We highlight here some exciting recent preprints. At the end of this page, you can find the full list of publications.
You can find open-source software for most recent papers on our group’s Github.
Multiscale systems are difficult to simulate since fine-scale dynamics must be linked to emergent bulk physics. Coarse-graining high-dimensional dynamics inevitably produces dissipative, history-dependent, and stochastic behavior. We propose a metriplectic-bracket-based framework that learns such coarse-grained dynamics from particle trajectories while guaranteeing thermodynamic consistency, momentum conservation, and fluctuation–dissipation balance. A self-supervised strategy identifies emergent structural variables without labels. We validate the method on benchmark systems, star polymers, and colloidal suspensions, and release open-source implementations in PyTorch and LAMMPS for large-scale inference.
Quercus Hernandez, Max Win, Thomas C. O’Connor, Paulo E. Arratia, Nathaniel Trask
arXiv preprint, arXiv:2508.12569 (Under review)
We propose a framework for real-time digital twins using structure-preserving reduced finite element models conditioned on a latent variable. Conditional attention learns both a reduced basis and nonlinear conservation laws within finite element exterior calculus, ensuring well-posedness and exact conservation despite sparse data. The method enables real-time calibration to sensor data, integrates seamlessly with standard finite element tools, and handles complex geometries. Benchmarks from advection–diffusion to battery thermal runaway show accurate predictions with only 25 LES simulations and real-time inference (~0.1s, 3.1×10^8 speedup).
Brooks Kinch, Benjamin Shaffer, Elizabeth Armstrong, Michael Meehan, John Hewson, Nathaniel Trask
arXiv preprint, arXiv:2508.06981 (Under review)
Data-driven particle dynamics: Structure-preserving coarse-graining for emergent behavior in non-equilibrium systems
Quercus Hernandez, Max Win, Thomas C. O’Connor, Paulo E. Arratia, Nathaniel Trask
arXiv preprint, arXiv:2508.12569 (Under review)
Structure-Preserving Digital Twins via Conditional Neural Whitney Forms
Brooks Kinch, Benjamin Shaffer, Elizabeth Armstrong, Michael Meehan, John Hewson, Nathaniel Trask
arXiv preprint, arXiv:2508.06981 (Under review)
A discontinuous piecewise polynomial generalized moving least squares scheme for robust finite element analysis on arbitrary grids
Paul Kuberry, Pavel Bochev, Jeffrey Koester, Nathaniel Trask
Engineering with Computers 41(3):1535–1554 (2025)
Unsupervised physics-informed disentanglement of multimodal materials data
Nathaniel Trask, Carianne Martinez, Trevor Shilt, Ethan Walker, Kookjin Lee, Andrew Garland, David P. Adams, John F. Curry, Matthew T. Dugger, Seth R. Larson, Bradley L. Boyce
Materials Today 80:286–296 (2024)
Efficiently Parameterized Neural Metriplectic Systems
Anthony Gruber, Kookjin Lee, Hyungjin Lim, Namsu Park, Nathaniel Trask
arXiv:2405.16305 (Accepted to ICLR 2025)
Spiking Physics-Informed Neural Networks on Loihi 2
Benjamin H. Theilman, Qi Zhang, Ariel Kahana, Eric C. Cyr, Nathaniel Trask, James B. Aimone, George E. Karniadakis
NICE 2024, IEEE, pp. 1–6
A discontinuous piecewise polynomial generalized moving least squares scheme for robust finite element analysis on arbitrary grids
Paul Kuberry, Pavel Bochev, Jeffrey Koester, Nathaniel Trask
Engineering with Computers (2024):1–20
Flow-based parameterization for DAG and feature discovery in scientific multimodal data
Ethan Walker, Jacob A. Actor, Carianne Martinez, Nathaniel Trask
Frontiers in Mechanical Engineering 10:1408649 (2024)
Graph Convolutions Enrich the Self-Attention in Transformers!
Jeongwhan Choi, Hyowon Wi, Jayoung Kim, Yehjin Shin, Kookjin Lee, Nathaniel Trask, Noseong Park
arXiv:2312.04234 (Accepted to NeurIPS 2024)
Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation
Xinhui Wu, Nathaniel Trask, Jesse Chan
Numerical Methods for Partial Differential Equations 40(6):e23129 (2024)
Unsupervised physics-informed disentanglement of multimodal data
Ethan Walker, Nathaniel Trask, Carianne Martinez, Kookjin Lee, Jacob A. Actor, Sourav Saha, Trevor Shilt, Daniel Vizoso, Remi Dingreville, Bradley L. Boyce
Foundations of Data Science (2024)
Reversible and irreversible bracket-based dynamics for deep graph neural networks
Anthony Gruber, Kookjin Lee, Nathaniel Trask
NeurIPS 36 (2024)
Data-driven Whitney forms for structure-preserving control volume analysis
Jacob A. Actor, Xiaozhe Hu, Andy Huang, Scott A. Roberts, Nathaniel Trask
Journal of Computational Physics 496:112520 (2024)
A stable mimetic finite-difference method for convection-dominated diffusion equations
James H. Adler, Casey Cavanaugh, Xiaozhe Hu, Andy Huang, Nathaniel Trask
SIAM Journal on Scientific Computing 45(6):A2973–A3000 (2023)
Special issue of computational mechanics on machine learning theories, modeling, and applications to computational materials science, additive manufacturing, mechanics of materials, design and optimization
Wing Kam Liu, Miguel A. Bessa, Francisco Chinesta, Shaofan Li, Nathaniel Trask
Computational Mechanics (2023) 72:1–2 Editorial
Probabilistic partition of unity networks for high-dimensional regression problems
Tiffany Fan, Nathaniel Trask, Marta D’Elia, Eric Darve
International Journal for Numerical Methods in Engineering 124(10):2215–2236 (2023)
Design of experiments for the calibration of history-dependent models via deep reinforcement learning and an enhanced Kalman filter
R. Villarreal, N. N. Vlassis, N. N. Phan, T. A. Catanach, R. E. Jones, Nathaniel A. Trask, S. L. Kramer, W. Sun
Computational Mechanics 72(1):95–124 (2023)
Accurate compression of tabulated chemistry models with Partition-of-Unity Networks
Elizabeth Armstrong, Michael A. Hansen, Robert C. Knaus, Nathaniel A. Trask, John C. Hewson, James C. Sutherland
Combustion Science and Technology (2022)
Structure-preserving sparse identification of nonlinear dynamics for data-driven modeling
Kookjin Lee, Nathaniel Trask, Panos Stinis
Mathematical and Scientific Machine Learning (PMLR 190), 2022
Efficient optimization-based quadrature for variational discretization of nonlocal problems
M. Pasetto, Z. Shen, Marta D’Elia, Xiaochuan Tian, Nathaniel Trask, Daniel Kamensky
Computer Methods in Applied Mechanics and Engineering 396:115104 (2022)
Scalable algorithms for physics-informed neural and graph networks
Khemraj Shukla, Minglang Xu, Nathaniel Trask, George Karniadakis
Data-Centric Engineering 3:e24 (2022)
Hierarchical Partition-of-Unity Networks: fast multilevel training
Nathaniel Trask, Amelia Henriksen, Carianne Martinez, Eric Cyr
Proceedings of Machine Learning Research 145 (2022):1–18
A general-purpose, inelastic, rotation-free Kirchhoff–Love shell formulation for peridynamics
Masoud Behzadinasab, Mert Alaydin, Nathaniel Trask, Yuri Bazilevs
Computer Methods in Applied Mechanics and Engineering 389:114422 (2022)
Enforcing exact physics in scientific machine learning: a data-driven exterior calculus on graphs
Nathaniel Trask, Andrew Huang, Xujiao Hu
Journal of Computational Physics 110969 (2022)
Coupling of IGA and Peridynamics for air-blast fluid–structure interaction using an immersed approach
Masoud Behzadinasab, Georgios Moutsanidis, Nathaniel A. Trask, John T. Foster, Yuri Bazilevs
Forces in Mechanics 4:100045 (2021)
Machine learning structure-preserving brackets for forecasting irreversible processes
Kookjin Lee, Nathaniel A. Trask, Panos Stinis
NeurIPS 34 (2021)
An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture
Yue Yu, Huaiqian You, Nathaniel A. Trask
Computer Methods in Applied Mechanics and Engineering 377:113691 (2021)
A unified, stable and accurate meshfree framework for peridynamic correspondence modeling — Part I: core methods
Masoud Behzadinasab, Nathaniel Trask, Yuri Bazilevs
Journal of Peridynamics and Nonlocal Modeling 3(1):24–45 (2021)
Thermodynamically consistent physics-informed neural networks for hyperbolic systems
Ravi G. Patel, Indu Manickam, Nathaniel A. Trask, Mitchell A. Wood, Minseok Lee, Isaac Tomas, Eric C. Cyr
Journal of Computational Physics 449:110754 (2022)
Data-driven learning of nonlocal physics from high-fidelity synthetic data
Huaiqian You, Yue Yu, Nathaniel A. Trask, Mamikon Gulian, Marta D’Elia
Computer Methods in Applied Mechanics and Engineering 374:113553 (2021)
Physics-Informed Graph Neural Network for circuit compact model development
Xujiao Gao, Andy Huang, Nathaniel Trask, Shahed Reza
SISPAD 2020, pp. 359–362
Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion
Yu Leng, Xiaochuan Tian, Nathaniel Trask, John T. Foster
SIAM Journal on Numerical Analysis 59(1):88–118 (2021)
Interface flux recovery coupling method for the ocean–atmosphere system
K. C. Sockwell, K. Peterson, Paul Kuberry, Pavel Bochev, Nathaniel A. Trask
Results in Applied Mathematics 7:100116 (2020)
Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation
Yu Leng, Xiaochuan Tian, Nathaniel A. Trask, John T. Foster
Computer Methods in Applied Mechanics and Engineering 370:113264 (2020)
Robust training and initialization of deep neural networks: An adaptive basis viewpoint
Eric C. Cyr, Mamikon Gulian, Ravi G. Patel, Mauro Perego, Nathaniel A. Trask
Mathematical and Scientific Machine Learning (PMLR 107):512–536 (2020)
An asymptotically compatible approach for Neumann-type boundary conditions on nonlocal problems
Huaiqian You, Xiantao Lu, Nathaniel A. Trask, Yue Yu
ESAIM: M2AN 54(4):1373–1413 (2020)
Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares approach
Benjamin J. Gross, Nathaniel A. Trask, Paul Kuberry, Peter J. Atzberger
Journal of Computational Physics 409:109340 (2020)
A conservative, consistent, and scalable meshfree mimetic method
Nathaniel A. Trask, Pavel Bochev, Mauro Perego
Journal of Computational Physics 409:109187 (2020)
Compatible meshfree discretization of surface PDEs
Nathaniel A. Trask, Paul Kuberry
Computational Particle Mechanics 7(2):271–277 (2020)
Optimization Based Particle-Mesh Algorithm for High-Order and Conservative Scalar Transport
J. M. Maljaars, R. J. Labeur, Nathaniel A. Trask, Deborah L. Sulsky
Lecture Notes in Computational Science and Engineering 132:265–275 (2020)
Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation–dissipation balance
William Pazner, Nathaniel A. Trask, Peter J. Atzberger
Results in Applied Mathematics 4:100014 (2019)
A spatially adaptive high-order meshless method for fluid–structure interactions
Weiming Hu, Nathaniel A. Trask, Xujiao Hu, Wenxiao Pan
Computer Methods in Applied Mechanics and Engineering 355:112657 (2019)
Mitigation of the self-force effect in unstructured PIC codes using generalized moving least squares
Nathaniel A. Trask, Pavel Bochev, Mauro Perego
Computers & Mathematics with Applications 78(2):688–705 (2019)
Mesh-hardened finite element analysis through a Generalized Moving Least-Squares approximation of variational problems
Pavel Bochev, Nathaniel A. Trask, Paul Kuberry, Mauro Perego
Large-Scale Scientific Computing, LNCS 11958:67–75 (2019)
Conservative, high-order particle–mesh scheme with applications to advection-dominated flows
J. M. Maljaars, R. J. Labeur, Nathaniel A. Trask, Deborah Sulsky
Computer Methods in Applied Mechanics and Engineering 348:443–465 (2019)
An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics
Nathaniel A. Trask, Huaiqian You, Yue Yu, Michael L. Parks
Computer Methods in Applied Mechanics and Engineering 343:151–165 (2019)
A compatible high-order meshless method for the Stokes equations with applications to suspension flows
Nathaniel A. Trask, Michael Maxey, Xujiao Hu
Journal of Computational Physics 355:310–329 (2018)
A high-order staggered meshless method for elliptic problems
Nathaniel A. Trask, Mauro Perego, Pavel Bochev
SIAM Journal on Scientific Computing 39(2):A479–A502 (2017)
Compact moving least squares: an optimization framework for generating high-order compact meshless discretizations
Nathaniel A. Trask, Michael Maxey, Xujiao Hu
Journal of Computational Physics 326:596–611 (2016)
Intercomparison of 3D pore-scale flow and solute transport simulation methods
Xin Yang, Yashar Mehmani, William A. Perkins, Alberto Pasquali, Matthias Schonherr, Kang Kim, Mauro Perego, Michael L. Parks, Nathaniel A. Trask, Matthew T. Balhoff, Michael C. Richmond
Advances in Water Resources 95:176–189 (2016)
Smoothed particle hydrodynamics and its applications for multiphase flow and reactive transport in porous media
Alexander M. Tartakovsky, Nathaniel A. Trask, Kang Pan, Benjamin Jones, Wenxiao Pan, James R. Williams
Computational Geosciences 20(4):807–834 (2016)
A scalable consistent second-order SPH solver for unsteady low Reynolds number flows
Nathaniel A. Trask, Michael Maxey, et al.
Computer Methods in Applied Mechanics and Engineering 296:305–326 (2015)
Diesel spray CFD simulations based on the Sigma–Y Eulerian atomization model
José M. García-Oliver, José M. Pastor, Ankur Pandal, Nathaniel A. Trask, Eric Baldwin, David P. Schmidt
Atomization and Sprays 23(1):71–95 (2013)
Compressible modeling of the internal two-phase flow in a gas-centered swirl coaxial fuel injector
Nathaniel A. Trask, David P. Schmidt, Matthew Lightfoot, Stephen Danczyk
Journal of Propulsion and Power 28(4):685–693 (2012)
Multidimensional modeling of condensing two-phase ejector flow
Michael Colarossi, Nathaniel A. Trask, David P. Schmidt, Michael J. Bergander
International Journal of Refrigeration 35(2):396–408 (2012)
