George Karniadakis

Ehsan Haghighat, Umair bin Waheed, George Karniadakis

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.

Simin Shekarpaz, Fanhai Zeng, George Karniadakis

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.

Nazanin Ahmadi Daryakenari, Shupeng Wang, George Karniadakis

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.

Qile Jiang, George Karniadakis

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.

Zhiwei Gao, George Karniadakis

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.

Minglei Lu, Chensen Lin, Martian Maxey et al.

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.

Jiawen Wei, Aniruddha Bora, Vivek Oommen et al.

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.

Chayan Banerjee, Kien Nguyen, Clinton Fookes et al.

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.

Marta D'Elia, Hang Deng, Cedric Fraces et al.

The "Workshop on Machine learning in heterogeneous porous materials" brought together international scientific communities of applied mathematics, porous media, and material sciences with experts in the areas of heterogeneous materials, machine learning (ML) and applied mathematics to identify how ML can advance materials research. Within the scope of ML and materials research, the goal of the workshop was to discuss the state-of-the-art in each community, promote crosstalk and accelerate multi-disciplinary collaborative research, and identify challenges and opportunities. As the end result, four topic areas were identified: ML in predicting materials properties, and discovery and design of novel materials, ML in porous and fractured media and time-dependent phenomena, Multi-scale modeling in heterogeneous porous materials via ML, and Discovery of materials constitutive laws and new governing equations. This workshop was part of the AmeriMech Symposium series sponsored by the National Academies of Sciences, Engineering and Medicine and the U.S. National Committee on Theoretical and Applied Mechanics.

Somdatta Goswami, Minglang Yin, Yue Yu et al.

Failure trajectories, identifying the probable failure zones, and damage statistics are some of the key quantities of relevance in brittle fracture applications. High-fidelity numerical solvers that reliably estimate these relevant quantities exist but they are computationally demanding requiring a high resolution of the crack. Moreover, independent intensive simulations need to be carried out even for a small change in domain parameters and/or material properties. Therefore, fast and generalizable surrogate models are needed to alleviate the computational burden but the discontinuous nature of fracture mechanics presents a major challenge to developing such models. We propose a physics-informed variational formulation of DeepONet (V-DeepONet) for brittle fracture analysis. V-DeepONet is trained to map the initial configuration of the defect to the relevant fields of interests (e.g., damage and displacement fields). Once the network is trained, the entire global solution can be rapidly obtained for any initial crack configuration and loading steps on that domain. While the original DeepONet is solely data-driven, we take a different path to train the V-DeepONet by imposing the governing equations in variational form and we also use some labelled data. We demonstrate the effectiveness of V-DeepOnet through two benchmarks of brittle fracture, and we verify its accuracy using results from high-fidelity solvers. Encoding the physical laws and also some data to train the network renders the surrogate model capable of accurately performing both interpolation and extrapolation tasks, considering that fracture modeling is very sensitive to fluctuations. The proposed hybrid training of V-DeepONet is superior to state-of-the-art methods and can be applied to a wide array of dynamical systems with complex responses.

José del Águila Ferrandis, Michael Triantafyllou, Chryssostomos Chryssostomidis et al.

Predicting motions of vessels in extreme sea states represents one of the most challenging problems in naval hydrodynamics. It involves computing complex nonlinear wave-body interactions, hence taxing heavily computational resources. Here, we put forward a new simulation paradigm by training recurrent type neural networks (RNNs) that take as input the stochastic wave elevation at a certain sea state and output the main vessel motions, e.g., pitch, heave and roll. We first compare the performance of standard RNNs versus GRU and LSTM neural networks (NNs) and show that LSTM NNs lead to the best performance. We then examine the testing error of two representative vessels, a catamaran in sea state 1 and a battleship in sea state 8. We demonstrate that good accuracy is achieved for both cases in predicting the vessel motions for unseen wave elevations. We train the NNs with expensive CFD simulations offline, but upon training, the prediction of the vessel dynamics online can be obtained at a fraction of a second. This work is motivated by the universal approximation theorem for functionals [1], and it is the first implementation of such theory to realistic engineering problems.

Liu Yang, Sean Treichler, Thorsten Kurth et al.

Uncertainty quantification for forward and inverse problems is a central challenge across physical and biomedical disciplines. We address this challenge for the problem of modeling subsurface flow at the Hanford Site by combining stochastic computational models with observational data using physics-informed GAN models. The geographic extent, spatial heterogeneity, and multiple correlation length scales of the Hanford Site require training a computationally intensive GAN model to thousands of dimensions. We develop a hierarchical scheme for exploiting domain parallelism, map discriminators and generators to multiple GPUs, and employ efficient communication schemes to ensure training stability and convergence. We developed a highly optimized implementation of this scheme that scales to 27,500 NVIDIA Volta GPUs and 4584 nodes on the Summit supercomputer with a 93.1% scaling efficiency, achieving peak and sustained half-precision rates of 1228 PF/s and 1207 PF/s.

Mamikon Gulian, Maziar Raissi, Paris Perdikaris et al.

Data-driven discovery of "hidden physics" -- i.e., machine learning of differential equation models underlying observed data -- has recently been approached by embedding the discovery problem into a Gaussian Process regression of spatial data, treating and discovering unknown equation parameters as hyperparameters of a modified "physics informed" Gaussian Process kernel. This kernel includes the parametrized differential operators applied to a prior covariance kernel. We extend this framework to linear space-fractional differential equations. The methodology is compatible with a wide variety of fractional operators in $\mathbb{R}^d$ and stationary covariance kernels, including the Matern class, and can optimize the Matern parameter during training. We provide a user-friendly and feasible way to perform fractional derivatives of kernels, via a unified set of d-dimensional Fourier integral formulas amenable to generalized Gauss-Laguerre quadrature. The implementation of fractional derivatives has several benefits. First, it allows for discovering fractional-order PDEs for systems characterized by heavy tails or anomalous diffusion, bypassing the analytical difficulty of fractional calculus. Data sets exhibiting such features are of increasing prevalence in physical and financial domains. Second, a single fractional-order archetype allows for a derivative of arbitrary order to be learned, with the order itself being a parameter in the regression. This is advantageous even when used for discovering integer-order equations; the user is not required to assume a "dictionary" of derivatives of various orders, and directly controls the parsimony of the models being discovered. We illustrate on several examples, including fractional-order interpolation of advection-diffusion and modeling relative stock performance in the S&P 500 with alpha-stable motion via a fractional diffusion equation.

Maziar Raissi, George Karniadakis

We develop a novel multi-fidelity framework that goes far beyond the classical AR(1) Co-kriging scheme of Kennedy and O'Hagan (2000). Our method can handle general discontinuous cross-correlations among systems with different levels of fidelity. A combination of multi-fidelity Gaussian Processes (AR(1) Co-kriging) and deep neural networks enables us to construct a method that is immune to discontinuities. We demonstrate the effectiveness of the new technology using standard benchmark problems designed to resemble the outputs of complicated high- and low-fidelity codes.