Surrogate Modeling

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21 papers in the last four weeks, up 200% on the four weeks before. 0.2% of all new papers.

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Latest papers 139

Jun 1, 2026physics.comp-ph

Machine Learning-Driven Chemical Reactor Network Modeling of the Sandia-D Flame

Turbulent combustion simulations are crucial for many scientific and engineering systems. However, the high cost to fully resolve the complex multiscale and multiphysics behavior makes direct simulation typically infeasible. The equivalent reactor network (ERN) approach attempts to improve computational efficiency by replacing a multidimensional turbulent simulation with a series of much cheaper 0-D and 1-D chemical reactors, providing a surrogate model that retains detailed chemistry at the cost of simplified flow physics. However, their development remains a challenge, often requiring either expert analysis, or automated approaches that sacrifice accuracy. In this work, we develop an automated machine-learning-assisted framework for constructing ERNs of the Sandia-D turbulent methane/air flame. Principal component analysis is first used to reduce high-dimensional thermochemical computational fluid dynamics (CFD) data to a low-dimensional latent space, where k-means clustering identifies physically interpretable flame regions used to initialize a reactor-network graph. This initialization is then refined using finite-difference gradient descent wrapped around non-differentiable Cantera reactor simulations. Across 30 RANS simulations spanning a range of pilot temperatures and inlet methane compositions, the optimized 7-reactor ERN achieves a maximum-temperature

R^2

score of 0.7945 while preserving a

\sim6000\times

speedup over the CFD solver. Outlet CO prediction remains more challenging, with a final

R^2

score of

-0.4183

, but improves substantially from the unoptimized clustering initialization. These results show that unsupervised thermochemical feature extraction can provide effective physics-informed initializations for ERN construction, while gradient-based refinement can significantly improve predictive accuracy without manual reactor-network design.

Nicolas J. Tricard, Benjamin C. Koenig, Sili Deng

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Surrogate Modeling Computational Fluid Dynamics

Jun 1, 2026stat.ML

Scalable Derivative Gaussian Processes via Exact Gradient Reduction

Gradient observations can substantially improve Gaussian process (GP) surrogates, particularly in high-dimensional settings where function evaluations are expensive. However, exact inference with

n

function values and

n

full gradients in

d

dimensions scales cubically in the joint state size, imposing an intractable

\mathcal{O}(n^3 d^3)

computational bottleneck. We introduce TERA, a highly scalable derivative GP method based on target-specific exact gradient reduction. We prove that for stationary kernels, the gradient components orthogonal to the directions connecting the target and conditioning points are conditionally independent of the target function value; consequently, the exact conditional density is fully characterized by at most

m^2

directional derivatives once a conditioning set of size

m

is specified. By using these reduced, dimension-free conditionals as local factors in a Vecchia approximation, TERA effectively decouples

n

and

d

from the dense matrix inversion. This reduces the per-target evaluation cost to

\mathcal{O}(dm^2 + m^6)

time and

\mathcal{O}(dm^2 + m^4)

memory, leaving the underlying derivative GP model mathematically unchanged. Empirical evaluations demonstrate that TERA achieves state-of-the-art predictive accuracy while operating orders of magnitude faster than standard derivative GPs. Crucially, both computation time and peak GPU memory remain essentially flat with respect to

d

, enabling highly scalable inference in high-dimensional spaces.

Hyunseok Seung, Matthias Katzfuss

Department of Statistics · University of Wisconsin–Madison · Madison, WI, 53706

Surrogate Modeling Gaussian Processes Gaussian Process Regression

Jun 1, 2026cs.LG

TN-SHAP-G: Graph-Structured Tensor Network Surrogates for Shapley Values and Interactions

Shapley values are a widely used tool for attributing importance and interactions among input variables in black-box models, but their computation involves a function defined over an exponentially large space of subsets. We propose TN-SHAP-G, a framework that exploits structure in graph-structured inputs to compute Shapley values and higher-order interaction indices efficiently. Given a predictor and a fixed masking scheme, TN-SHAP-G learns a compact, graph-aligned multilinear surrogate that approximates the masked-input behavior, represented as a tensor network whose topology mirrors the input graph. Once trained from a small number of oracle queries, the surrogate enables deterministic recovery of first- and higher-order Shapley indices via the multilinear extension, without additional model queries or Monte Carlo variance. Experiments on molecular benchmarks show that the learned factorization closely matches exact Shapley values on small graphs and scales efficiently to larger graphs where sampling-based methods become infeasible.

Farzaneh Heidari, Guillaume Rabusseau

Universit´e de Montr´eal, Montr´eal, Quebec, Canada · Mila, Quebec AI Institute, Montreal, Quebec, Canada · CIFAR AI Chair.

Tensor Networks Surrogate Modeling Feature Attribution Shapley Value Attribution

May 30, 2026cs.NE

Meta-Black-Box Optimization with Ensemble Surrogate Modeling for Robustness-Accuracy Trade-off within SAEA

Surrogate-assisted evolutionary algorithms (SAEAs) have been widely used for expensive black-box optimization problems. However, their reliance on rigid and manually designed components limits their flexibility and generalization across tasks. Meta-black-box optimization (MetaBBO) provides a promising paradigm for adaptively configuring algorithmic components. Nevertheless, existing MetaBBO methods usually control only a single component, and few studies have investigated the unified control of multi-component optimizers such as SAEAs. Moreover, the robustness-accuracy trade-off in surrogate modeling, which is crucial for stable early-stage exploration and accurate late-stage exploitation, has rarely been explicitly considered. To address these issues, we propose AdaE-SAEA, an adaptive ensemble surrogate-assisted evolutionary algorithm for expensive multi-objective optimization. AdaE-SAEA embeds SAEA as the low-level optimizer within the MetaBBO framework and jointly controls the infill criterion and ensemble-based surrogate modeling. Specifically, bagging and boosting are designed as surrogate modeling modules to adaptively balance robustness and accuracy across different search phases, while the meta-policy simultaneously selects the infill criterion to enable adaptive sampling decisions. The meta-policy is trained through reinforcement learning with parallel sampling and centralized training, improving both training efficiency and transferability. Experiments on synthetic and real-world problems demonstrate that AdaE-SAEA outperforms state-of-the-art baselines and MetaBBO-based methods. We further verify the effectiveness of TabPFN as the base surrogate model for ensemble learning. To the best of our knowledge, this is the first work to unify the control of surrogate modeling and infill criteria in SAEAs while explicitly addressing the robustness--accuracy trade-off.

Xiao Jin, Yongxiong Wang, Haobo Liu +2

School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, 200093, Shanghai, China · Xinxiang University, 453003, Xinxiang, China · College of System Engineering, National University of Defense Technology, 410073, Changsha, China

Surrogate Modeling Surrogate-Assisted Optimization Learning to Optimize Adaptive Sampling Multi-Objective Optimization

May 30, 2026cs.LG

Extending Causal Metamodeling to a non-Markovian Queue

Metamodels for discrete-event simulations approximate the behavior of simulation models without running expensive simulations. Prior work introduced modular dynamic Bayesian networks (MDBNs) -- a class of metamodels that can estimate a range of probabilistic and causal queries (PCQs) using a single, trained model -- but the method was limited to Markovian systems. In this paper, we initiate an extension of MDBNs to non-Markovian queues by approximating non-exponential distributions using phase-type distributions. This approach raises novel challenges, including balancing metamodeling accuracy and tractability when choosing the number of phases, efficiently learning metamodel parameters, and choosing the sampling interval that is used to approximate a continuous-time simulation by a discrete-time MDBN. We provide preliminary solutions to these challenges, yielding the first causal metamodeling technique for non-Markovian systems. Experiments on a G/M/1 queue demonstrate that the MDBN can produce accurate answers to PCQs with orders-of-magnitude speedup of inference times relative to direct simulation.

Pracheta Amaranath, Anant Bhide, David Jensen +1

Manning College of Information and Computer Sciences, University of Massachusetts Amherst, MA, USA

Surrogate Modeling Bayesian Networks Causal Inference

May 29, 2026physics.comp-ph

Physically Constrained Ensemble Gaussian Process Modelling for Expensive Quantum Systems with Heteroskedastic Noise

Accurate modeling of quantum many-body systems often requires computationally expensive simulations such as Density Matrix Renormalization Group (DMRG) or Quantum Monte Carlo (QMC) calculations. These methods, while precise, impose significant time and resource constraints, limiting their use in exhaustive parameter exploration. Moreover, these expensive simulations can contain variable errors over the large unknown parameter space, which needs to be quantified and propagated. Thus, predictive modelling is required to estimate the functional space accurately over scarcely sampled data with heteroskedastic noise, while preserving the physical relevance of the estimation. Therefore, we present a Physically Constrained Ensemble Gaussian Process (pc-EGP) framework designed to efficiently model complex and noisy quantum systems under physical consistency constraints. The proposed method first enforces physical constraints as a user controlled weighted penalty to the data-driven loss function of the Gaussian Process (GP) surrogates. Then an ensemble of such GP models is trained with variable noisy simulations via numerical quadrature method where these multiple GP(s) at different nodes is integrated as a quadrature weighted average. We first demonstrate the framework on synthetically generated data before applying to quantum systems. In the first case study, we leverage DMRG simulations of the Bose-Hubbard Model to predict the critical interaction parameter Uc governing the superfluid-to-Mott-insulator transition. In the second case study, we demonstrate our method on QMC simulations, of a quantum liquid confined inside a nanoporous silicate with the goal of optimizing a chemical environment to realize a one-dimensional superfluid. Compared to conventional GP, pc-EGP achieves a better balance of accuracy and physically meaningful predictions.

Arpan Biswas, Sutirtha Paul, Joseph Agada +2

University of Tennessee-Oak Ridge Innovation Institute, University of Tennessee, Knoxville, TN 37996, USA · Institute for Advanced Materials and Manufacturing, University of Tennessee, Knoxville, Tennessee 37996, USA · Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA. +2

Surrogate Modeling Uncertainty Quantification Heteroscedastic Regression Physics-Informed ML Quantum Many-Body Physics Gaussian Process Regression

May 29, 2026cs.LG

The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling

Dynamical systems reconstruction (DSR) aims to learn surrogate models that capture the dynamics underlying time-series data. Reliably deploying these surrogates requires uncertainty estimates consistent with the learned dynamics. We expose a dynamic-probabilistic consistency (DPC) gap: the pursuit of finite-horizon probabilistic objectives can degrade dynamics or decouple predictive uncertainty from the local tangent dynamics it ought to reflect. We isolate three mechanisms behind this gap: core collapse, noise masking, and blind uncertainty. Specifically, we show that open-loop Gaussian rollout objectives can penalize Jacobian-generated covariance growth in chaotic systems, encouraging optimization shortcuts that weaken physical expansion or decouple uncertainty from it. To mitigate this gap, we propose KAFFEE (Kalman-Aware Framework For Ergodic Emulation), a differentiable extended Kalman filter-based training framework that evaluates likelihood on local predictive residuals (innovations) while transporting covariance through learned local Jacobians. On stochastic hyperchaotic Lorenz-96, KAFFEE reduces the identified failure modes, improves reconstruction of dynamical invariants relative to open-loop objectives, and maintains competitive predictive scores. We further show that the DPC gap appears when probabilistically adapting a DSR foundation model across 13 chaotic systems, where KAFFEE enables in-context Bayesian filtering while largely preserving zero-shot dynamics.

Andre Herz, Matthijs Pals, Daniel Durstewitz +1

1Interdisciplinary Center for Scientific Computing, Heidelberg University, Germany · Faculty of Mathematics and Computer Science, Heidelberg University, Germany · 3Dept. of Theoretical Neuroscience, Central Institute of Mental Health (CIMH), Mannheim, Germany +3

Dynamical Systems Surrogate Modeling Uncertainty Quantification Chaotic Dynamical Systems Extended Kalman Filtering

May 27, 2026cs.LG

Hybrid Neural World Models

Neural surrogates promise large speedups over classical solvers for physical dynamics but fail silently at sharp dynamical events such as shocks, fronts, and contact. We present hybrid neural world models for physical dynamics: a recipe for training and deploying multi-horizon surrogates in physical state space, where a single network with continuous horizon conditioning is trained with direct supervision against textbook reference solvers to predict any future state at horizon T in one forward pass. Although no part of the training data, loss function, or architecture supervises discontinuity location, the trained surrogate encodes it implicitly, recoverable from its forward passes alone as a per-trajectory error map that concentrates on shocks, fronts, and contacts, and stays small elsewhere. The map is competitive with or better than standard label-free baselines including deep ensembles, learned error heads, gradient-magnitude indicators, and locally-adaptive conformal prediction, while using only a single trained network and requiring no calibration set or governing-equation knowledge. The recipe supports two operating points. Mode 1 runs the surrogate alone for maximum throughput, with same-hardware CPU speedups of 26x to 72x against textbook solvers on the PDE environments. Mode 2 uses the error map to gate a reference-solver fallback, deferring uncertain trajectories and roughly halving the surrogate's residual error at the default operating point. The recipe applies without modification across reaction-diffusion, compressible Euler, and rigid-body collision dynamics.

Pranav Lakshmanan, Paras Chopra

Lossfunk

Neural Surrogate Modeling Surrogate Modeling World Model Learning PDE Surrogate Modeling

May 26, 2026cs.LG

High-Fidelity Industrial Crash Dynamics Prediction via Geometry-Aware Operator Learning with Memory-Efficient Low-Rank Attention

Automotive crashworthiness optimization remains a safety-critical challenge, requiring the management of large-scale nonlinear structural deformations and energy dissipation through iterative, high-fidelity simulations. While traditional finite element solvers are computationally prohibitive, emerging operator learning frameworks provide rapid surrogate predictions; however, applying them to industrial-scale crash analysis, where complex geometry, contact nonlinearities, and rapidly evolving transient deformation coexist, remains an open challenge. In this paper, we demonstrate that the GeoTransolver framework provides a viable solution for accurate, high-fidelity crash dynamics prediction at industrial scale. Benchmarked on complex bumper beam and full-vehicle crash datasets, GeoTransolver captures multi-scale geometric context and accurately resolves plastic deformation patterns as well as acceleration profiles at critical occupant locations. Beyond the architecture itself, we propose and systematically evaluate a suite of temporal prediction recipes, including one-shot, time-conditional, and autoregressive rollout strategies, demonstrating that the one-shot approach achieves state-of-the-art accuracy with significantly reduced training overhead and inference latency. As a secondary contribution, we introduce a Fast Low-rank Attention Routing Engine (FLARE)-based modification to the GeoTransolver attention backbone that reduces memory overhead by approximately 2x while further improving predictive accuracy for O(N) long-range, high-frequency transients, preserving the geometry-aware cross-attention strengths of the base framework. Our results highlight the practical viability of geometry-aware operator learning for high-fidelity surrogate modeling of complex, safety-critical automotive dynamics.

Deepak Akhare, Mohammad Amin Nabian, Corey Adams +2

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN · NVIDIA, San Jose, CA · General Motors, Detroit, MI

Low-Rank Attention Surrogate Modeling Attention Mechanisms Operator Learning Neural Operators

May 26, 2026stat.ML

Accelerating Reinforcement Learning Training Using Simulation Surrogate Models

High-fidelity simulation models are widely used to analyze complex stochastic systems, but their high computational cost motivates the development of cheaper surrogate models that approximate the simulation model's input-output relationship. In parallel, reinforcement learning (RL) has emerged as a powerful framework for making online decisions in stochastic environments, with increasing attention being given to the use of simulation models as training environments for RL models. We investigate a class of surrogate models suitable for accelerating RL training in settings where the reward structure, model parameters, or system dynamics change over time and explore their interactions with simulation models and RL models. Through numerical experiments on a stochastic service system modeled via discrete-event simulation, we demonstrate that leveraging surrogate models can substantially accelerate RL training and re-training.

Mohammadmahdi Ghasemloo, David J. Eckman, Yaxian Li

Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX, USA · Intuit AI, Mountain View, CA, USA

Surrogate Modeling Reinforcement Learning Model-Based RL

May 25, 2026cs.AI

FLOATBench: A Dataset and Benchmark for Floating Offshore Wind Turbine Tower Fatigue

Most of the world's offshore wind resource lies in waters too deep for fixed-bottom foundations, making floating offshore wind turbines (FOWTs) essential for deep-water deployment. As the industry scales toward

22

MW class designs, tower fatigue becomes increasingly critical because larger structures amplify the coupled aero-hydro-servo-elastic loads induced by continuous wind and wave excitation. Accurate fatigue-damage prediction is therefore central to certification, design optimization, and cost reduction. Yet the field lacks a shared surrogate benchmark: studies report different simulations, splits, and metrics, making methods difficult to compare. We present FLOATBench, a public tabular benchmark with

582{,}120

per-section fatigue-damage labels across three

22

MW FOWT tower geometries, derived from

19{,}404

high-fidelity OpenFAST simulations across the three towers (

6{,}468

per tower:

1{,}078

aligned wind/wave operating points

\times

six turbulence seeds), labeled at

30

cross-sections per tower. FLOATBench includes a regime-aware alpha-shape partition of the joint wind/wave operating envelope, stratifying test points into in-train, interpolation, and extrapolation regimes. It is paired with a reproducible evaluation harness covering three protocol levels: random validation (E1), within-tower regime-aware evaluation (E2), and cross-tower transfer (E3). The regime-aware protocol reveals rank shifts between global and extrapolation performance that random-split leaderboards cannot detect. To the authors' knowledge, FLOATBench is the first FOWT fatigue benchmark for tabular surrogate modeling, and offers an evaluation protocol that generalizes to engineering surrogates defined over physical operating envelopes. Dataset and code available at: https://github.com/Joao97ribeiro/FLOATBench.

João Alves Ribeiro, Bruno Alves Ribeiro, Francisco Pimenta +2

Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA 02139 USA · CONSTRUCT, Faculty of Engineering University of Porto Porto, 4200-465 Portugal · Center for Mechanical Technology and Automation University of Aveiro Aveiro, 3810-193 Portugal +2

Surrogate Modeling Benchmark Construction

May 19, 2026cs.AI

Explainable Wastewater Digital Twins: Adaptive Context-Conditioned Structured Simulators with Self-Falsifying Decision Support

Operators of safety-critical industrial processes increasingly rely on digital twins to screen control interventions, but such simulators rarely carry certified safety guarantees. Wastewater treatment plants exemplify the gap: operators face a daily safety-efficiency trade-off where aerating too little risks effluent violations and nitrous-oxide (N2O) spikes, and aerating too much wastes energy. We develop an explainable digital twin for aeration and dosing setpoints. CCSS-IX, the simulator, is a bank of interpretable locally linear state-space "experts" adaptively mixed by a context-aware gating network, building on a continuous-time regime-switching scaffold. A runtime decision layer applies conformal risk control to abstain, reopen, or return a falsifying temporal witness for any operator-proposed action that cannot be statistically certified. The artificial-intelligence contribution is twofold: an identifiable, context-conditioned structured surrogate that retains operator-readable dynamics, and a self-falsifying decision rule with finite-sample coverage guarantees. The engineering contribution is a validated, end-to-end decision-support pipeline, tested on a 1000-step slice of the Avedøre full-scale plant (42.6% sensor missingness, 2-minute sampling), the Agtrup/BlueKolding full-scale plant in Denmark, and the Benchmark Simulation Model No. 2 (BSM2) international benchmark, under a matched ten-seed protocol. The static structured ensemble lies within 0.78% root-mean-square error of an unconstrained black-box reference, and the adaptive variant within 1.08%. The calibrated reopen rule cuts aggregate two-plant regret by 43.6% at an unsafe-action cost weight of 4 and eliminates unsafe chosen actions on the BSM2 main slice. Event-aligned temporal witnesses prevent 93 of 187 false-safe N2O approvals, about 4.65x the dyadic baseline (paired McNemar p < 1e-21).

Gary Simethy, Daniel Ortiz Arroyo, Petar Durdevic

Department of Energy, Aalborg University, Niels Bohrs Vej 8, Esbjerg, 6700, Denmark

Surrogate Modeling Safety-Critical Control Conformal Risk Control Digital Twins

May 19, 2026cs.LG

A Family of Divergence Measures for Evaluating the Reconstruction Quality of Explainable Ensemble Trees

Validating interpretable surrogate models for ensemble learners requires measuring agreement between the ensemble's internal representation and its surrogate approximation, rather than mere association. Correlation-based approaches are scale-invariant and fail to detect systematic discrepancies in co-occurrence structure. We propose a statistical framework grounded in the agreement-association distinction, centered on the normalized Loss of Interpretability (nLoI). Rooted in the Cressie-Read power divergence family with lambda equal to 2, the nLoI admits a closed-form decomposition into within-node and between-node components, providing a unique diagnostic capability to identify precisely where and why reconstruction fails. The framework incorporates four complementary measures capturing distinct structural facets of approximation quality. A unified permutation testing procedure delivers valid inference for all measures within a single resampling pass. Theoretical properties, including boundedness and symmetry, are established for each metric. Monte Carlo simulations and empirical evaluations confirm exact Type I error control and demonstrate that these measures detect reconstruction fidelity gradients invisible to correlation-based alternatives. The framework is developed and illustrated in the context of Explainable Ensemble Trees (E2Tree), and empirical evaluation on three benchmark datasets illustrates the practical utility of the framework.

Massimo Aria, Agostino Gnasso, Carmela Iorio

Department of Economics and Statistics, University of Naples2026 Federico II, Naples, Italy

Surrogate Modeling Ensemble Learning Explainability Evaluation

May 16, 2026cs.LG

Emulating the Forced Response of Climate Models with Generative Machine Learning

Global climate models are essential tools to simulate past and potential future pathways of climate change, as well as associated climate impacts. Shared Socioeconomic Pathways (SSPs) describe a range of future scenarios of global economic and demographic development. These SSPs are intrinsically linked to changes in climate forcings -- the external drivers, such as greenhouse gas and aerosol emissions, which change Earth's energy balance over time. These forcings act as boundary conditions in Earth System Models (ESM), providing insight into the potential climatic impacts of each SSP. Running an ESM, however, is extremely computationally expensive, conflicting with the need for large ensemble runs to provide robust estimates in the presence of internal variability and scenario uncertainty. Machine Learning emulators provide a promising avenue towards fast and cheap scenario generation, but until recently lacked uncertainty quantification and the ability to condition on external forcings. Here, we build upon recent work and extend it by training on multiple SSPs. We successfully generate scenarios of IPSL -- CM6A -- LR unseen during training and remain physically consistent with the underlying climate model, even under strong extrapolation scenarios. Our emulator is validated against MESMER -- M, a statistical emulator of land surface temperature. Our research demonstrates that our model, ArchesClimate -- SSP, does not simply imitate scenarios seen during training, but is actually capable of modeling the response of a climate state to diverse forcings. This is an important step towards reliable and rapid climate model scenario generation.

Graham Clyne, Julia Kaltenborn, Peer Nowack +2

1INRIA, Paris, France · 2MILA, Montreal, Canada · 3Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

Neural Surrogate Modeling Surrogate Modeling Climate Science Climate Modeling Conditional Generative Modeling

May 15, 2026cs.LG

Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutions

The source distribution in conditional flow matching is a design parameter that can be calibrated to data, not a default isotropic prior. We exploit this in Multi-Fidelity Flow Matching (MFFM), a cascade refinement framework for parametric PDE solutions: the source is calibrated to the empirical low-to-high-fidelity residual scale with local Gaussian-blur correlation, and the velocity network is conditioned on the low-fidelity solution. Conditioning makes the residual refinement problem substantially easier than unconditional field generation, while residual-calibrated source noise improves the flow-matching training geometry. A multi-resolution cascade applies the same construction independently between adjacent fidelities. After level-wise flow-matching pretraining, we fine-tune the composed cascade end-to-end with a deterministic one-step rollout, which makes one velocity evaluation per cascade level the optimized operating point at inference. The result is a learned analog of multigrid refinement that reaches the finest grid in

L

deterministic network evaluations per query. We validate MFFM on eight benchmarks: two super-resolution problems and six spatiotemporal forecasting tasks from PDEBench, The Well, and the FNO Navier--Stokes dataset.

Sipeng Chen, Junliang Liu, Hewei Tang +1

Department of Computer Science, Florida State University · Department of Materials Science and Engineering, Florida State University · Hildebrand Department of Petroleum and Geosystems Engineering, The University of Texas at Austin

Surrogate Modeling Conditional Flow Matching PDE Surrogate Modeling Multifidelity Modeling

May 15, 2026cs.LG

CTF4Nuclear: Common Task Framework for Nuclear Fission and Fusion Models

The demand for clean energy is ever increasing, with new nuclear technologies presenting a complementary solution to renewable energies. However, designing and operating these systems is exceptionally difficult, given the complexity of the physical phenomena that interact to form the system dynamics. While high-fidelity simulations help to understand the non-linear, multi-physics interactions within a reactor, they are computationally expensive and rarely suitable for real-time applications. Furthermore, model-based approaches are inherently sensitive to simplifying assumptions required to derive their governing equations and parameters, leading to inevitable discrepancies with real-world measurements. In contrast, Machine Learning (ML) methods have the potential to generate reliable surrogate models which may be able to quickly predict the system's behaviour. However, the number of data-driven methods that can potentially be used for this task is large and diverse. In a safety-critical setting such as nuclear engineering, a fair comparison of different ML methods, and a clear understanding of their advantages and limitations, is of paramount importance. To address this, we introduce a Common Task Framework (CTF) for ML in nuclear engineering, building upon previous efforts in dynamical systems and seismology. This CTF considers a curated set of datasets from different nuclear and nuclear-adjacent systems. The CTF evaluates the performance of a method on 12 established metrics, alongside a new paradigm focused on system monitoring from sparse measurements only. We illustrate the framework by benchmarking standard ML baselines against these datasets, revealing current method limitations. Our vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets, raising the bar for rigour and reproducibility in scientific ML for the nuclear industry.

Stefano Riva, Carolina Introini, Antonio Cammi +13

Autodesk Research, London, UK · Department of Energy, Nuclear Engineering Division, Politecnico di Milano, Milan, Italy · Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 +10

Surrogate Modeling Scientific ML

May 14, 2026cs.LG

Watch your neighbors: Training statistically accurate chaotic systems with local phase space information

Chaotic systems pose fundamental challenges for data-driven dynamics discovery, as small modeling errors lead to exponentially growing trajectory discrepancies. Since exact long-term prediction is unattainable, it is natural to ask what a good surrogate model for chaotic dynamics is. Prior work has largely focused either on reproducing the Jacobian of the underlying dynamics, which governs local expansion and contraction rates, or on training surrogate models that reproduce the ground-truth dynamics' long-term statistical behavior. In this work, we propose a new framework that aims to bridge these two paradigms by training surrogate dynamics models with accurate Jacobians and long-term statistical properties. Our method constructs a local covering of a chaotic attractor in phase space and analyzes the expansion and contraction of these coverings under the dynamics. The surrogate model is trained by minimizing the maximum mean discrepancy between the pushforward distributions of the coverings under the surrogate and ground-truth dynamics. Experiments show that our method significantly improves Jacobian accuracy while remaining competitive with state-of-the-art statistically accurate dynamics learning methods. Our code is fully available at https://anonymous.4open.science/r/neighborwatch.

Joon-Hyuk Ko, Andrus Giraldo, Deok-Sun Lee

Center for AI and Natural Sciences Korea Institute for Advanced Study Seoul, Korea 02455 · School of Computational Sciences Korea Institute for Advanced Study Seoul, Korea 02455

Surrogate Modeling Nonlinear System Identification Chaotic Dynamical Systems

May 13, 2026cs.LG

StampFormer: A Physics-Guided Material-Geometry-Coupled Multimodal Model for Rapid Prediction of Physical Fields in Sheet Metal Stamping

Traditional sheet metal forming relies on time-consuming and expensive Finite Element Analysis (FEA) for design validation, a process that significantly prolongs design cycles. While surrogate models offer faster iteration, current approaches have limitations: scalar-based methods cannot capture comprehensive field-based FEA results, while existing image-based models often ignore the critical role of material properties by focusing solely on geometry. To address this gap, we develop a physics-guided deep learning framework, namely StampFormer, which simultaneously uses component geometry and material stress-strain responses to predict FEA outcomes. The StampFormer framework uses three core components to process data. A Material-Augmented Geometric Network (MAGN) first fuses geometric and material data. This information is then integrated at various levels by a Hierarchical Material Embedding Injection Unit (HMEIU) before being processed by the primary network backbone, an adapted Swin-UNet. We evaluated our model on the stamping of a crossmember panel with two simulation datasets for steel and aluminium panels, and results demonstrate that StampFormer provides high-fidelity predictions of critical physical fields - including thinning, major strain, minor strain, plastic strain, and displacement - in under a second. Compared with ground truth FEA, our model achieved an average relative error of less than 8.5% on the four 2D fields and a mean squared error of less than 1.2 mm2 for the 3D displacement field. In summary, we introduce a practical and efficient framework that integrates multimodal information, namely geometry and material properties, to provide fast and accurate predictions, enabling designers to perform real-time manufacturability assessments.

Jiajie Luo, Mohamed Mohamed, Osama Hassan +8

Neural Surrogate Modeling Surrogate Modeling Physics-Informed ML Multimodal Fusion

May 12, 2026cs.LG

Efficient Conditioning Why Pseudo Observation Batch Bayesian Optimization Works When It Does not

Constant Liar (CL), Kriging Believer (KB), and fantasy models are widely used for batch selection in parallel Bayesian Optimization, yet a unified theory explaining their effectiveness and conditions under which they fail has been lacking. We identify efficient conditioning as the key surrogate property the ability to update predictions in closed form when data is augmented. We prove that Gaussian Processes satisfy this requirement, producing provably distinct batch points with separation of order l, and that this holds for any acquisition function monotonically non decreasing in posterior uncertainty (EI, UCB, PI), with qualitatively similar behavior for Thompson Sampling. We unify CL, KB, and fantasy models as instances of a single conditioning mechanism differing only in the lie value distribution, and draw quantitative connections to Local Penalization (LP) and qualitative connections to Determinantal Point Processes (DPPs). To disentangle model structure from optimizer randomness, we introduce the Structural Diversity Diagnostic (SDD), a reusable methodology for testing surrogate compatibility. Experiments on Hartmann6D, Ackley 8D, Levy10D, and SVM hyperparameter tuning validate all theoretical predictions: CL or KBs implicit penalty matches or outperforms explicit LP greedy conditioning achieves convergence on par with joint qEI efficient conditioning extends to Multiquadric RBF networks; and parametric surrogates produce degenerate batches even when fully retrained (random forests), while neural networks regain diversity only at 15x the wall clock cost of GP conditioning. Robustness is confirmed across multiple initial datasets and under observation noise.

Kumbha Nagaswetha, Rabi Pathak

Surrogate Modeling Bayesian Optimization Gaussian Processes

May 11, 2026gr-qc

Discovery of Interpretable Surrogates via Agentic AI: Application to Gravitational Waves

Fast surrogate models for expensive simulations are now essential across the sciences, yet they typically operate as black boxes. We present \texttt{GWAgent}, a large language model (LLM)-based workflow that constructs interpretable analytic surrogates directly from simulation data. Surrogate modeling is well suited to agentic workflows because candidate models can be quantitatively validated against ground-truth simulations at each iteration. As a demonstration, we build a surrogate for gravitational waveforms from eccentric binary black hole mergers. We show that providing the agent with a physics-informed domain ansatz substantially improves output model accuracy. The resulting analytic surrogate attains a median Advanced LIGO mismatch of

6.9\times10^{-4}

together with an

\sim 8.4\times

speedup in waveform evaluation, surpassing both symbolic regression and conventional machine learning baselines. Beyond producing an accurate model, the workflow identifies compact physical structure from the learned representation. As an astrophysical application, we use \texttt{GWAgent} to analyze the eccentricity of GW200129 and infer

e_{20\mathrm{Hz}}=0.099^{+0.063}_{-0.044}

. These results show that validation-constrained agentic workflows can produce accurate, fast, and interpretable surrogates for scientific simulations and inference.

Tousif Islam, Digvijay Wadekar, Tejaswi Venumadhav +4

Kavli Institute for Theoretical Physics, University of California Santa Barbara, Kohn Hall, Lagoon Rd, Santa Barbara, CA 93106, USA. · Center for Gravitational Physics, University of Texas at Austin, Austin, TX 78712, USA. · Department of Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA. +5

Surrogate Modeling Gravitational-Wave Astronomy Agentic Workflows Large Language Model-Guided Scientific Discovery

May 11, 2026cs.LG

Support-Proximity Augmented Diffusion Estimation for Offline Black-Box Optimization

Offline black-box optimization aims to discover novel designs with high property scores using only a static dataset, a task fundamentally challenged by the out-of-distribution (OOD) extrapolation problem. Existing approaches typically bifurcate into inverse methods, which struggle with the ill-posed nature of mapping scores to designs, and forward methods, which often lack the distributional expressivity to quantify uncertainty effectively. In this work, we propose SPADE (Support-Proximity Augmented Diffusion Estimation), a novel framework that reimagines forward surrogate modeling through the lens of conditional generative modeling. SPADE models the forward likelihood p(y|x) using a diffusion model, but with two critical enhancements to tailor it for optimization: (1) a Calibrated Diffusion Estimation module that enforces global consistency in statistical moments and pairwise rankings, and (2) a Support-Proximity Regularization mechanism that implicitly internalizes the data manifold constraint p(x) via kNN-based density estimation. Theoretically, we prove that our regularization is first-order equivalent to maximizing a Bayesian posterior with a valid design prior. Empirically, SPADE achieves state-of-the-art performance across Design-Bench tasks and an LLM data mixture optimization benchmark.

Yonghan Yang, Ye Yuan, Zipeng Sun +5

MBZUAI - Mohamed bin Zayed University of Artificial Intelligence · McGill University · Mila - Quebec AI Institute +1

Surrogate Modeling Surrogate-Assisted Optimization Black-Box Optimization OOD Generalization

May 11, 2026stat.ML

Multifidelity Gaussian process regression for solving nonlinear partial differential equations

Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work, as the available information is inherently multifidelity, we propose a kernel learning approach based on cokriging, leveraging empirical information from multifidelity simulations. In the first step, we fit a differentiable non-stationary kernel to an empirical kernel obtained from low-fidelity simulations. In the second step, we derive a high-fidelity kernel with estimated hyperparameters, and construct a corresponding high-fidelity mean using the multifidelity framework. These components can then be used within a Gaussian process framework for solving PDEs. Finally, we demonstrate the performance of the proposed physics-informed method on the Burgers' equation.

Fatima-Zahrae El-Boukkouri, Josselin Garnier, Olivier Roustant

Institut de Mathématiques de Toulouse, Université de Toulouse, INSA, Toulouse, 31077, France · CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, Palaiseau, 91120, France

Surrogate Modeling PDE Solving Multifidelity Modeling Physics-Informed ML Gaussian Process Regression

May 7, 2026cs.LG

Physics-based Digital Twins for Integrated Thermal Energy Systems Using Active Learning

Real-time supervisory control of thermal energy distribution systems requires digital twins that are accurate, interpretable, and uncertainty-aware, yet remain data and computationally efficient. High-fidelity simulations alone are costly, while purely data-driven surrogates often lack robustness. To address these challenges, this work proposes an active learning (AL) framework that couples system-level Modelica simulations with four simpler physics-informed and data-driven surrogate modeling approaches: deterministic Sparse Identification of Nonlinear Dynamics with Control (SINDyC), its probabilistic multivariate-Gaussian extension (MvG-SINDyC), feedforward neural network (FNN), and gated recurrent unit (GRU) network. Tailored to each surrogate, model-specific AL query strategies are employed, including Mahalanobis-distance sampling in coefficient space for MvG-SINDyC and error-based sampling in prediction space for SINDyC, FNN, and GRU, allowing the learning process to prioritize dynamically informative trajectories. The proposed approach is demonstrated on the glycol heat exchanger (GHX) subsystem of the Thermal Energy Distribution System (TEDS) at Idaho National Laboratory. Across key GHX outputs--the bypass mass flow rate

\dot{m}_{\mathrm{GHX}}

and heat transfer rate

Q_{\mathrm{GHX}}

-the AL framework achieves comparable predictive accuracy using as few as one-fifth of the simulation trajectories required by random sampling. Among the evaluated surrogates, the GRU achieves the highest predictive fidelity, while SINDyC remains the most computationally efficient and interpretable. The probabilistic MvG-SINDyC surrogate further enables uncertainty quantification and exhibits the largest computational gains under AL.

Umme Mahbuba Nabila, Paul Seurin, Linyu Lin +1

Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, United States · Nuclear Science & Technology Division, Idaho National Laboratory, Idaho Falls, ID 83415, United States · Department of Computer Science and Engineering, University of Michigan, Ann Arbor, MI 48109, United States

Surrogate Modeling Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Active Learning Physics-Informed ML Digital Twins

May 6, 2026cs.CL

Elicitation Matters: How Prompts and Query Protocols Shape LLM Surrogates under Sparse Observations

Large language models are increasingly used as surrogate models for low-data optimization, but their optimizer-facing prediction and its uncertainty remain poorly understood. We study the surrogate belief elicited from an LLM under sparse observations, showing that it depends strongly on prompt text and query protocol. We introduce an uncertainty-alignment criterion that measures whether model uncertainty tracks residual ambiguity among sample-consistent functions. Across controlled inference tasks and Bayesian optimization studies, we find that structural prompts act as effective priors, POINTWISE and JOINT querying induce different beliefs, and sequential evidence leads to non-monotonic, order-sensitive confidence updates. These effects change downstream acquisition decisions and regret, showing that elicitation protocol is part of the LLM surrogate specification, not a formatting detail.

Ge Lei, Samuel J. Cooper

Dyson School of Design Engineering Imperial College London Exhibition Road, London SW7 2AZ

Surrogate Modeling Bayesian Optimization LLM Prompting Large Language Model-Guided Optimization LLM Uncertainty Estimation

May 5, 2026cs.LG

Enabling Real-Time Training of a Wildfire-to-Smoke Map with Multilinear Operators

Wildfires are a major producer of fine particulate matter, impacting human health and the electrical grid. Accurately forecasting smoke impacts over long time scales incorporates fuel treatment strategies, natural fuel succession, and stochastic events like lightning strikes. However, predicting smoke for each fuel distribution with a forward simulation of a coupled fire-atmosphere model is computationally infeasible. Moreover, relatively simple fire models are tractable to run in many long-time scenarios but do not capture smoke transport. We use data-driven multilinear operators to predict a smoke concentration field from knowledge of the time since ignition for two quantities of interest: aerosol optical depth and smoke detection. Our method first computes the principal components of time-since-ignition and smoke concentration fields and then learns a map from powers of the input coefficients to the output coefficients. We apply our learned operator to smoke prediction in the Upper Rio Grande Watershed. After collecting training data, learning the approximation weights on a CPU takes less than 30 seconds, and each forward call takes less than 1 ms. On a proxy for aerosol optical depth, we obtain equal accuracy to Monte Carlo sampling with fewer than half as many coupled model calls. For smoke detection, we obtain an intersection-over-union (IoU) of 0.64 and an area under the receiver operating characteristic curve (AUC) of 0.95 on holdout data. Our method is significantly more accurate than the most similar published smoke classifier, which obtains an IoU and AUC of 0.15 and 0.61, respectively, on a 2015 bushfire in Australia.

Zachary Morrow, Joseph Crockett, John D. Jakeman +1

Sandia National Laboratories 1515 Eubank Blvd SE, Albuquerque, NM 87123

Surrogate Modeling Wildfire Forecasting Spatiotemporal Forecasting Operator Learning

May 4, 2026physics.ao-ph

Multi-Fidelity Emulation of Atmospheric Correction Coefficients with Physics-Guided Kolmogorov-Arnold Networks

Atmospheric correction is a critical preprocessing step in optical remote sensing, but repeated high-fidelity radiative transfer simulations remain computationally expensive for dense look-up-table generation, sensitivity analysis, retrieval support, and operational preprocessing. This study presents a physics-aware multi-fidelity surrogate framework for emulating atmospheric correction coefficients using paired 6S and libRadtran simulations. Atmospheric and geometric states are sampled using Latin Hypercube Sampling, and both radiative transfer models are evaluated under matched conditions for Sentinel-2 bands using spectral-response-function-aware coefficient generation. The high-fidelity targets are path reflectance, total transmittance, and spherical albedo. A physics-guided Kolmogorov-Arnold Network, termed pKANrtm, receives the atmospheric state and low-fidelity 6S coefficients, predicts the residual relative to libRadtran, and reconstructs the high-fidelity coefficients. The pKANrtm model uses an Efficient-KAN architecture and is trained with a physics-consistency penalty applied in the original coefficient space. The proposed model is evaluated against state-of-the-art regression-based RTM surrogates. Across both standard and out-of-distribution evaluation settings, pKANrtm achieves the strongest overall predictive performance among the compared models. Runtime benchmarking demonstrates substantial acceleration relative to libRadtran, with GPU inference providing approximately four orders of magnitude single-sample speedup and batched inference reaching tens of thousands of samples per second. These results indicate that physics-aware multi-fidelity pKANrtm emulation provides an accurate, physically structured, and computationally efficient strategy for atmospheric correction coefficient generation.

Md Abdullah Al Mazid, Naphtali Rishe

Knight Foundation School of Computing and Information Sciences, Florida International University, Miami, FL 33199, USA

Surrogate Modeling Remote Sensing Kolmogorov-Arnold Networks

May 4, 2026physics.comp-ph

Multi-fidelity surrogates for mechanics of composites: from co-kriging to multi-fidelity neural networks

Composite materials exhibit strongly hierarchical and anisotropic properties governed by coupled mechanisms spanning constituents, plies, laminates, structures, and manufacturing history. This intrinsic complexity makes predictive modeling of composites expensive, because repeated experiments and high-fidelity simulations are needed to cover large design spaces of material, structure, and manufacturing. Multi-fidelity surrogate modeling addresses this challenge by combining abundant, less expensive data with limited high-accuracy data to recover reliable high-fidelity predictions. This review presents a structured overview of multi-fidelity modeling for composite mechanics, covering Gaussian-process or Kriging-based methods, including co-Kriging, coregionalization models, autoregressive formulations, nonlinear autoregressive Gaussian processes, multi-fidelity deep Gaussian processes, and multi-fidelity neural networks. Their distinctions are examined in terms of cross-fidelity correlation, discrepancy representation, uncertainty quantification, and scalability. Selected examples of their applications to composites are introduced according to the roles that multi-fidelity surrogates play in engineering problems, including forward prediction for rapid exploration of material design spaces, inverse optimization for composite parameter identification and design search under limited high-fidelity access, and workflow integration, where heterogeneous data sources, constraints, and validation requirements determine model utility. Open question discussions highlight recurring challenges specific to composites, such as regime-dependent fidelity gaps associated with nonlinear damage and manufacturing history, mismatches between simulations and experiments, and uncertainty propagation across multi-fidelity models.

Haizhou Wen, Elham Kiyani, Gang Li +3

Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA · Division of Applied Mathematics, Brown University, Providence, RI 02912, USA · Center for Composite Materials, University of Delaware, Newark, DL 19716, USA +4

Surrogate Modeling Surrogate-Assisted Optimization Multifidelity Modeling Materials Science

Apr 27, 2026cs.RO

Generalizable Friction Coefficient Estimation via Material Embedding and Proxy Interaction Modeling

Accurately estimating friction coefficients between arbitrary material pairs is critical for robotics, digital fabrication, and physics-based simulation, but exhaustive pairwise testing scales quadratically with the number of materials. We introduce a proxy-based modeling framework that approximates any pairwise friction

f(A,B)

from a small, fixed set of proxy materials

C=[c_1,\dots,c_k]

by learning a per-material embedding

z_A = g(f(A,c1),\dots,f(A,ck))

and a fusion function

p

such that

f(A,B)\approx p\big(z_A,z_B\big)

. We present deterministic and probabilistic realizations of

g

and

p

, procedures for selecting diverse proxy sets, and mechanisms for handling missing or noisy proxy measurements. The learned embeddings are compact, interpretable, and enable calibrated uncertainty estimates for downstream decision making. On simulated and measured friction datasets, our approach achieves high predictive accuracy, robust performance with partial observations, and substantial experimental savings by significantly reducing pairwise testing.

Zhendong Wang, Huamin Wang

Style3D Research Hangzhou, China

Surrogate Modeling Materials Property Prediction

Apr 26, 2026cs.LG

WISE-FM:Operation-Aware, Engineering-Informed Foundation Model for Multi-Task Well Design

Deploying machine learning models across diverse well portfolios requires generalisation to wells with design parameters outside the training distribution. Current data-driven approaches to virtual flow metering (VFM) and bottomhole estimation typically treat each well independently or ignore the influence of well design on operational behaviour. We present WISE (Well Intelligence and Systems Engineering Foundation Model), a design-aware, physics-informed multi-task model that integrates three complementary mechanisms: Feature-wise Linear Modulation (FiLM) and cross-modal attention to condition operational embeddings on well design parameters; multi-task learning for simultaneous prediction of flow rates, bottomhole conditions, and flow regime classification; and structural mass conservation with soft physics constraints derived from well engineering principles. Evaluation on the ManyWells benchmark (2000 simulated wells,

10^6

data points) demonstrates that design-aware models reduce VFM prediction error by up to

13\times

compared to design-unaware baselines, and that physics constraints reduce negative flow predictions by 65%. Flow regime classification achieves 97.7% bottomhole accuracy, providing continuous well integrity monitoring without additional sensors. The methodology transfers to real operational data from five Equinor Volve producers (oil rate

R^2 = 0.89

, bottomhole pressure

R^2 = 0.98

, water rate

R^2 = 0.97

). The trained model additionally serves as a fast surrogate for integrity-aware well design optimisation over a 24-dimensional design space, with more than

1000\times

speedup over drift-flux simulations. These results demonstrate that design awareness, physics enforcement, and multi-task learning are essential and complementary ingredients for foundation models intended to operate across large well portfolios.

Carine de Menezes Rebello, Anderson Rapello dos Santos, Idelfonso B. R. Nogueira

Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, Norway · Petróleo Brasileiro S.A. (PETROBRAS), Wells/Well Engineering, Rio de Janeiro, Brazil

Surrogate Modeling Multi-Task Learning Physics-Informed ML

Apr 24, 2026cs.LG

Predicting Wind Loads on Container Ships in Harbor Environments through Multi-Fidelity Modeling

Modern container ships face higher wind loads due to increased windage areas, making accurate predictions of wind loads essential for mooring design. Existing empirical models, largely developed for container ships with smaller windage areas and simpler geometrical configurations than those of modern large-scale vessels, often lack accuracy and do not account for the influence of nearby structures. This study proposes a multi-fidelity surrogate modelling framework for the prediction of wind-load coefficients, combining empirical correlations with simplified and detailed CFD models for ships in open-sea and harbor environments. The approach relies on recursive co-kriging to consistently fuse information across fidelity levels, enabling accurate predictions at a reduced computational cost. A sensitivity analysis is used to identify the most influential geometric parameters, and the resulting reduced parameter space is explored through sequential sampling to efficiently construct the training database. The surrogate models are validated over a wide range of loading configurations and for two distinct harbor environments. The results demonstrate that the multi-fidelity approach significantly improves prediction accuracy compared to single-fidelity models, while substantially reducing the reliance on high-fidelity simulations. In particular, the proposed framework captures the dependence of wind loads on key geometric parameters and consistently outperforms traditional empirical correlations, providing a robust and efficient tool for engineering applications.

Matilde Fiore, Andrea Bresciani, Miguel Alfonso Mendez +1

Environmental and Applied Fluid Dynamics Department, von Karman Institute for Fluid Dynamics, Waterloosesteenweg 72, Sint-Genesius-Rode, B-1640, Belgium · Aero-Thermo-Mechanics Laboratory, École Polytechnique de Bruxelles, Université Libre de Bruxelles, Av. Franklin Roosevelt 50, Brussels, 1050, Belgium · Aerospace Engineering Research Group, Universidad Carlos III de Madrid, Av. de la Universidad 30, Leganés, 28911, Spain

Surrogate Modeling