strip, ADC, and TDC tokens. We demonstrate that the model faithfully reproduces hit multiplicity, spatial distributions, energy deposits, and the energy-momentum response of the electromagnetic calorimeter. The generator achieves inference rates exceeding 700 events per second on a single GPU, providing a substantial speedup over traditional Geant4-based simulations while maintaining physics fidelity essential for high-luminosity experimental programs.

Multi-objective optimization (MOO) has emerged as a powerful approach to solving complex optimization problems involving multiple objectives. In many practical scenarios, function evaluations are unavailable or prohibitively expensive, necessitating optimization solely based on a fixed offline dataset. In this setting, known as offline MOO, the goal is to find out the Pareto set without access to the true objective functions. This setting suffers from the out-of-distribution (OOD) issue, where the surrogate model is not accurate for unseen designs. Due to the OOD issue, surrogate errors may cause the optimizer to select solutions that do not lie on the true Pareto front and are biased toward its extremes. To address this, this paper proposes Diversity-driven Offline Multi-Objective Optimization (DOMOO), which aims to find out a diverse and high-quality set of solutions. First, DOMOO incorporates an accumulative risk control module that estimates the potential risk of candidate solutions and alleviates the OOD issue between the training data and the generated solutions. In addition, a nested Pareto set learning (PSL) strategy is proposed to jointly learn preference and PSL parameters, then optimize them, enabling adaptation to diverse Pareto front geometries. To further enhance solution quality, we design a diversity-driven selection strategy that extracts a representative and well-distributed set of final solutions. To achieve this diversity-driven selection strategy, we propose $\text{IGD}_\text{offline}$, a tailored indicator for the offline setting that considers both diversity and convergence, and avoids the bias of hypervolume indicator. Extensive experiments on synthetic and real-world benchmarks show that DOMOO achieves the best average rank across tasks in both convergence and diversity among the compared methods.

Scientific machine learning (SciML) has emerged as a promising approach for accelerating simulations of complex physical systems, yet achieving physically consistent and generalizable predictions for nonlinear, history-dependent problems remains a central challenge. In this study, we propose a hybrid GNN--FEM framework for efficient and generalizable phase-field fracture modeling. While phase-field approaches provide a robust variational framework for simulating complex crack evolution, their high computational cost limits practical applications because they require solving coupled, nonlinear, and history-dependent systems within an incremental finite element procedure. To address this challenge, a graph neural network surrogate is integrated into the conventional staggered scheme, replacing the phase-field update at each load increment while retaining the FEM-based displacement solver to enforce mechanical equilibrium and boundary conditions. By preserving the incremental solution structure, the framework remains consistent with history-dependent fracture evolution without requiring the surrogate to approximate the full solution trajectory. This selective surrogate strategy emphasizes the identification of a physically meaningful and incrementally structured learning target, rather than relying on brute-force data generation to learn the full fracture process. The proposed framework achieves strong generalization across varying geometries, loading conditions, material properties, and discretizations through dimensionless feature design, a graph-based formulation on mesh-based domains, and a physics-informed loss derived from the governing phase-field equation. Numerical experiments demonstrate that the hybrid approach reduces computational cost while maintaining accuracy compared with conventional FEM, and exhibits robust predictive performance across diverse problem settings.