PDE Surrogate Modeling

PDE: Partial Differential Equation

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32 papers in the last four weeks, up 146% on the four weeks before. 0.3% of all new papers.

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Latest papers 208

Jun 30, 2026cs.CV

Temperature Field Reconstruction of Tungsten Monoblock Divertor on EAST using Physics-aware Neural Operator Transformer

Accurate modeling of the divertor temperature field is essential for preventing material melting and damage and for extending the service life of fusion devices. However, conventional numerical methods, such as the Finite Element Method (FEM), are computationally expensive and therefore unsuitable for real-time applications. Therefore, a fast and generalizable method is required for real-time reconstruction of the divertor temperature field and subsequent real-time control. To address the above issue, we propose a Physics-aware Neural Operator Transformer (PNOT) to characterize the spatiotemporal evolution of the divertor temperature field. It models boundary heat-flux relations as a structured graph and employs graph attention to explicitly capture spatial physical dependencies. Inspired by physics-aware attention, we further develop a physics-aware neural operator module to aggregate query points with similar physical conditions via slicing and model heat diffusion, while a gradient-constrained Sobolev regularization loss enforces consistency between function values and their derivatives. Experimental results show that these physical constraints improve prediction accuracy while preserving physical consistency. The source code of this paper will be released on https://github.com/Event-AHU/OpenFusion

Zikang Yan, Xiao Wang, Qingquan Yang +6

School of Computer Science and Technology, Anhui University, Hefei, China · Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, China · Tongling University, Tongling 244000, China +2

PDE Surrogate Modeling Physics-Informed Neural Operators

Jun 28, 2026cs.LG

Randomized neural operator for parametric PDEs with fast training and conformal uncertainty quantification

Repeatedly solving parametric PDEs is essential for uncertainty quantification, design optimization and inverse problems, but conventional neural operators require expensive non-convex training. We introduce PCA--RaNN, a randomized latent neural operator that combines PCA-based dimensionality reduction with fixed random features and a closed-form least-squares readout. It recasts latent operator learning as fixed-feature linear regression, reducing training time by one to three orders of magnitude across benchmarks while maintaining competitive accuracy. We introduce an energy-matched scaling rule and a lightweight two-parameter BFGS refinement to correct suboptimal feature scales. Ensemble averaging reduces predictive variance. On Burgers, Darcy, Navier--Stokes and backward heat equation benchmarks, PCA--RaNN provides a favorable speed--accuracy trade-off against operator-learning baselines. The ensemble supports split-conformal prediction intervals, and the linear readout enables rapid online adaptation via recursive least squares without retraining hidden features. This provides an efficient, uncertainty-aware surrogate for many-query scientific workflows.

Zirui Deng, Jingbo Sun, Deyu Meng +1

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

Uncertainty Quantification PDE Surrogate Modeling PDE Operator Learning Neural Operators

Jun 28, 2026cs.LG

PCGD: Physics-Guided Conditional Graph Diffusion for TCAD Device Simulation

Technology computer-aided design (TCAD) semiconductor device simulation is fundamentally constrained by the high computational cost of iteratively solving coupled drift-diffusion equations. Existing ML surrogates either reduce internal physics to macroscopic scalar regressions, or rely on single-step mappings that lack the iterative refinement required to resolve stiff, coupled fields. To address this, we introduce PCGD, a Physics-Guided Conditional Graph Diffusion framework operating natively on unstructured TCAD meshes to predict coupled electrostatic and carrier density fields. PCGD employs a Condition-Aware MeshGraphNet denoiser that explicitly injects boundary conditions and device structure context via global cross-attention. By augmenting data-driven denoising with a physics-guided hybrid objective that integrates exponent-free quasi-Fermi gradient matching with noise-aware PDE residuals, PCGD progressively enforce physical constraints in the iterative diffusion trajectory. This strategy successfully bypasses the numerical instabilities typical of stiff drift-diffusion equations. Evaluated on a challenging mixed PN/MOS benchmark, PCGD significantly outperforms deterministic one-step regression (1.207% error) and local diffusion (1.585% error) baselines by achieving a sub-percent mean relative field error of 0.835%, while concurrently reducing maximum PDE residual errors by nearly three orders of magnitude compared to pure diffusion. It also transfers robustly to unseen SOI topologies (0.815% error) via LoRA adaptation, using 5.30

\times

less data and 14.34

\times

fewer parameters than full fine-tuning. Ultimately, PCGD bridges the computational efficiency of generative surrogates with the rigorous physical fidelity of traditional TCAD, unlocking highly scalable, field-level analysis for robust device engineering.

Yihan Zhang, Zhiteng Zhang, Kun Chen +1

PDE Surrogate Modeling Physics-Informed ML Conditional Diffusion Models Physics-Based Simulation Graph Diffusion

Jun 26, 2026cs.LG

A Trainable-by-Parts Operator Learning Framework: Bridging DeepONet and Karhunen-Loeve Expansions for Large-Scale Applications

Training operator-learning models for large-scale problems governed by partial differential equations (PDEs) is challenging due to the curse of dimensionality, memory constraints, and limited training data. These challenges arise in many scientific and engineering applications, including subsurface flow, climate modeling, and geological carbon storage (GCS). In this work, we propose a scalable operator-learning framework based on the Karhunen-Loeve Deep Neural Network (KL-DNN) and demonstrate its performance for modeling GCS. The model is trained on a dataset comprising 100 samples of large-scale simulations in a three-dimensional domain with 1.7 million cells and 50 time steps. The KL-DNN method constructs latent spaces using low-rank singular value decomposition of static properties and a nested Karhunen-Loeve expansion for dynamic pressure fields, enabling full-resolution predictions without subsampling or spatial coarsening. The KL-DNN model achieves an average root mean square error (RMSE) of 1.1 psi for pressure (0.04% relative error with respect to the average pressure in the domain) and RMSE of 0.0146 for CO2 saturation (5% relative error with respect to the average saturation inside the plume). The model requires 20 minutes of training on a single GPU, representing a 19% reduction in the pressure errors, 7% reduction in the saturation error, and a two-order-of-magnitude speedup compared to DeepONet trained on the same dataset. These results, along with inference time of less than one minute, establish the proposed model as a practical and accurate solution for large-scale PDE problems, enabling rapid uncertainty quantification, history matching, and real-time decision support.

Christian Munoz, Alexandre Tartakovsky

Civil Engineering Department University of Illinois Urbana-Champaign Urbana, IL 61801 · Pacific Northwest National Laboratory,Jun Richland, WA 99352

PDE Surrogate Modeling Operator Learning

Jun 26, 2026cs.LG

Physics-Informed Neural Network with Transfer Learning for State Estimation in Lithium-Ion Batteries using the Single Particle Model with Electrolyte

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving nonlinear partial differential equations (PDEs), including battery electrochemical models. They typically en-force conservation laws within the loss function to ensure physically consistent solutions. Tradi-tional numerical methods such as finite difference, finite volume, and finite element techniques, re-ly on discretization and can be computationally expensive for nonlinear systems. To address this challenge, PINNs offer improved scalability, particularly for reduced-order models like the single particle model with electrolyte (SPMe). The SPMe describes lithium-ion battery dynamics through coupled diffusion, transport, reaction kinetics, and voltage equations. Despite these advantages, training SPMe-based PINNs from scratch for different battery chemistries or operating conditions is demanding and often leads to slow convergence. To overcome this limitation, this work introduces a transfer learning framework for SPMe-PINNs. The model is first pretrained to learn general elec-trochemical dynamics and then adapted to a target battery by transferring weights, freezing se-lected layers, and fine tuning the remaining parameters, including estimating key electrochemical variables. Validation using PyBaMM demonstrates accurate voltage prediction, indicating that the proposed approach preserves electrochemical consistency while reducing training time and ena-bling efficient generalization across batteries.

Gift Modekwe, Qiugang Lu

Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA

PDE Surrogate Modeling Transfer Learning Physics-Informed ML Lithium-Ion Batteries

Jun 25, 2026cs.LG

Error-Conditioned Neural Solvers

Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each iteration, enabling it to read the spatial structure of its own errors and learn an update policy to iteratively correct its predictions. Across four PDE families, ENS attains the highest prediction accuracy in the large majority of settings, with gains reaching

10\times

on turbulent Kolmogorov flow, while avoiding the expensive compute cost of hybrid methods. ENS's learned correction policy generalizes under distribution shift, including zero-shot parameter changes and cross-equation transfer, where its relative advantage is largest in the ill-conditioned regimes where residual minimization is least reliable. Project website: https://neuralsolver.github.io/.

Haina Jiang, Liam Wang, Peng-Chen Chen +4

University of Michigan · KAIST AI · Los Alamos National Laboratory

Neural PDE Solvers PDE Surrogate Modeling

Jun 23, 2026physics.flu-dyn

A Physics-Informed Fourier-Wavelet Transformer for Multiscale Computational Fluid Dynamics Surrogate Modeling

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.

Somyajit Chakraborty, Ming Pan, Xizhong Chen

Shanghai Jiao Tong University, Department of Chemistry and Chemical Engineering 800 Dongchuan Road, Minhang District, Shanghai, 200240, China

Surrogate Modeling PDE Surrogate Modeling Wavelet Methods Physics-Informed ML Computational Fluid Dynamics

Jun 22, 2026physics.flu-dyn

Prediction of Viscoelastic Droplet Impact Dynamics Using a Vision Transformer-Based Approach

Droplet impact on solid surfaces is a complex fluid dynamics problem with applications in spray cooling, inkjet printing, and pharmaceutical processing. Although numerical simulations are widely used to investigate these dynamics, their computational cost becomes significant when multiple parametric variations are considered. In this work, we investigate the use of a Video Vision Transformer (ViViT) architecture to predict the temporal evolution of viscoelastic droplets impacting solid surfaces using volume fraction fields obtained from the Volume of Fluid (VOF) method. In Newtonian fluids, impact dynamics are mainly characterized by the Reynolds number

Re

, representing the ratio of inertial to viscous forces, and the Weber number

We

, representing the ratio of inertial to surface tension forces. For viscoelastic fluids, additional parameters are required to account for elastic effects, namely the solvent viscosity ratio

β

and the Weissenberg number

Wi

, increasing simulation complexity and cost. Instead of simulating the entire droplet dynamics, the proposed approach uses only the initial 10% to 20% of the simulation to predict the remaining evolution. Depending on the prediction configuration, this strategy reduces computational cost by approximately 80% to 90% compared to full numerical simulations. The ViViT produces physically consistent predictions across different parameters and prediction horizons, successfully capturing both spreading and bouncing regimes while preserving geometric features and structural similarity. Since volume fraction fields can also be extracted from experimental videos, the proposed framework could be extended to incorporate experimental data during training, potentially improving the physical fidelity of the predicted dynamics.

Diego A. de Aguiar, Cassio M. Oishi

Departamento de Matem´atica e Computa¸c˜ao, Faculdade de Ciˆencias e Tecnologia, Universidade Estadual Paulista “J´ulio de Mesquita Filho”, Presidente Prudente, Brazil

PDE Surrogate Modeling Video Prediction Computational Fluid Dynamics

Jun 22, 2026cs.CE

Attention mechanism for scalable mesh-based neural surrogates of free-surface fluids

High-fidelity simulations of free-surface flows using Lagrangian methods such as the Particle Finite Element Method (PFEM) are computationally demanding due to continuous domain updates and repeated solution of the governing equations. This challenge is further amplified by non-Newtonian rheologies, where material nonlinearities increase computational cost. These limitations motivate the development of efficient surrogate models to approximate PFEM dynamics at reduced cost. While data-driven deep learning approaches are promising, a key challenge is designing models that operate on arbitrary and evolving geometries. We propose a self-attention-based neural surrogate for PFEM simulations of free-surface flows. The architecture leverages attention mechanisms to model node interactions and capture complex spatial dependencies, while preserving the PFEM mesh discretization. This provides a geometric and topological framework for remeshing and node redistribution, maintaining high-quality spatial discretization during rollouts, improving long-term stability, and enabling reconstruction of derived mechanical quantities via standard finite element operators. Two attention formulations are considered: a standard self-attention mechanism and a linear variant that reduces computational cost and improves scalability. The models are evaluated on two- and three-dimensional free-surface flow benchmarks with evolving geometries, varying material parameters, and non-Newtonian fluids. Results show accurate prediction of transient dynamics and final configurations, with significantly improved scalability. The mesh-based formulation also enables direct reconstruction of quantities such as stress fields. Overall, the framework provides an accurate and scalable surrogate strategy for PFEM simulations in engineering-scale applications.

Federico Lanteri, Massimiliano Cremonesi

Department of Civil and Environmental Engineering, Politecnico di Milano Piazza Leonardo da Vinci, 32, 20133, Milano, Italy

Neural Surrogate Modeling Self-Attention PDE Surrogate Modeling Finite Element Method Computational Fluid Dynamics

Jun 22, 2026cs.LG

One-Step Flow Matching for Generative Modeling of Path-Dependent Physical Fields

Physical simulations for intricate geometries with path-dependent constitutive models face difficulties due to the enormous computational cost they require. Recently, the emergence of generative AI models, which succeed in image and video synthesis tasks, has provided a promise to further improve simulations. Although U-Net-based denoising diffusion probabilistic models (DDPMs) have been adopted for elastic stress field generation, they typically require hundreds of sampling steps, and applications of generative models to path-dependent, e.g. plastic, stress fields remain very limited. In this work, we propose a novel flow matching (FM) model based on a transformer backbone for high-resolution path-dependent stress field generation with stochastic loading-unloading paths and geometry. The proposed model operates within the latent space of a variational autoencoder (VAE) and formulates the simulation of plastic fields as a video synthesis task, directly generating the stress fields across all time steps. Meanwhile, we design a non-Gaussian source distribution for flow matching, such that crossings among conditional transport paths are reduced during training. This enables our model to generate satisfactory samples in one step without relying on distillation. In addition, we introduce token-level loading embeddings and two auxiliary networks to further enhance the model performance in path-dependent simulation. The results demonstrate that, even with a limited training dataset, our model can accurately generate high-resolution path-dependent fields. It is much more computationally efficient than finite element analysis, providing a speedup of 6 to 7 times over FEM on CPUs and approximately two orders of magnitude speedup on consumer-grade GPUs.

Yijing Zhou, Jasmin Jelovica

aDepartment of Mechanical Engineering, The University of British Columbia, Vancouver, Canada · bDepartment of Civil Engineering, The University of British Columbia, Vancouver, Canada

Flow Matching Structural Mechanics PDE Surrogate Modeling One-Step Generative Modeling Flow-Based Generative Modeling

Jun 18, 2026cs.LG

Neural network surrogates with uncertainty quantification for inverse problems in partial differential equations

Inverse problems for differential equations arise throughout science and engineering, where one seeks to infer unknown model parameters from noisy or incomplete observations. Traditional numerical methods for these problems are often computationally expensive, particularly in Bayesian settings where evaluating the likelihood becomes costly for complex forward models and high-dimensional parameter spaces. To address this challenge, we introduce DeepGaLA, a neural-network surrogate for differential equation solvers that provides uncertainty-aware predictions, reducing overconfident inference when training data are limited. To evaluate the fidelity of the surrogate-induced posterior approximations in practice, we show that a short run of delayed-acceptance Markov chain Monte Carlo can serve as an effective diagnostic. Across a range of numerical experiments, DeepGaLA delivers forward-model approximations with accuracy comparable to established Gaussian-process surrogates, while better maintaining efficiency as parameter dimension grows. Moreover, it can incorporate differential-equation constraints, including in nonlinear settings. Overall, these results indicate that uncertainty-quantified neural surrogates can enable scalable and reliable Bayesian inference for inverse problems in complex systems.

Christian Jimenez-Beltran, Aretha L. Teckentrup, Antonio Vergari +1

School of Mathematics and Maxwell Institute for Mathematical Sciences University of Edinburgh · School of Informatics University of Edinburgh

Neural Surrogate Modeling Uncertainty Quantification Bayesian Inverse Problems PDE Surrogate Modeling

Jun 18, 2026cs.LG

Comparative Study of Neural Surrogate Architectures for Autoregressive Prediction of Internal Battery States

The Doyle-Fuller-Newman (DFN) model resolves internal electrochemical states in lithium-ion batteries with high fidelity. However, the numerical solution of its governing equations is computationally prohibitive for real-time deployment, limiting scalability from individual cells to pack and fleet-scale applications. While machine learning surrogates can substantially reduce inference latency through GPU acceleration, most existing approaches learn solution approximations tied to specific operating conditions rather than learning generalizable state-evolution dynamics. This work presents a systematic comparison of four neural network architectures (MLP, ResNet, U-Net, FNO) formulated as autoregressive state-transition operators that predict full DFN internal states across a wide range of operating conditions. To ensure a controlled architectural comparison, all models are trained under a unified framework using multi-step unrolling and current-conditioning, isolating the impact of spatial inductive bias. Results demonstrate that the U-Net's multi-scale feature hierarchy achieves a mean final-step nRMSE of 3% averaged across all internal state variables after 300-step autoregressive rollouts, while providing a 5.38x speed-up over the numerical solver. These findings highlight spatial inductive bias as a critical determinant of surrogate performance, advancing the development of surrogates for internal state observability for next-generation battery management systems and digital twins.

Gihyun Lee, Thorben Menne, Simon Olma +2

Technische Universität Berlin, Berlin, Germany · IAV GmbH, Berlin, Germany

Neural Surrogate Modeling PDE Surrogate Modeling Lithium-Ion Batteries Neural Operators Autoregressive Modeling

Jun 17, 2026cs.LG

Advances in Scientific Machine Learning for Coupled Fluid Flow and Transport

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.

Gabriel F. Barros, Rômulo M. Silva, Alvaro L. G. A. Coutinho

COPPE −Federal University of Rio de Janeiro - UFRJ, Rio de Janeiro, Brazil

PDE Surrogate Modeling Reduced-Order Modeling Physics-Informed ML Computational Fluid Dynamics

Jun 16, 2026math.NA

Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems

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.

Kuilin Qin, Lianfang Wang, Xu Sun +4

PDE Surrogate Modeling PDE Operator Learning PDE Inverse Problems Neural Operators

Jun 16, 2026cs.LG

Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learning

Neural operators provide fast surrogates for PDEs but their deterministic predictions limit their use in tasks requiring uncertainty quantification (UQ), especially under geometric variability. Existing approaches primarily model uncertainty in network parameters, largely overlooking the geometry-aware representations learned by the operator itself. We propose REEF-GP (Residual on Embedded Features Gaussian Process), a post-hoc UQ framework that fits a GP to the residuals of a frozen neural operator whose internal embeddings define the kernel feature space. Rather than learning a separate feature map, REEF-GP adapts the operator's intrinsic coordinate-feature representations to construct geometry-aware uncertainties. To ensure stability and scalability on unstructured domains, REEF-GP incorporates spectral-normalized projections, heteroscedastic geometry-aware noise, and efficient subset-based training that avoids restrictive low-rank approximations. Across five PDE benchmarks with varying geometries, REEF-GP preserves predictive accuracy while achieving calibrated uncertainty estimates competitive with deep ensembles but at a fraction of their cost. Our approach remains robust under geometric distribution shift, with uncertainty concentrating in physically meaningful regions (e.g., shock fronts). Our results demonstrate that accurate and scalable post-hoc UQ for neural operators can be achieved directly in their learned feature space, offering a practical alternative to parameter-centric approaches.

Oriol Vendrell-Gallart, Nima Negarandeh, Ramin Bostanabad

Department of Mechanical and Aerospace Engineering University of California, Irvine Irvine, CA 92617

Uncertainty Quantification PDE Surrogate Modeling Neural Operators Gaussian Process Regression

Jun 16, 2026cs.LG

Operator Boosting Produces Pareto-Efficient PDE Surrogates

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.

Lennon J. Shikhman

Georgia Institute of Technology College of Computing Atlanta, Georgia, USA

PDE Surrogate Modeling Deep Operator Networks Neural Network Compression Neural Operators

Jun 15, 2026cs.LG

Towards Fast GNN Surrogates for CO2 Migration in Complex Geological Formations

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.

Rodrigo S. Luna, Thiago H. N. Coelho, Luiz S. L. Neto +8

Systems and Computer Engineering and High Performance Computing Center, NACAD - COPPE, Federal University of Rio de Janeiro · Civil Engineering and High Performance Computing Center, NACAD - COPPE, Federal University of Rio de Janeiro · Mechanical Engineering and High Performance Computing Center, NACAD - COPPE, Federal University of Rio de Janeiro +1

Graph Neural Networks PDE Surrogate Modeling Spatiotemporal Forecasting

Jun 15, 2026cs.AI

MR-GVNO: A Geometry-Aware Variational Physics-Informed Neural Operator for Mindlin-Reissner Plates on Irregular Domains

Plate and shell structures are widely used in engineering, making rapid response prediction under varying geometries, materials, and loads highly desirable. However, conventional finite element methods require repeated modeling and solution, resulting in high computational costs. This study proposes a geometry-aware variational neural operator for Mindlin-Reissner plate problems, termed MR-GVNO. The method uses boundary point clouds to represent irregular geometries and employs separate encoders for spatially varying material fields, pressure loads, and scalar physical parameters. A cross-attention mechanism integrates these inputs with query point information to predict transverse deflections and rotations at arbitrary locations. MR-GVNO is trained without labeled solution data using a variational physics-informed loss derived from the discretized total potential energy. It directly processes irregular point clouds and allows different physical fields to be discretized independently, avoiding interpolation onto a common grid. Numerical experiments on single-hole, double-hole, and L-shaped plates demonstrate accurate response prediction under homogeneous and heterogeneous materials and uniform and random loads. The model also achieves millisecond-level full-field inference and favorable cross-geometry generalization.

Siqi Wang, Daobo Sun, Yizheng Wang +4

Institute of Computational Mechanics × AI & College of Intelligent Robotics and Advanced Manufacturing, Fudan University, Shanghai 200433, China · School of Aerospace Engineering, Xiamen University, Xiamen 361102, China · Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China +2

PDE Surrogate Modeling PDE Operator Learning Physics-Informed Neural Operators

Jun 15, 2026cs.LG

PhysGuard: Fisher-Guided Gradient Projection for Sim-to-Real Neural PDE Surrogates

Neural operator models trained on simulation data often lose accuracy when applied to experimental measurements due to the sim-to-real gap. Standard fine-tuning with limited real data can reduce this gap, but it may also damage the core physics-relevant representations learned during pretraining. Although knowledge-preserving adaptation has been widely investigated in vision or language tasks, it remains unclear whether these methods are suitable for neural operators whose architectures and protected knowledge are fundamentally different. Neural operators need to preserve core-scale physical structures rather than semantic or visual features. We propose PhysGuard, a physics-preserving framework for accurate sim-to-real adaptation of neural operators. Specifically, PhysGuard uses the empirical Fisher Information Matrix computed on simulation data to identify physics-critical parameter directions, then restricts fine-tuning updates to directions that do not interfere with them. A layer-wise Gram-matrix formulation makes this efficient for models with millions of parameters, while an adaptive threshold automatically determines the protected subspace size. A spectral probe experiment shows that the dominant Fisher directions are strongly associated with low-frequency output structures. Experiments on benchmark across four neural operator architectures and different physical systems show that PhysGuard performs strongly on most evaluation metrics compared to baselines. The benefits are most evident under severe domain shift, where it reduces low-frequency error by up to 32% compared to standard fine-tuning while maintaining adaptability. Our code is available at https://github.com/ZhouChaunge/PhysGuard.

Changjian Zhou, Junfeng Fang, Negin Yousefpour +3

Faculty of Engineering and IT, University of Melbourne · School of Computing, National University of Singapore · Artificial Intelligence Research Institute, IFLYTEK Co., Ltd.

Synthetic-to-Real Domain Adaptation PDE Surrogate Modeling Neural Operators

Jun 15, 2026physics.flu-dyn

Learning Interface Breakup: A Geometry-Conditioned Latent Surrogate for Spray Formation

Designing spray nozzles requires predicting how geometry shapes transient two-phase breakup, but high-fidelity volume-of-fluid (VOF) simulations with adaptive mesh refinement (AMR) are too expensive for iterative design exploration. Standard surrogate models are also challenged by this setting because both the liquid--gas interface and the underlying adaptive discretization evolve across time and geometries. We introduce a geometry-conditioned latent surrogate trained on 797 two-phase nozzle simulations that addresses this by encoding the AMR cell-density field, rather than the full multi-channel flow state, as a compact proxy for where the solver concentrates resolution. From this representation, the model reconstructs transient density evolution and nozzle geometry, and a lightweight second stage recovers the remaining flow variables. On held-out simulations, the method accurately captures key interface dynamics while reducing inference time to 0.045 seconds per trajectory, corresponding to a speed-up of more than

6\times10^4

relative to Basilisk CFD. These results suggest that AMR refinement structure can serve as a compact and learnable representation for geometry-conditioned surrogate modeling of transient two-phase flows.

Julius H Ramlau, Friedrich Hastedt, Tolga Birdal +3

Department of Chemical Engineering, Imperial College London, London, United Kingdom · CMAC, CEDAR, an EPSRC Center for Doctoral Training in Cyber-Physical Systems for Medicines Development and Manufacturing, Imperial College London, London, United Kingdom · Sargent Centre for Process Systems Engineering, Imperial College London, London, United Kingdom +2

PDE Surrogate Modeling Adaptive Mesh Refinement Computational Fluid Dynamics

Jun 13, 2026cs.LG

Physics-conforming Latent Twins

Surrogate models are central to scientific machine learning, where they enable fast prediction, simulation, inference, and control for complex physical systems. For time-dependent problems, however, accurate interpolation of training trajectories is not sufficient: reliable surrogates should also respect the conservation laws, invariants, admissibility conditions, and dissipative structures that give those trajectories physical meaning. We introduce Physics-conforming Latent Twins, a framework for learning latent surrogate solution operators whose dynamics satisfy selected physical principles by design. The method builds on the Latent Twin formulation by jointly learning an encoder, a decoder, and a latent flow map between arbitrary time-indexed states, while constraining the latent dynamics to preserve or dissipate prescribed structural quantities. We develop a constraint-transfer viewpoint that connects physical structure in the original state space with compatible constraints in latent space, and prove structure-preservation bounds showing how latent enforcement improves control of physical defects after decoding. We also derive algebraic conditions for latent flow maps that preserve linear and quadratic invariants or enforce dissipative inequalities. Numerical experiments on representative ODE and PDE benchmarks demonstrate improved constraint satisfaction, structural fidelity, and qualitative long-time behavior while maintaining accurate surrogate prediction.

Matthias Chung, Yutong Bu, Deepanshu Verma

Department of Mathematics, Emory University, 400 Dowman Drive, Atlanta, 30322, GA, USA · School of Mathematical and Statistical Sciences, Clemson University, 220 Parkway Drive, Clemson, 29634, SC, USA

Neural Surrogate Modeling Surrogate Modeling Latent Dynamics Modeling PDE Surrogate Modeling Physics-Informed ML

Jun 12, 2026cs.LG

Separable Neural Architectures as Physical World Models: from Mathematical Theory to Applications

This work introduces the Separable Neural Architecture (SNA), a function representational class combining neural approximation with tensor decomposition. The SNA decouples localized coordinate functions (atoms) from global interactions governed by a sparse, low-rank interaction object. This architecture possesses a compact and smooth inductive bias well-suited for solving partial differential equations (PDEs). When viewed as a Galerkin trial space under the variational SNA (VSNA) framework, the formulation satisfies classical variational guarantees under Lax-Milgram: well-posedness, quasi-optimality, convergence, and stability. In high-dimensional spatiotemporal--parametric PDEs, the VSNA mitigates the curse of dimensionality by scaling algebraically rather than exponentially. Exploiting an entirely factorized, tensor-native alternating least squares (ALS) optimization framework reduces this cost to linear in dimension. The VSNA is validated across elliptic, hyperbolic, and parabolic systems, demonstrating close alignment with predicted algebraic and spectral scaling rates. We showcase the SNA as a "solve once, query anywhere" physical world model via two engineering case studies: a 7D parametric manufacturing simulation and an experimental thermal-to-property inversion pipeline for Inconel 718. The VSNA executes a 1,000,000-query Monte Carlo sweep in 102s on a standard laptop CPU, yielding a 150,000x speedup over a full-grid finite element baseline hosted on an NVIDIA A100 GPU. It further enables real-time generative inverse-mode reconstructions under 100ms. These results demonstrate that the SNA serves as a compact mathematical substrate for continuous parameter manifolds to enable real-time inversion, optimization loops, and rapid uncertainty propagation.

Reza T Batley, Andrew Kichline, Sourav Saha

Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, 1600 Innovation Drive, Blacksburg, 24060, VA, United States

Neural PDE Solvers PDE Solving PDE Surrogate Modeling

Jun 12, 2026cs.LG

A Hybrid GNN-FEM Framework for Phase-Field Fracture Simulation. Physics-Preserving Hybridization for Generalizable Surrogate Modeling

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.

Hyeonbin Moon, Yongjin Choi, Seunghwa Ryu

Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, 34141, Republic of Korea · KAIST InnoCORE PRISM-AI Center, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, 34141, Republic of Korea · Department of AX, College of AI, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, 34141, Republic of Korea

Structural Mechanics PDE Surrogate Modeling Physics-Informed ML Computational Materials Science

Jun 10, 2026cs.LG

Structure-Preserving Neural Surrogates with Tractable Uncertainty Quantification

Recent advances in scientific machine learning provide a means of near-real-time solution to partial differential equations (PDEs), but lack the theoretical underpinnings of conventional simulators that support contemporary verification and validation. In this work, we construct data-driven reduced-order models that serve as structure-preserving, real-time surrogates. Remarkably, the exterior calculus that imposes physical conservation structure also exposes topological structure that we use to build a Gaussian process (GP) representation of uncertainty in state-flux relationships, ultimately yielding a Dirichlet-to-Neumann map for quantities of interest with closed-form expressions for posterior uncertainty. We specifically propose structure-preserving

H(\mathrm{div})

--

L^2

subspaces of conventional Raviart--Thomas and

dgP_0

elements prescribed by a lightweight transformer. Reduced-order dynamics consistent with this subspace are learned by posing a conservation law in which a GP describes the fluxes between volumes. This work hinges on a novel interface between mixed FEM spaces and GP regression; when training is posed as the optimal recovery problem (ORP), the resulting GP regression can be written as an optimization problem with equality constraints that impose a conservation structure, amenable to a fast Schur-complement training strategy. The trained model can then be solved in real time with closed-form estimators for boundary fluxes driven by prescribed Dirichlet data. The paper includes RKHS posterior error bounds for linear functionals to support uncertainty quantification, as well as numerical experiments demonstrating the accuracy of the posterior distribution as a surrogate for error estimation.

Handi Zhang, Adrienne M. Propp, Brooks Kinch +2

Applied Mathematics and Computational Science, University of Pennsylvania, Philadelphia, PA, USA · Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA · Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, USA +1

Uncertainty Quantification PDE Surrogate Modeling Reduced-Order Modeling Finite Element Method Gaussian Process Regression

Jun 9, 2026cs.LG

First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems

We introduce First-Order Trajectory Matching (FTM), a surrogate-modeling method that learns the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM learns the probability current velocity, whose flow preserves time marginals to match ensemble averages, while also capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents. FTM learns the current velocity directly from trajectories, avoiding drift, diffusion, and score estimation. Our stability analysis separates discretization error from sampling variance and shows that the one-step simulation-free FTM loss is stable when temporal resolution and sample size are properly balanced. Across stochastic dynamical systems and PDE examples, we empirically demonstrate that FTM provides trajectory-aware ensemble predictions at low, deterministic-rollout cost.

Shreya Jha, Timo Schorlepp, Nicholas Geissler +2

Courant Institute of Mathematical Sciences, New York University

Dynamical Systems Surrogate Modeling Stochastic Linear Dynamical Systems PDE Surrogate Modeling Probabilistic Forecasting

Jun 9, 2026cs.LG

Least-Action-Guided Diffusion for Physical Extrapolation

Reliable extrapolation remains a central challenge for generative models in computational physics, because models trained over finite ranges of time, parameters, or geometries may produce physically inconsistent predictions outside the training distribution. We introduce a least-action-principle-guided diffusion, LAPG, a framework that promotes physical consistency during inference rather than relying solely on constraints imposed during training. The method combines a conditional score-based diffusion model with an action-derived physical guidance score. In the first stage, the learned score model generates an in-distribution proposal; in the second, an action-based variational prior refines this proposal toward the target out-of-distribution condition. This formulation turns the principle of least action into a differentiable inference-time correction mechanism and provides an alternative to pointwise residual penalties that often require empirical loss balancing. We evaluate LAPG on representative ordinary- and partial-differential-equation systems, including free fall, conservative and dissipative spring-mass dynamics, interacting point vortices, and potential flow over parameterized airfoils. In temporal, parameter, and geometric extrapolation tests, LAPG reduces phase drift, preserves dissipative decay, captures vortex motion, and improves the lift response of airfoil flows compared with training-time physics-informed baselines.

Zhongxin Yang, Yuanwei Bin, Xiang I. A. Yang +1

College of Engineering, Peking University, Beijing 100871, China · Ningbo Institute for Digital Twin, Eastern Institute of Technology, Ningbo 315200, Zhejiang, China · Eastern Institute for Advanced Study, Eastern Institute of Technology, Ningbo 315200, Zhejiang, China +2

Physics-Informed Diffusion Models Physics-Informed Generative Modeling Diffusion Guidance PDE Surrogate Modeling OOD Generalization Physics-Informed ML

Jun 9, 2026cs.LG

Physics-Constrained Neural Surrogate for Domain Growth Prediction in Systems with Conserved Kinetics

The spatiotemporal evolution of many physical, chemical, and biological systems is described by nonlinear partial differential equations (PDEs). Recently, deep neural network-based surrogate models have emerged as efficient alternatives to computationally expensive numerical PDE solvers. In this work, we propose a physics-constrained deep neural network as a surrogate model to learn the microstructural evolution of a binary mixture, in which conservation of the order parameter is imposed directly on the network output as a hard constraint. We train the model to accurately predict the time-evolution of phase separation in binary mixtures governed by the Cahn-Hilliard equation. We show that predictions from our trained surrogate model remain stable and accurate over long-time rollouts for both critical and off-critical mixtures and preserve the mixture composition throughout evolution. In contrast, a variant in which conservation is enforced only via a penalty term in the loss function drifts away from the initial composition and significantly loses predictive accuracy over the same rollout. This establishes that the hard constraint is essential for long-time stability and order parameter conservation. We also show that our model accurately captures the growth of domain size and is consistent with the Lifshitz-Slyozov domain-growth law. These results demonstrate the effectiveness of the proposed framework for modeling systems with conserved kinetics, and the construction extends directly to other systems with conserved quantities.

Vijay Yadav, Pallvi Pandey, Madhu Priya +2

Department of Physics, Indian Institute of Technology Jodhpur, Jodhpur, Rajasthan 342 030, India · Department of Physics, Birla Institute of Technology Mesra, Ranchi, Jharkhand 835 215, India · Department of Physics, Central University of Rajasthan, Ajmer, Rajasthan 305 817, India

Dynamical Systems PDE Surrogate Modeling Physics-Informed ML

Jun 8, 2026physics.comp-ph

Input-schema identifiability limits in physics-informed surrogates for mechanics-governed flow

Physics-informed and data-driven surrogates are increasingly used to approximate mechanics-governed flow fields, but the target quantities assigned to such models are not always identifiable from the input variables available at prediction time. We introduce an input-schema identifiability certificate for computational surrogates. Starting from a reduced physical model, the certificate decomposes a target field into components that are measurable from geometry, components that require boundary-condition information, and components identifiable only up to a symmetry quotient. This yields a pre-training audit: it predicts which oracle-channel interventions should reduce error, which should fail, and which ambiguity cannot be removed by changing the architecture, loss, optimizer, or sample size. We instantiate the framework for incompressible tubular flow using a Cosserat-rod reduction, where lumen velocity separates into a mesh-measurable tangent direction, a boundary-condition-dependent magnitude, and a signed-orientation ambiguity. Controlled experiments on patient-specific aortic CFD geometries, analytic Womersley flows, and an advection-diffusion transfer problem confirm the predicted pattern: supplying signed direction collapses angular error to the oracle regime, whereas supplying magnitude without orientation leaves the predicted sign ambiguity and yields 16-33 percent per-node sign flips. The results provide a mechanics-based diagnostic for deciding whether a surrogate modelling task is physically identifiable before training, and expose failure modes that aggregate error metrics can hide.

Daniel Cieslak, Andrzej Czyzewski

Gdańsk University of Technology, Gdańsk, Poland

PDE Surrogate Modeling Physics-Informed ML Computational Fluid Dynamics

Jun 8, 2026cs.LG

Learning Where to Simulate: Generative Active Sampling for Online PDE Surrogate Training

Data-driven PDE surrogates are trained with data produced by numerical PDE solvers. However, when the surrogate's goal is to generalize across a wide range of PDE configurations (e.g., initial conditions and physical coefficients), generating a representative training set is non-trivial. Uniform sampling of configuration parameters often under-represents trajectories exhibiting challenging dynamics, leading to high prediction errors and large error variance in the trained surrogate. Online training, where data generation and surrogate training are coupled, offers a natural advantage by allowing solver parameters to be steered on-the-fly. To efficiently exploit this capability, we introduce Online Generative Active Sampling (OGAS), an active learning method that reactively learns the relationship between configuration parameters and surrogate performance to control the sampling distribution. OGAS trains a fast diffusion model in parallel to the surrogate to act as a conditional sampler, mapping a surrogate-derived difficulty signal (e.g., loss or uncertainty) to configuration parameters. By actively drawing target signals from a prior biased toward high difficulty, OGAS continuously steers data generation toward challenging regimes without delaying the training workflow. We evaluate OGAS across 2D PDEs with distinct challenging dynamics (Kuramoto-Sivashinsky, Navier-Stokes, Gray-Scott) and up to 308 parameters, using multiple surrogate architectures. Across all settings, OGAS consistently improves tail statistics, yielding substantial reductions in errors above the 99th percentile and overall error dispersion compared to uniform sampling. While prioritizing challenging trajectories introduces a trade-off with average error, OGAS effectively ensures worst-case reliability of trained surrogates with negligible wall-time overhead.

Pierre Cesar, Sofya Dymchenko, Abhishek Purandare +1

DATAMOVE · Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LIG, Grenoble, France

PDE Surrogate Modeling Active Learning Adaptive Sampling

Jun 7, 2026cs.LG

Conformal Prediction for Neural Operators: Distribution-Free Uncertainty Quantification in Physics Simulation

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

Michael Chin

Independent Researcher

Uncertainty Quantification PDE Surrogate Modeling Conformal Prediction Neural Operators Regression Prediction Intervals