arXiv subjects

George Karniadakis

Publications and source records attributed to George Karniadakis.

At least 19 recordsLinked to original sources

arXiv2026

Site-scale, Multi-resolution WRF Wind Modeling with Observational Nudging for Methane Emission Monitoring in the Permian Basin

Accurate methane source localization and emission-rate estimation at oil and gas facilities require wind fields that resolve site-scale spatial and temporal variability, which cannot be characterized by a single anemometer or coarse operational weather products. We develop a multiscale Weather Research and Forecasting (WRF) framework for a Permian Basin facility that dynamically downscales hourly, 3 km High-Resolution Rapid Refresh (HRRR) fields through four nested domains with grid spacings of 3 km, 1 km, 200 m, and 40 m. The two outer domains use planetary-boundary-layer parameterization, whereas the two inner domains operate in large-eddy-simulation mode. One-minute wind observations from two on-site anemometers are assimilated through wind-only observational nudging in the innermost domain, and the effects of nesting feedback and turbulence-closure choices are examined. Simulations for winter and summer 2025 periods are evaluated using near-surface wind-speed time series, power spectra, spatial fields, and statistical metrics. The baseline simulation reproduces the broad evolution of observed wind events but drifts during weak-wind periods. Observational nudging with one- and two-way nesting reduces the absolute bias of wind-speed by %65 at one sensor and %87 at the other, while reducing the corresponding root-mean-square errors by %33 and %44. Nudging also increases high-frequency energy and resolved spatial gradients, although improvements in turbulence statistics are not uniform. These results demonstrate a practical physics-based pathway for generating high-resolution wind fields for methane-plume modeling while emphasizing the need for independent spatial observations to validate accuracy away from assimilated sensors.

math-ph↗

arXiv2026

Proposal-Guided Greedy Surrogate Refinement for PDE-Driven High-Dimensional Rare-Event Estimation

Accurate surrogate construction for PDE-driven high-dimensional rare-event simulation is challenging when performance evaluations are expensive. Since a globally accurate surrogate may require many high-fidelity evaluations, adaptive importance sampling provides a natural localization tool: its evolving proposal distribution progressively identifies the failure-relevant region. Motivated by this observation, we propose a surrogate-assisted adaptive importance sampling framework that refines the surrogate locally along the evolving proposal, rather than over the entire input space. The surrogate combines an encoder with a neural network, providing a low-dimensional latent representation for both prediction and sample selection. At each adaptive iteration, candidates drawn from the current proposal are selected by a greedy latent-space rule balancing proximity to the estimated failure boundary and sample diversity. The selected samples are evaluated by the high-fidelity model and used to refine the surrogate, which then guides the subsequent cross-entropy-type adaptive proposal update. We establish one-step proposal stability bounds under local surrogate errors, together with surrogate-induced misclassification and finite-sample estimation error bounds. Numerical experiments on multimodal benchmarks and PDE-driven rare-event problems up to 100 dimensions show that the proposed method achieves accuracy comparable to true-model adaptive importance sampling while requiring substantially fewer high-fidelity evaluations.

math.NA↗

arXiv2026

AI-enabled cardiac shape reconstruction from routine magnetic resonance imaging

Computational models of cardiac structure and function are increasingly central to the development of subject-specific cardiac digital twins, enabling improved characterization of contractile dysfunction, pathological remodeling, and electrical abnormalities. A critical prerequisite for these models is the accurate reconstruction of three-dimensional (3D) cardiac anatomy from medical imaging. Multi-planar magnetic resonance imaging, particularly when combined with artificial intelligence, offers a clinically feasible alternative to conventional reconstruction techniques. In this study, we present a neural field-based reconstruction framework that recovers 3D cardiac geometries from sparse planar contour data by learning continuous shape representations. Reconstruction performance was evaluated using complementary in-silico and in vivo datasets spanning variations in sampling density and geometric complexity. Across both datasets, reconstructed meshes closely matched reference geometries, demonstrating that the neural field approach faithfully captures cardiac planar contours. Compared with traditional local interpolation methods, the proposed framework exhibited improved geometric fidelity in anatomically challenging regions, including the left ventricular apex and basal segments, particularly under sparse sampling conditions. Collectively, these findings demonstrate that neural field-based reconstruction provides a robust and efficient pathway for multi-planar cardiac shape recovery, with particular relevance for AI-driven modeling pipelines and data-limited settings such as small-animal and time-resolved cardiac imaging.

physics.med-ph↗

arXiv2026

AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learning

Scientific Machine Learning (SciML) integrates data-driven inference with physical modeling to solve complex problems in science and engineering. However, the design of SciML architectures, loss formulations, and training strategies remains an expert-driven research process, requiring extensive experimentation and problem-specific insights. Here we introduce AgenticSciML, a collaborative multi-agent system in which over 10 specialized AI agents collaborate to propose, critique, and refine SciML solutions through structured reasoning and iterative evolution. The framework integrates structured debate, retrieval-augmented method memory, and ensemble-guided evolutionary search, enabling the agents to generate and assess new hypotheses about architectures and optimization procedures. Across physics-informed learning and operator learning tasks, the framework discovers solution methods that outperform single-agent and human-designed baselines by up to four orders of magnitude in error reduction. The agents produce novel strategies -- including adaptive mixture-of-expert architectures, decomposition-based PINNs, and physics-informed operator learning models -- that do not appear explicitly in the curated knowledge base. These results show that collaborative reasoning among AI agents can yield emergent methodological innovation, suggesting a path toward scalable, transparent, and autonomous discovery in scientific computing.

cs.AI↗

arXiv2026

Nonlinear parametrization solver for fractional Burgers equations

Fractional Burgers equations pose substantial challenges for classical numerical methods due to the combined effects of nonlocality and shock-forming nonlinear dynamics. In particular, linear approximation frameworks-such as spectral, finite-difference, or discontinuous Galerkin methods-often suffer from Gibbs-type oscillations or require carefully tuned stabilization mechanisms, whose effectiveness degrades in transport-dominated and long-time integration regimes. In this work, we introduce a sequential-in-time nonlinear parametrization (STNP) for solving fractional Burgers equations, including models with a fractional Laplacian or with nonlocal nonlinear fluxes. The solution is represented by a nonlinear parametric ansatz, and the parameter evolution is obtained by projecting the governing dynamics onto the tangent space of the parameter manifold through a regularized least-squares formulation at each time step. This yields a well-posed and stable time-marching scheme that preserves causality and avoids global-in-time optimization. We provide a theoretical analysis of the resulting projected dynamics, including a stability estimate and an a posteriori error bound that explicitly decomposes the total error into contributions from initial condition fitting, projection residuals, and discretization of fractional operators. Our analysis clarifies the stabilizing role of regularization and quantifies its interaction with the nonlocal discretization error. Numerical experiments for both fractional Burgers models demonstrate that STNP achieves oscillation-free shock resolution and accurately captures long-time dynamics. The method consistently outperforms high-order spectral schemes augmented with spectral vanishing viscosity, while requiring significantly fewer degrees of freedom and avoiding ad hoc stabilization.

math.NA↗

arXiv2025

PINNs4Drops: Video-conditioned physics-informed neural networks for two-phase flow reconstruction

Two-phase flow phenomena underpin critical technologies such as hydrogen fuel cells, spray cooling, and combustion, where droplet dynamics govern performance and efficiency. Conventional optical diagnostics, including shadowgraphy and particle image velocimetry, provide valuable insights but are limited to two-dimensional projections of inherently three-dimensional flows. We employ a specialized optical technique that encodes droplet surface information through color-coded glare points, enabling enhanced reconstruction of gas-liquid interfaces. To interpret these measurements, we introduce video-conditioned physics-informed neural networks (VcPINNs), which integrate experimental observations with governing fluid dynamics equations. This hybrid framework leverages the strengths of both data-driven learning and physical constraints, allowing accurate volumetric flow reconstruction from limited input images. Applied to droplet impingement experiments, our method yields highly resolved and physically consistent 3D interface and flow dynamics. The combined imaging and PINN reconstruction strategy provides a powerful platform for advancing multiphase-flow analysis, with broad potential impact across energy, cooling, and propulsion applications.

physics.flu-dyn↗

arXiv2025

Safe cross-entropy-based importance sampling for rare event simulations

The Improved Cross-Entropy (ICE) method is a powerful tool for estimating failure probabilities in reliability analysis. Its core idea is to approximate the optimal importance-sampling density by minimizing the forward Kullback-Leibler divergence within a chosen parametric family-typically a mixture model. However, conventional mixtures are often light-tailed, which leads to slow convergence and instability when targeting very small failure probabilities. Moreover, selecting the number of mixture components in advance can be difficult and may undermine stability. To overcome these challenges, we adopt a weighted cross-entropy-penalized expectation-maximization (EM) algorithm that automatically prunes redundant components during the iterative process, making the approach more stable. Furthermore, we introduce a novel two-component mixture that pairs a light-tailed distribution with a heavy-tailed one, enabling more effective exploration of the tail region and thus accelerating convergence for extremely small failure probabilities. We call the resulting method Safe-ICE and assess it on a variety of test problems. Numerical results show that Safe-ICE not only converges more rapidly and yields more accurate failure-probability estimates than standard ICE, but also identifies the appropriate number of mixture components without manual tuning.

math.NA↗

arXiv2025

XAI4Extremes: An interpretable machine learning framework for understanding extreme-weather precursors under climate change

Extreme weather events are increasing in frequency and intensity due to climate change. This, in turn, is exacting a significant toll in communities worldwide. While prediction skills are increasing with advances in numerical weather prediction and artificial intelligence tools, extreme weather still present challenges. More specifically, identifying the precursors of such extreme weather events and how these precursors may evolve under climate change remain unclear. In this paper, we propose to use post-hoc interpretability methods to construct relevance weather maps that show the key extreme-weather precursors identified by deep learning models. We then compare this machine view with existing domain knowledge to understand whether deep learning models identified patterns in data that may enrich our understanding of extreme-weather precursors. We finally bin these relevant maps into different multi-year time periods to understand the role that climate change is having on these precursors. The experiments are carried out on Indochina heatwaves, but the methodology can be readily extended to other extreme weather events worldwide.

cs.LG↗

arXiv2024

Pharmacometrics Modeling via Physics-Informed Neural Networks: Integrating Time-Variant Absorption Rates and Fractional Calculus for Enhancing Prediction Accuracy

We present a novel method to improve pharmacokinetics modeling, an essential step of drug development. Conventional models frequently fail to fully represent the intricacies of drug absorption and distribution, which limits their predictive abilities required for personalized treatment strategies. Our methodology introduces two innovations to enhance modeling accuracy: 1. Time-varying parameters: this approach is designed to accommodate the dynamic nature of drug absorption rates. 2. Fractional calculus in representing delayed drug response. This approach effectively captures anomalous diffusion phenomena, surpassing traditional models in describing drug delayed response without the need for extensive compartmentalization.

q-bio.QM↗

arXiv2024

CMINNs: Compartment Model Informed Neural Networks -- Unlocking Drug Dynamics

In the field of pharmacokinetics and pharmacodynamics (PKPD) modeling, which plays a pivotal role in the drug development process, traditional models frequently encounter difficulties in fully encapsulating the complexities of drug absorption, distribution, and their impact on targets. Although multi-compartment models are frequently utilized to elucidate intricate drug dynamics, they can also be overly complex. To generalize modeling while maintaining simplicity, we propose an innovative approach that enhances PK and integrated PK-PD modeling by incorporating fractional calculus or time-varying parameter(s), combined with constant or piecewise constant parameters. These approaches effectively model anomalous diffusion, thereby capturing drug trapping and escape rates in heterogeneous tissues, which is a prevalent phenomenon in drug dynamics. Furthermore, this method provides insight into the dynamics of drug in cancer in multi-dose administrations. Our methodology employs a Physics-Informed Neural Network (PINN) and fractional Physics-Informed Neural Networks (fPINNs), integrating ordinary differential equations (ODEs) with integer/fractional derivative order from compartmental modeling with neural networks. This integration optimizes parameter estimation for variables that are time-variant, constant, piecewise constant, or related to the fractional derivative order. The results demonstrate that this methodology offers a robust framework that not only markedly enhances the model's depiction of drug absorption rates and distributed delayed responses but also unlocks different drug-effect dynamics, providing new insights into absorption rates, anomalous diffusion, drug resistance, peristance and pharmacokinetic tolerance, all within a system of just two (fractional) ODEs with explainable results.

q-bio.QM↗

arXiv2024

A time-dependent symplectic network for non-convex path planning problems with linear and nonlinear dynamics

We propose a novel neural network architecture (TSympOCNet) to address high--dimensional optimal control problems with linear and nonlinear dynamics. An important application of this method is to solve the path planning problem of multi-agent vehicles in real time. The new method extends our previous SympOCNet framework by introducing a time-dependent symplectic network into the architecture. In addition, we propose a more general latent representation, which greatly improves model expressivity based on the universal approximation theorem. We demonstrate the efficacy of TSympOCNet in path planning problems with obstacle and collision avoidance, including systems with Newtonian dynamics and non-convex environments, up to dimension 512. Our method shows significant promise in handling efficiently both complex dynamics and constraints.

math.OC↗

arXiv2024

Hypersonic Boundary Layer Transition and Heat Loading

Hypersonic boundary layer transition using high-order methods for direct numerical simulations (DNS) is largely unexplored, although a few references exist in the literature. Experimental data in the hypersonic regime are scarce, while almost all existing hypersonic codes have low-order accuracy, which could lead to erroneous results in long-time integration and at high Reynolds numbers. Here, we focus on the transition from laminar to turbulent flow, where the Nusselt number may be five times or higher than the Nusselt number in the turbulent regime. The hypersonic flow regime must be modeled accurately using realistic chemistry to predict heat flux on the surface correctly. In this study, we simulate hypersonic boundary layer transition on a flat plate to compute the thermal and shear stresses on the wall. The domain is initialized with a laminar Blasius solution of compressible boundary layer equations. The transition to turbulence is induced using suction and blowing from a thin strip on the wall. Sponge regions are defined at the inflow and outflow boundaries to eliminate the solution contamination due to the reflections of the boundary conditions. In the spanwise direction, periodic boundary conditions are imposed. The numerical method applied here is a sixth-order compact finite difference in the interior of the domain coupled with a fourth-order Runge-Kutta time-stepping scheme for a structured Cartesian grid using staggered variables. The base finite difference code can perform simulations of non-equilibrium hypersonic flows. The numerical scheme is stabilized using the approximate deconvolution model (ADM) and artificial diffusion coefficients. We investigated the effect of non-equilibrium chemistry on the non-linear instability growth with hypersonic boundary layers. We also quantify thermal loads during the boundary layer laminar to turbulent transition for Mach number 10.

physics.flu-dyn↗

arXiv2024

Deep operator learning-based surrogate models for aerothermodynamic analysis of AEDC hypersonic waverider

Neural networks are universal approximators that traditionally have been used to learn a map between function inputs and outputs. However, recent research has demonstrated that deep neural networks can be used to approximate operators, learning function-to-function mappings. Creating surrogate models to supplement computationally expensive hypersonic aerothermodynamic models in characterizing the response of flow fields at different angles of attack (AoA) is an ideal application of neural operators. We investigate the use of neural operators to infer flow fields (volume and surface quantities) around a geometry based on a 3D waverider model based on experimental data measured at the Arnold Engineering Development Center (AEDC) Hypervelocity Wind Tunnel Number 9. We use a DeepONet neural operator which consists of two neural networks, commonly called a branch and a trunk network. The final output is the inner product of the output of the branch network and the output of the trunk net. Because the flow field contains shocks across the entire volume, we conduct a two-step training approach of the DeepONet that facilitates accurate approximation of solutions even in the presence of discontinuities. We train various DeepONet models to understand and predict pressure $(p)$, density $(ρ)$, velocity $(u)$, heat flux $(Q_w)$, and total shear stress $(τ_{w})$ for the AEDC waverider geometry at Ma=7.36 across AoA that range from $-10^{\circ}$ to $10^{\circ}$ for surface quantities and from $-14^{\circ}$ to $14^{\circ}$ for volume quantities.

physics.flu-dyn↗

arXiv2024

Physics-Informed Computer Vision: A Review and Perspectives

The incorporation of physical information in machine learning frameworks is opening and transforming many application domains. Here the learning process is augmented through the induction of fundamental knowledge and governing physical laws. In this work, we explore their utility for computer vision tasks in interpreting and understanding visual data. We present a systematic literature review of more than 250 papers on formulation and approaches to computer vision tasks guided by physical laws. We begin by decomposing the popular computer vision pipeline into a taxonomy of stages and investigate approaches to incorporate governing physical equations in each stage. Existing approaches in computer vision tasks are analyzed with regard to what governing physical processes are modeled and formulated, and how they are incorporated, i.e. modification of input data (observation bias), modification of network architectures (inductive bias), and modification of training losses (learning bias). The taxonomy offers a unified view of the application of the physics-informed capability, highlighting where physics-informed learning has been conducted and where the gaps and opportunities are. Finally, we highlight open problems and challenges to inform future research. While still in its early days, the study of physics-informed computer vision has the promise to develop better computer vision models that can improve physical plausibility, accuracy, data efficiency, and generalization in increasingly realistic applications.

eess.IV↗

arXiv2024

Bridging scales in multiscale bubble growth dynamics with correlated fluctuations using neural operator learning

The intricate process of bubble growth dynamics involves a broad spectrum of physical phenomena from microscale mechanics of bubble formation to macroscale interplay between bubbles and surrounding thermo-hydrodynamics. Traditional bubble dynamics models including atomistic approaches and continuum-based methods segment the bubble dynamics into distinct scale-specific models. In order to bridge the gap between microscale stochastic fluid models and continuum-based fluid models for bubble dynamics, we develop a composite neural operator model to unify the analysis of nonlinear bubble dynamics across microscale and macroscale regimes by integrating a many-body dissipative particle dynamics (mDPD) model with a continuum-based Rayleigh-Plesset (RP) model through a novel neural network architecture, which consists of a deep operator network for learning the mean behavior of bubble growth subject to pressure variations and a long short-term memory network for learning the statistical features of correlated fluctuations in microscale bubble dynamics. Training and testing data are generated by conducting mDPD and RP simulations for nonlinear bubble dynamics with initial bubble radii ranging from 0.1 to 1.5 micrometers. Results show that the trained composite neural operator model can accurately predict bubble dynamics across scales, with a 99% accuracy for the time evaluation of the bubble radius under varying external pressure while containing correct size-dependent stochastic fluctuations in microscale bubble growth dynamics. The composite neural operator is the first deep learning surrogate for multiscale bubble growth dynamics that can capture correct stochastic fluctuations in microscopic fluid phenomena, which sets a new direction for future research in multiscale fluid dynamics modeling.

physics.flu-dyn↗

arXiv2023

An enrichment approach for enhancing the expressivity of neural operators with applications to seismology

The Eikonal equation plays a central role in seismic wave propagation and hypocenter localization, a crucial aspect of efficient earthquake early warning systems. Despite recent progress, real-time earthquake localization remains challenging due to the need to learn a generalizable Eikonal operator. We introduce a novel deep learning architecture, Enriched-DeepONet (En-DeepONet), addressing the limitations of current operator learning models in dealing with moving-solution operators. Leveraging addition and subtraction operations and a novel `root' network, En-DeepONet is particularly suitable for learning such operators and achieves up to four orders of magnitude improved accuracy without increased training cost. We demonstrate the effectiveness of En-DeepONet in earthquake localization under variable velocity and arrival time conditions. Our results indicate that En-DeepONet paves the way for real-time hypocenter localization for velocity models of practical interest. The proposed method represents a significant advancement in operator learning that is applicable to a gamut of scientific problems, including those in seismology, fracture mechanics, and phase-field problems.

cs.LG↗

arXiv2023

Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron models

We introduce a new approach for solving forward systems of differential equations using a combination of splitting methods and physics-informed neural networks (PINNs). The proposed method, splitting PINN, effectively addresses the challenge of applying PINNs to forward dynamical systems and demonstrates improved accuracy through its application to neuron models. Specifically, we apply operator splitting to decompose the original neuron model into sub-problems that are then solved using PINNs. Moreover, we develop an $L^1$ scheme for discretizing fractional derivatives in fractional neuron models, leading to improved accuracy and efficiency. The results of this study highlight the potential of splitting PINNs in solving both integer- and fractional-order neuron models, as well as other similar systems in computational science and engineering.

math.NA↗

arXiv2022

High-Order Methods for Hypersonic Flows with Strong Shocks and Real Chemistry

We compare high-order methods including spectral difference (SD), flux reconstruction (FR), the entropy-stable discontinuous Galerkin spectral element method (ES-DGSEM), modal discontinuous Galerkin methods, and WENO to select the best candidate to simulate strong shock waves characteristic of hypersonic flows. We consider several benchmarks, including the Leblanc and modified shock-density wave interaction problems that require robust stabilization and positivity-preserving properties for a successful flow realization. We also perform simulations of the three-species Sod problem with simplified chemistry with the chemical reaction source terms introduced in the Euler equations. The ES-DGSEM scheme exhibits the highest stability, negligible numerical oscillations, and requires the least computational effort in resolving reactive flow regimes with strong shock waves. Therefore, we extend the ES-DGSEM to hypersonic Euler equations by deriving a new set of two-point entropy conservative fluxes for a five-species gas model. Stabilization for capturing strong shock waves occurs by blending high-order entropy conservative fluxes with low-order finite volume fluxes constructed using the HLLC Riemann solver. The hypersonic Euler solver is verified using the non-equilibrium chemistry Sod problem. To this end, we adopt the Mutation++ library to compute the reaction source terms, thermodynamic properties, and transport coefficients. We also investigate the effect of real chemistry versus ideal chemistry, and the results demonstrate that the ideal chemistry assumption fails at high temperatures, hence real chemistry must be employed for accurate predictions. Finally, we consider a viscous hypersonic flow problem to verify the transport coefficients and reaction source terms determined by the Mutation++ library.

physics.flu-dyn↗