The PhILMs team measures its success, in part, on its fundamental contributions to scientific literature, focusing primarily on mathematics and materials science, as well as the generation and dissemination of algorithms and open-source software.

2023

He Q, M Perego, AA Howard, GE Karniadakis, and P Stinis. 2023. "A Hybrid Deep Neural Operator/Finite Element Method for Ice-Sheet Modeling." Journal of Computational Physics, Vol 492. DOI: 10.1016/j.jcp.2023.112428.

Psaros AF, X Meng, Z Zou, L Guo, and GM Karniadakis. 2023. “Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons." Journal of Computational Physics, 477: 111902, DOI: 10.1016/j.jcp.2022.111902.

Shin, Y, J Darbon, and GE Karniadakis. 2023. “Accelerating Gradient Descent and Adam via Fractional Gradients." Neural Networks, 161: 185–201, DOI: 10.1016/j.neunet.2023.01.002.

2022

Buczkowski NE, MD Foss, ML Parks, and P Radu. 2022. “Sensitivity Analysis for Solutions to Heterogeneous Nonlocal Systems. Theoretical and Numerical Studies.” Journal of Peridyn Nonlocal Model, 4: 367–397, DOI: 10.1007/s42102-022-00081-6.

Darve E, M D’Elia, R Garrappa, A Giusti, and NL Rubio. 2022.  “On the fractional Laplacian of variable order.” Fractional Calculus and Applied Analysis, 25:15–28, DOI: 10.1007/s13540-021-00003-1.

Goswami S, K Kontolati, MD Shields, and GE Karniadakis.2022. “Deep transfer operator learning for partial differential equations under conditional shift." Nature Machine Intelligence, 4: 1155–1164, DOI: 10.1038/s42256-022-00569-2.

Goswami S, M Yin, Y Yu, and GE Karniadakis. 2022. “A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials.” Computer Methods in Applied Mechanics and Engineering, Volume 391, March 1 2022, 114587, DOI: 10.1016/j.cma.2022.114587.

Hanson J, B Paskalev, E Keller, P Bochev, C Hembree. 2022. "A hybrid analytic-numerical compact model for radiation induced photocurrent effects." IEEE Transactions on Nuclear Science, volume 69, no. 2, 160–168, DOI: 10.1109/TNS.2022.3144069. 

Howard, A, M Maxey, and S Gallier. 2022. “Bidisperse Suspension Balance Model.” Physical Review Fluids, Volume 7, 124301, DOI: 10.1103/PhysRevFluids.7.124301.

Howard AA, M Perego, GE Karniadakis, and P Stinis. 2022. “Multifidelity Deep Operators Networks.” arXiv preprint arXiv:2204.09157.

Hu, Z, AD Jagtap, GE Karniadakis, and K Kawaguchi. 2022. “Augmented Physics-Informed Neural Networks (APINNs): A Gating Network-based Soft Domain Decomposition Methodology.” arXiv preprint arXiv:2211.08939

Jagtap AD, Y Shin, K Kawaguchi, and GE Karniadakis. 2022. “Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functions.” Neurocomputing, Volume 468, January 2022, Pages 165-180, ISSN 0925-2312, DOI: 10.1016/j.neucom.2021.10.036.

Jin P, Z Zhang, IG Kevrekidis, and GE Karniadakis. 2022. "Learning Poisson Systems and Trajectories of Autonomous Systems via Poisson Neural Networks." IEEE Transactions on Neural Networks and Learning Systems, DOI: 10.1109/TNNLS.2022.3148734.

Lu L, X Meng, S Cai, Z Mao, S Goswami, Z Zhang, and GE Karniadakis. 2022. “A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data.” Computer Methods in Applied Mechanics and Engineering, Volume 393, Apri 1 2022, 114778, DOI: 10.1016/j.cma.2022.114778.

Mahmoudabadbozchelou M, GE Karniadakis, and S Jamali. 2022. “nn-PINNs: Non-Newtonian physics-informed neural networks for complex fluid modeling.” Royal Society of Chemistry, Soft Matter, 18:172-185, DOI: 10.1039/D1SM01298C.

Meng X, L Yang, Z Mao, J del Á Ferrandisc, and GE Karniadakis. 2022. “Learning functional priors and posteriors from data and physics.” Journal of Computational Physics, Volume 457, May 15 2022, 111073, DOI: 10.1016/j.jcp.2022.111073.

Pasetto M, Z Shen, M D’Elia, X Tian, N Trask, and D Kamensky. 2022. “Efficient optimization-based quadrature for variational discretization of nonlocal problems.” arXiv preprint arxiv:2201.12391.

Xu X, M D’Elia, J Forster, and C Glusa. 2022. “Machine-learning of nonlocal kernels for anomalous subsurface transport from breakthrough curves.” arXiv preprint arxiv:2201.11146.

Xua K and E Darve. 2022. “Physics constrained learning for data-driven inverse modeling from sparse observations.” Journal of Computational Physics Volume 453, March 15, 2022, 110938, DOI: 10.1016/j.jcp.2021.110938.

Yang L, C Daskalkis, and GE Karniadakis. 2022. “Generative ensemble regression: learning particle dynamics from observations of ensembles with physics-informed deep generative models.” SIAM Journal of Scientific Computing., Society for Industrial and Applied Mathematics, Vol. 44, No. 1, pp. B80-B99, DOI: 10.1137/21M1413018.

You H, Y Yu, M D’Elia, T Gao, and S Siling. 2022. “Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network.” arXiv:2201.02217.

Yu J, L Lu, X Meng, and GE Karniadakis. 2022. “Gradient-enhanced physics-informed neutral networks for forward and inverse PDE problems.” Computer Methods in Applied Mechanics and Engineering, Volume 393, April 2022, 114823. DOI: 10.1016/j.cma.2022.114823.

Zhu W, K Xu, E Darve, B Biondi, and GC Beroza. 2022. “Integrating deep neural networks with full-waveform inversion: Reparameterization, regularization, and uncertainty quantification”, Geophysics, Volume 87, Issue 1. DOI: 10.1190/geo2020-0933.1. 

2021

Ainsworth M and J Dong. 2021. "Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control." SIAM Journal of Scientific Computing, 43, A2474–A2501. DOI: 10.1137/20M1366587.

Ainsworth M and J Dong. 2021. "Least Squares Galerkin Neural Networks: A Framework for High Resolution Approximation of Multiscale Differential Equations." Submitted to NeuRIPS.

Ainsworth M and Y Shin. 2021. "Plateau Phenomenon in Gradient Descent Training of ReLU networks: Explanation, Quantification and Avoidance." arXiv preprint arxiv: 2007.07213.

Atzberger PJ. "MLMOD Package: Machine Learning Methods for Data-Driven Modeling in LAMMPS." arXiv preprint arXiv: 2107.14362.

Aulisa E, G Capodaglio, A Chierici, M D'Elia. 2021. "Efficient quadrature rules for finite element discretizations of nonlocal equations." Numer. Methods Partial Differ. Eq., 1– 27. DOI: 10.1002/num.22833.

Behzadinasab M, G Moutsanidis, N Trask, JT Foster, and Y Bazilevs. 2021. "Coupling of IGA and Peridynamics for Air-Blast Fluid-Structure Interaction Using an Immersed Approach." arXiv preprint arXiv: 2108.11265.

Behzadinasab M, M Alaydin, M, N Trask, and Y Bazilevs. 2021. "A general-purpose, inelastic, rotation-free Kirchhoff-Love shell formulation for peridynamics." arXivpreprint arXiv: 2107.13062.

Behzadinasab M, N Trask, and Y Bazilevs. 2021. "A Unified, Stable and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part I: Core Methods." Journal of Peridynamics and Nonlocal Modeling 3, 24–45. DOI: 10.1007/s42102-020-00040-z. 

Bochev P and B Paskaleva. 2021. "Development of data-driven exponential integrators with application to modeling of delay photocurrents." Num. Meth. PDE. Accepted

Buczkowski N, M Foss, M Parks, P Rady, and J Trageser. 2021. "Two Nonlocal Biharmonic Operators" in Computer Science Research Institute Summer Proceedings 2021, D. Smith and M. Wolf, eds., To Appear, Sandia National Laboratories. 

Burghardt J, T Bao, K Xu, AM Tartakovsky, and E Darve. 2021. “Autonomous Inversion of In Situ Deformation Measurement Data for CO2 Storage Decision Support.” arXiv preprint arXiv:2109.12203.

Burkovska O, C Glusa, MD'Elia. 2021. "An optimization-based approach to parameter learning for fractional type nonlocal models." Computers & Mathematics with Applications. DOI: 10.1016/j.camwa.2021.05.005.

Cai M, E Kharazmi, C Li, and GE Karniadakis. 2021. "Fractional Buffer Layers: Absorbing Boundary Conditions for Wave Propagation." arXIv preprint arXiv:2101.02355v1.

Cai S, H Li, F Zheng, F Kong, M Dao, GE Karniadakis, and S Suresh. 2021. "Artificial intelligence velocimetry and microaneurysm-on-a-chip for three-dimensional analysis of blood flow in physiology and disease." Proceedings of the National Academy of Sciences, 118 (13) e2100697118. DOI: 10.1073/pnas.2100697118.

Cai S, Z Mao, Z Wang, M Yin, and GE Karniadakis. 2021. “Physics-informed Neural Networks (PINNs) for Fluid Mechanics: A Review.” arXiv preprint arXiv:2105.09506.

Cai S, Z Wang, F Fuest, YJ Jeon, C Gray, and GE  Karniadakis. "Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks." Journal of Fluid Mechanics, 915, A102. DOI: 10.1017/jfm.2021.135. 

Cai S, Z Wang, L Lu, TA Zaki, and GE Karniadakis. 2021. “DeepM&Mnet: Inferring the Electroconvection Multiphysics Fields based on Operator Approximation by Neural Networks.” Journal of Computational Physics, 436: 110296. DOI: 10.1016/j.jcp.2021.110296.

Capodaglio G, M D'Elia, M Gunzburger, P Bochev, M Klar, C Vollmann. 2021. "A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions." Numer. Methods Partial Differ. Eq., 1– 29. DOI: 10.1002/num.22832.

Chen X, J Duan, and GE Karniadakis. 2021. "Learning and meta-learning of stochastic advection–diffusion– reaction systems from sparse measurements." European Journal of Applied Mathematics, 32, 397–420. DOI: 10.1017/S0956792520000169.

Chen X, L Yang, J Duan, GE Karniadakis. 2021. "Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker–Planck Equation and Physics-Informed Neural Networks." SIAM Journal on Scientific Computing, 43, B811–B830. DOI: 10.1137/20M1360153.

Christia F, MJ Curry, C Daskalakis, ED Demaine, JP Dickerson, M Hajiaghayi, A Hesterberg, M Knittel, and A Milliff. 2021. "Scalable Equilibrium Computation in Multi-agent Influence Games on Networks" in Proceedings of the 35th AAAI Conference on Artificial Intelligence.

Dagan Y, C Daskalakis, N Dikkala, and AV Kandiros. 2021. "Learning Ising models from one or multiple samples" in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC).

Daskalakis C. 2021. "Technical perspective: The quest for optimal multi-item auctions." Communications of the ACM, 64, 108. DOI: 10.1145/3470440.

Daskalakis C, S Skoulakis, and M Zampetakis. 2021. "The complexity of constrained min-max optimization" in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC).

Daskalakis C, M Fishelson, and N Golowich. 2021. "Near-Optimal No-Regret Learning in General Games" in Proceedings of the the 35th Annual Conference on Neural Information Processing Systems (NeurIPS).

Daskalakis C, P Stefanou, R Yao, and E Zampetakis. 2021. "Efficient Truncated Linear Regression with Unknown Noise Variance" in Proceedings of the the 35th Annual Conference on Neural Information Processing Systems (NeurIPS).

Daskalakis C, V Kontonis, C Tzamos, and E Zampetakis. 2021. "A Statistical Taylor Theorem and Extrapolation of Truncated Densities" in Proceedings of the 34th Annual Conference on Learning Theory (COLT).

Daskalakis C and Q Pan. 2021. "Sample-optimal and efficient learning of tree Ising models" in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC).

Diakonikolas J, C Daskalakis, and MI Jordan. 2021. "Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization" in Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS).

D’Elia M and P Bochev. 2021. "Formulation, analysis and computation of an optimization-based local-to-nonlocal coupling method." Results in Applied Mathematics, 9, 100129. 

D’Elia M, JC De Los Reyes, and A Miniguano-Trujillo. 2021. "Bilevel parameter learning for nonlocal image denoising models." Journal of Mathematical Imaging and Vision, 63, 753–775. DOI: 10.1007/s10851-021-01026-2.

D’Elia M and C Glusa. 2021. "A fractional model for anomalous diffusion with increased variability. Analysis, algorithms and applications to interface problems." arXiv preprint arXiv:2101.11765.

D'Elia M and M Gulian. 2021. "Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anoma-lous Transport." arXiv preprint arXiv:2101.04289.  

D'Elia M, M Gulian, H Olson, and GE Karniadakis. 2021. “A Unified Theory of Fractional, Nonlocal, and Weighted Nonlocal Vector Calculus.” arXic preprint arXiv:2005.0768.

Deng B, Y Shin, L Lu, Z Zhang, and GE Karniadakis. 2021. "Convergence rate of DeepONets for learning operators arising from advection-diffusion equations." arXiv preprint arXiv:2102.10621. 

Di Leoni PC, L Lu, C Meneveau, G Karniadakis, and TA Zaki. 2021. “DeepONet Prediction of Linear Instability Waves in High-speed Boundary Layers." arXiv preprint arXiv:2105.08697.

Ding M, C Daskalakis, S Feizi. 2021. "GANs with Conditional Independence Graphs: On Subadditivity of Probability Divergences" in Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS).

Forghani M, Y Qian, J Lee, M Farthing, T Hesser, P Kitanidis, and E Darve. 2021. “Application of Deep Learning to Large Scale Riverine Flow Velocity Estimation.” Stochastic Environmental Research and Risk Assessment, 35: 1069-1088. DOI: 10.1007/s00477-021-01988-0.

Glusa C, H Antil, M D’Elia, B van Bloemen Waanders, and CJ Weiss. 2021. "A Fast Solver for the Fractional Helmholtz Equation." SIAM Journal on Scientific Computing, 43, A1362–A1388. DOI: 10.1137/19M1302351. 

Gross BJ, P Kuberry, and PJ Atzberger. 2021. "First-Passage Time Statistics on Surfaces of General Shape: Surface PDE Solvers using Generalized Moving Least Squares (GMLS)." arXiv preprint arXiv: 2102.02421.

Ha QT, PA Kuberry, NA Trask, and EM Ryan. 2021. "Parallel implementation of a compatible high-order meshless method for the Stokes’ equations." arXiv preprint arXiv:2104.14447.

Hanson J, P Bochev, B Paskaleva, E Keiter, CE Hembree. 2021. "A hybrid analytic-numerical compact model for radiation induced photocurrent effects." IEEE Transactions on Nuclear Science. Submitted.

Hanson J, P Bochev, E Keiter, C Hembree, and B Paskaleva. 2021. "Nonlinear numerical transient compact photocurrent model for high dose rates." JRERE OUO/ECI. Submitted.

Hu X, A Huang, N Trask, and C Brissette. 2021. "Greedy Fiedler Spectral Partitioning for Data-driven Discrete Exterior Calculus Accepted" in AAAI Spring Symposium: MLPS. 

Hu Z, AD Jagtap, GE Karniadakis, K Kawaguchi. 2021. "When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?" arXiv preprint arXiv:2109.09444.

Jin X, S Cai, H Li, GE Karniadakis. 2021. "NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations." Journal of Computational Physics, 426, 109951. DOI: 10.1016/j.jcp.2020.109951.

Kandiros V, Y Dagan, N Dikkala, S Goel, and C Daskalakis. 2021. "Statistical Estimation from Dependent Data" in Proceedings of the 38th International Conference on Machine Learning (ICML). 

Karniadakis GE, JG Kevrekidis, L Lu, P Perdikaris, S Wang, and L Yang. 2021. “Physics-informed Machine Learning.” Nature Review Physics, 3: 422-440. DOI: 10.1038/s42254-021-00314-5.

Kharazmi E, M Cai, X Zheng, G Lin, and GE Karniadakis. 2021. “Identifiability and predictability of integer-and fractional-order epidemiological models using physics-informed neural networks." medRxiv. DOI: 10.1101/2021.04.05.21254919.

Kharazmi E, Z Zhang, and GE Karniadakis. 2021. "Hp-vpinns: Variational physics-informed neural networks with domain decomposition." Computer Methods in Applied Mechanics and Engineering, 374, 113547. DOI: 10.1016/j.cma.2020.113547.

Lee K, N Trask, and P Stinis. 2021. “Machine learning structure preserving brackets for forecasting irreversible processes.” arXiv preprint arXiv:2106.12619.

Lee K, N Trask, and P Stinis. 2021. "Structure-preserving Sparse Identification of Nonlinear Dynamics for Datadriven Modeling." arXiv preprint arXiv: 2109.05364.

Lee K, N Trask, RG Patel, MA Gulian, and EC Cyr. 2021. "Partition of unity networks: Deep hp-approximation." arXiv preprint arXiv:2101.11256.

Lin C, Z Li, M Maxey, and G Karniadakis. 2021. "A seamless multiscale operator neural network for inferring bubble dynamics." Journal of Fluid Mechanics, 929, A18. DOI: 10.1017/jfm.2021.866.

Lopez R and PJ Atzberger. 2021. "Variational Autoencoders for Learning Nonlinear Dynamics of Physical Systems" accepted AAAI-MLPS Proceedings, arXiv preprint arXiv: 2012.03448. 

Lou Q, X Meng, and GE Karniadakis. 2021. "Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation." Journal of Computational Physics, 110676. DOI: 10.1016/j.jcp.2021.110676.

Lu L, P Jin, G Pang, Z Zhang, and GE Karniadakis. 2021. "Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators." Nature Machine Intelligence, 3, 218–229. DOI: 10.1038/s42256-021-00302-5.

Lu L, X Meng, Z Mao, and GE Karniadakis. 2021. "DeepXDE: A Deep Learning Library for Solving Differential Equations." SIAM Rev., 63(1), 208–228. DOI: 10.1137/19M1274067.

Mahmoudabadbozchelou M, M Caggioni, S Shahsavari, WH Hartt, GE Karniadakis, and S Jamali. 2021. "Data-driven physics-informed constitutive metamodeling of complex fluids: a multifidelity neural network (MFNN) framework." Journal of Rheology, 65, 179–198. DOI: 10.1122/8.0000138.

Marsden A, JC Duchi, and G Valiant. 2021. "Misspecification in Prediction Problems and Robustness via Improper Learning" in The 24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021, April 13-15, 2021, Virtual Event 130 (PMLR, 2021), 2161–2169.

Meng X, H Babaee, and GE Karniadakis. 2021. “Multi-fidelity Bayesian Neural Networks: Algorithms and Applications.” Journal of Computational Physics, 438: 110361. DOI: 10.1016/j.jcp.2021.110361.

Meng X, L Yang, Z Mao, JDA Ferrandis, and GE Karniadakis. 2021. "Learning Functional Priors and Posteriors from Data and Physics." arXiv preprint arXiv: 2106.05863.

Meuris B, S Qadeer, and P Stinis. 2021. “Machine-learning custom-made basis functions for partial differential equations.” arXiv preprint arXiv:2111.05307.

Patel RG, NA Trask, MA Wood, and EC Cyr. 2021. “A physics-informed operator regression framework for extracting data-driven continuum models.” Computer Methods in Applied Mechanics and Engineering, 373, 113500.  DOI: 10.1016/j.cma.2020.113500.

Qiao M and G Valiant. 2021. "Exponential Weights Algorithms for Selective Learning" in Conference on Learning Theory, 3833–3858.

Qiao M and G Valiant. 2021. "Stronger calibration lower bounds via sidestepping" in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, 456–466.

Reyes B, AA Howard, P Perdikaris, and AM Tartakovsky. 2021. “Learning unknown physics of non-Newtonian fluids.” Physical Review Fluids, 6:073301. DOI: 10.1103/PhysRevFluids.6.073301.

Shukla K, AD Jagtap, and GE Karniadakis. 2021. “Parallel Physics-informed Neural Networks via Domain Decomposition.” Journal of Computational Physics, 447: 110683. DOI: 10.1016/j.jcp.2021.110683.

Shin Y, J Darbon, and GE Karniadakis. 2021. "A Caputo fractional derivative-based algorithm for optimization." arXiv preprint arXiv: 2104.02259.

Shukla K, AD Jagtap, JL Blackshire, D Sparkman, and GE Karniadakis. 2021. "A physics-informed neural network for quantifying the microstructure properties of polycrystalline Nickel using ultrasound data." arXiv preprint arXiv: 2103.14104.

Silling SA, M D’Elia, Y Yu, H You, and M Fermen-Coker. 2021. "Peridynamic Model for Single-Layer Graphene Obtained from Coarse Grained Bond Forces." arXiv preprint arXiv: 2109.07280.

Suzuki J, M Gulian, M Zayernouri, and M D’Elia. 2021. "Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials." SAND report SAND2021-11291 R (Sandia National Laboratories, Livermore, CA, and Albuquerque, NM).

Tai KS, P Bailis, and G Valiant. 2021. "Sinkhorn Label Allocation: Semi-Supervised Classification via Annealed Self-Training" in International Conference on Machine Learning (ICML).

Trask N, M Gulian, A Huang, and K Lee. 2021. "Probabilistic partition of unity networks: clustering based deep approximation." arXiv preprint arXiv: 2107.03066.

Wang Z, X Zheng, C Chryssostomidis, and GE Karniadakis. 2021. "A phase-field method for boiling heat transfer." Journal of Computational Physics, 435, 110239. DOI: 10.115/FEDSM2020-20176.

Xu K and E Darve. 2021. “Solving inverse problems in stochastic models using deep neural networks and adversarial training.” Computer Methods in Applied Mechanics and Engineering, 384, 113976. DOI: 10.1016/j.cma.2021.113976.

Xu K and E Darve. 2021. "Trust Region Method for Coupled Systems of PDE Solvers and Deep Neural Networks." arXiv preprint arXiv:2105.07552.

Xu X, M D'Elia, and JT Foster. 2021. "A machine-learning framework for peridynamic material models with physical constraints." Computer Methods in Applied Mechanics and Engineering, 384, 114062. DOI: 10.1016/j.cma.2021.114062.

Xu X, C Glusa, M D’Elia, and J Foster. 2021. “A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems.” Computer Methods in Applied Mechanics and Engineering, 387, 114148. DOI: 10.1016/j.cma.2021.114148.

Xu K, DZ Huang, and E Darve. 2021. “Learning constitutive relations using symmetric positive definite neural networks." Journal of Computational Physics, 428: 110072. DOI: 10.1016/j.jcp.2020.110072.

Xu K, AM Tartakovsky, J Burghardt, and E Darve. 2021. “Learning Viscoelasticity Models from Indirect Data using Deep Neural Networks.” Computer Methods in Applied Mechanics and Engineering, 387, 114124. DOI: 10.1016/j.cma.2021.114124.

Xu K, W Zhu, E Darve. 2021. "Learning Generative Neural Networks with Physics Knowledge." MSML21: Mathematical and Scientific Machine Learning. Submitted.

Yang L, T Meng, and GE Karniadakis. "Measure-conditional Discriminator with Stationary Optimum for GANs and Statistical Distance Surrogates." arXiv preprint arXiv:2101.06802.

Yang L, X Meng, and GE Karniadakis. "B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data." Journal of Computational Physics, 425, 109913. DOI: 10.1016/j.jcp.2020.109913.

You H, Y Yu, N Trask, M Gulian, and M D’Elia. 2021. "Data-driven learning of robust nonlocal physics from high fidelity synthetic data." Computer Methods in Applied Mechanics and Engineering, 374, 113553. DOI: 10.1016/j.cma.2020.113553.

You H, Y Yu, S Silling, and M D’Elia. 2021. "A data-driven peridynamic continuum model for upscaling molecular dynamics." arXiv preprint arXiv: 2108.04883.

You H, Y Yu, S Silling, and M D’Elia. 2021. "Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive laws" accepted in AAAI Spring Symposium: MLPS. 

You H, Y Yu, M D’Elia, T Gao, and S Silling. 2021. "Nonlocal Kernel Network (NKN): a Stable and Resolution- Independent Deep Neural Network." Preprint. Submitted.

Yu Y, H You, and N Trask. 2021. "An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture." Computer Methods in Applied Mechanics and Engineering 377, 113691. DOI: 10.1016/j.cma.2021.113691.

Zhang Z, Y Shin, and GE Karniadakis. 2021. "GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems." arXiv preprint arXiv: 2109.00092.

Zhao L, Z Li, Z Wang, B Caswell, J Ouyang, and GE Karniadakis. "Active-and transfer-learning applied to microscale-macroscale coupling to simulate viscoelastic flows." Journal of Computational Physics, 427, 110069. DOI: 10.1016/j.jcp.2020.110069.

Zhu, W, K Xu, E Darve, and GC Beroza. 2021. “A general approach to seismic inversion with automatic differentiation." Computers & Geosciences, 151 (2021): 104751. DOI: 10.1016/j.cageo.2021.104751.

2020

Aadithya K, P Kuberry, B Paskaleva, P Bochev, K Leeson, A Mar, T Mei, and E Keiter. 2020. “Data-driven Compact Models for Circuit Design and Analysis” in Proceedings of The First Mathematical and Scientific Machine Learning Conference, (eds J Lu and R Ward), vol. 107, pp. 555-569, PMLR, Princeton University, Princeton, NJ, USA. 

Aadithya K, P Kuberry, B Paskaleva, P Bochev, K Leeson, A. Mar, T Mei, and E Keiter. 2020. "Development, Demonstration, and Validation of Data-driven Compact Diode Models for Circuit Simulation and Analysis.” arXiv preprint arXiv:2001.01699.

Axelrod B, S Garg, V Sharan and G Valiant. 2020. "Sample Amplification: Increasing Dataset Size even when Learning is Impossible" in Proceedings of the 37th International Conference on Machine Learning, PMLR 119:442-451. 

Blanc G, N Gupta, G Valiant, and P Valiant. 2020. “Implicit regularization for deep neural networks driven by an Ornstein-Uhlenbeck like process in Conference on Learning Theory (COLT)” in Proceedings of Thirty Third Conference on Learning Theory, PMLR 125:483-513. 

Brustle J, Y Cai, and C Daskalakis. 2020. “Multi-Item Mechanisms without Item-Independence: Learnability via Robustness” in Proceedings of the 21st ACM Conference on Economics and Computation (EC), pp 715-761. DOI: 10.1145/3391403.3399541.

Capodaglio G, M D'Elia, P Bochev, and M Gunzburger. 2020. "An energy-based coupling approach to nonlocal interface problems." Computers & Fluids 207:104593. arXiv preprint arXiv:2001.03696v1.

Chen Y, L Lu, GE Karniadakis, and L Dal Negro. 2020. “Physics-informed neural networks for inverse problems in nano-optics and metamaterials.” Optics Express 28(8):11618-11633. https://doi.org/10.1364/OE.384875.

Chen JY, G Valiant, and P Valiant. 2020. "Worst-Case Analysis for Randomly Collected Data in NeurIPS".

Dagan Y, C Daskalakis, N Dikkala, and AV Kandiros. 2020. “Estimating Ising Models from One Sample.” arXiv preprint arXiv:2004.09370.

Daskalakis C, S Skoulakis, and M Zampetakis. 2020. “The Complexity of Constrained Min-Max Optimization.” arXiv preprint arXiv:2009.09623.

Daskalakis C, N Dikkala, and I Panageas. 2020. “Logistic regression with peer-group effects via inference in higher-order Ising models” in Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), Palermo, Italy. PMLR: volume 108. arXiv:2003.08259.

Daskalakis C, D Foster, and N Golowich. 2020. “Decoupled policy gradient methods for competitive reinforcement learning” in Proceedings of the 34th Annual Conference on Neural Information Processing Systems (NeurIPS). arXiv preprint arXiv:2101.04233

Daskalakis C and M Zampetakis. 2020. “More Revenue from Two Samples via Factor Revealing SDPs” in Proceedings of the 21st ACM Conference on Economics and Computation (EC). Virtual Event 2020, Hungary. Pages 257–272. DOI: 10.1145/3391403.3399543.

Daskalakis C, M Fishelson, B Lucier, V Syrgkanis, and S Velusamy. 2020. “Simple, Credible, and Approximately-Optimal Auctions” in Proceedings of the 21st ACM Conference on Economics and Computation (EC), Hungary. arXiv preprint arXiv:2002.06702.

Daskalakis C, D Rohatgi, and M Zampetakis. 2020. “Constant-Expansion Suffices for Compressed Sensing with Generative Priors” in Proceedings of the 34th Annual Conference on Neural Information Processing Systems (NeurIPS). arXiv preprint arXiv:2006.04237.

Daskalakis C, D Rohatgi, and M Zampetakis. 2020. “Truncated Linear Regression in High Dimensions” in Proceedings of the 34th Annual Conference on Neural Information Processing Systems (NeurIPS). arXiv preprint arXiv:2007.14539.

D'Elia M and C Glusa. 2020. “A fractional model for anomalous diffusion with increased variability.” Analysis, Algorithms and Applications to Interface Problems. arXiv preprint arXiv:2101.11765.

D'Elia M, M Gulian, H Olson, and GE Karniadakis. 2020. "A Unified Theory of Fractional, Nonlocal, and Weighted Nonlocal Vector Calculus." SAND2020-4869, Sandia National Laboratories. arXiv preprint arXiv:2005.07686.

D'Elia M and P Bochev. 2020. “Formulation, Analysis and Computation of an optimization-based local-to-nonlocal coupling method.” Accepted for publication in RINAM. arXiv preprint arXiv:1910.11214.

D'Elia M, GE Karniadakis, Pang G, and M Parks. 2020. “Nonlocal Physics-Informed Neural Networks - A unified theoretical and computational framework for nonlocal models” in Proceedings of the AAAI 2020 Spring Symposium on Combining Artificial Intelligence and Machine Learning with Physical Sciences, Stanford, CA, USA, March 23-25, 2020.

Ding M, C Daskalakis, and S Feizi. 2020. “When Do Local Discriminators Work? On Subadditivity of Probability Divergences.” arXiv preprint arXiv:2003.00652.

Fan D, L Yang, MS Triantafyllou, and GE Karniadakis. 2020. “Reinforcement Learning for Active Flow Control in Experiments.” arXiv preprint arXiv:2003.03419.

Fan T, K Xu, J Pathak, and E Darve. 2020. "Solving Inverse Problems in Steady State Navier-Stokes Equations using Deep Neural Networks." arXiv preprint arXiv: 2008.13074.

Fan T, J Xu Kailai Pathak, and E Darve. 2020. "Solving Inverse Problems in Steady State Navier-Stokes Equations using Deep Neural Networks" in AAAI 2020 Fall Symposium on Physics-Guided AI to Accelerate Scientific Discovery (PGAI-AAAI-20).

Gao P, X Yang, and AM Tartakovsky. 2020. "Learning Coarse-Grained Potentials for Binary Fluids." Journal of Chemical Information and Modeling, 60(8):3731-3745. DOI: 10.1021/acs.jcim.0c00337.

Golowich N, S Pattathil, C Daskalakis, and AE Ozdaglar. 2020. “Last Iterate is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems” in Proceedings of the 33nd Annual Conference on Learning Theory (COLT). arXiv preprint arXiv:2002.00057.

Golowich N, S Pattathil, and C Daskalakis. 2020. “Tight last-iterate convergence rates for no-regret learning in multi-player games” in Proceedings of the 34th Annual Conference on Neural Information Processing Systems (NeurIPS). arXiv preprint arXiv:2010.13724.

Gross BJ, N Trask, P Kuberry, and PJ Atzberger. 2020. “Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: a generalized moving least-squares (GMLS) approach.” Journal of Computational Physics 409, 109340. DOI: 10.1016.j.jcp.2020.109340.

Gulian M, M D'Elia, T Mengesha, and JM Scott. 2020. “A generalized nonlocal calculus for vector-valued functions: theory and applications.” In progress.

Hanson J, P Bochev, and B Paskaleva. 2020. "Learning Compact Physics-Aware Photocurrent Models Using Dynamic Mode Decomposition." arXiv preprint arXiv: 2008.12319.

He Q, D Barajas-Solano, G Tartakovsky, and A Tartakovsky. 2020. "Physics-informed neural networks for multiphysics data assimilation with application to subsurface transport." Advances in Water Resources 141:103610. DOI: 10.1016/j.advwatres.2020.103610.

He Q and AM Tartakovsky. 2020. "Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations." arXiv preprint arXiv:2012.11658. 

Huang DZ, K Xu, C Farhat, and E Darve. 2020. "Learning constitutive relations from indirect observations using deep neural networks." Journal of Computational Physics, 414:109491. DOI: 10.1016/j.jcp.2020.109491.

Ilyas A, E Zampetakis, and CA Daskalakis. 2020. “Theoretical and Practical Framework for Regression and Classification from Truncated Samples” in Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS) PMLR 108:4463-4473.

Jagtap A D and GE Karniadakis. 2020. "Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations." Communications in Computational Physics, 28, 2002–2041. DOI: 10.4208/cicp.OA-2020-0164.

Jagtap AD, K Kawaguchi, and GE Karniadakis. 2020. "Adaptive activation functions accelerate convergence in deep and physics-informed neural networks." Journal of Computational Physics, 404:109136. DOI: 10.1016/j.jcp.2019.109136.

Jagtap AD, K Kawaguchi, and GE Karniadakis. 2020. “Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks” in Proceedings of the Royal Society A 476, 20200334. DOI: 10.1098/rspa.2020.0334.

Jagtap AD, E Kharazmi, and GE Karniadakis. 2020. “Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems.” Computer Methods in Applied Mechanics and Engineering 365:113028. DOI: 10.1016/j.cma.2020.113028.

Jin P, Z Zhang, A Zhu, Y Tang, and GE Karniadakis. 2020. “SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems.” Neural Networks 132:166-179. DOI: 10.1016/j.neunet.2020.08.017.

Jin P, L Lu, Y Tang, and GE Karniadakis. 2020. “Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness.” Neural Networks 130:85-99. https://doi.org/10.1016/j.neunet.2020.06.024.

Kharazmi E, Z Zhang, and GE Karniadakis. 2020. “hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition.” Computer Methods in Applied Mechanics and Engineering, 374:113547, arXiv preprint arXiv:2003.05385.

Leng Y, X Tian, NA Trask, and JT Foster. 2020. "Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation." Computer Methods in Applied Mechanics and Engineering, 370, 113264. DOI: 10.1016/j.cma.2020.113264.

Li D, K Xu, JM Harris, and E Darve. 2020. “Coupled time-lapse full-waveform inversion for subsurface flow problems using intrusive automatic differentiation.” Water Resources Research 56:8. DOI: 10.1016/j.neunet.2020.06.024.

Lin C, Z Li, L Lu, S Cai, M Maxey, and GE Karniadakis. 2020. "Operator learning for predicting multiscale bubble growth dynamics." Journal of Chemical Physics, 154, 104118. DOI: 10.1063/5.0041203.

Lu L, M Dao, P Kumar, U Ramamurty, GE Karnadakis, and S Suresh. 2020. “Extraction of Mechanical Properties of Materials through Deep Learning from Instrumented Indentation.” PNAS, 117(13):7052-7062. DOI: 10.1073/pnas.1922210117.

Martinez C, JE Jones, D Levin, NA Trask, PD Finley. 2020. "Physics-Informed Machine Learning for Epidemiological Models." Sandia National Lab. (SNL-NM), Albuquerque, NM (United States).

Meng X and GE Karniadakis. 2020. “A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems.” Journal of Computational Physics 401:109020. DOI: 10.1016/j.jcp.2019.109020.

Meng X, Z Li, D Zhang, and GE Karniadakis. 2020. “PPINN: Parareal physics-informed neural network for time-dependent PDEs.” Computer Methods in Applied Mechanics and Engineering 370:113250. DOI: 10.1016/j.cma.2020.113250.

Meng X, Z Wang, D Fan, M Triantafyllou, and GE Karniadakis. 2020. "A fast multi-fidelity method with uncertainty quantification for complex data correlations: Application to vortex-induced vibrations of marine risers." arXiv preprint arXiv:2012.13481v1.

Meng X, H Babaee and GE Karniadakis. 2020. "Multi-fidelity Bayesian Neural Networks: Algorithms and Application." arXiv preprint arXiv:2012.13294v1.

Olson H, M Gulian, and M D'Elia. 2020. “On the equivalence of tempered fractional and truncated fractional operators.” CSRI summer proceedings 2020, Sandia National Laboratories. In progress.

Olson H, M Gulian, and M D’Elia. 2020. "The Tempered Fractional Laplacian As a Special Case of the Nonlocal Laplace Operator" in Computer Science Research Institute Summer Proceedings 2020, A.A. Rushdi and M.L. Parks, eds., Technical Report SAND2020-12580R, Sandia National Laboratories, 111–126. 

Pang G, M D'Elia, M Parks, and GE Karniadakis. 2020. "nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator." Sandia National Laboratories, SAND2020-3980. arXiv preprint arXiv:2004.04276.

Patel RG, NA Trask, MA Gulian, and EC Cyr. 2020. “A block coordinate descent optimizer for classification problems exploiting convexity.” arXiv preprint arXiv:2006.10123.

Patel RG, I Manickam, NA Trask, MA Wood, M Lee, I Tomas, and EC Cyr. 2020. "Thermodynamically consistent physics-informed neural networks for hyperbolic systems." arXiv preprint. (accepted to JCP). arXiv:2012.05343.

Pourzanjani A, RM Jiang, B Mitchell, PJ Atzberger, and L Petzold. 2020. “Bayesian Inference over the Stiefel Manifold via the Givens Transform.” Bayesian Analysis. DOI: 10.1214/20-BA1202.

Raissi M, A Yazdani, and GE Karnidakais. 2020. "Hidden Fluid Mechanics: Learning Velocity and Pressure Fields from Flow Visualizations." Science 367(6481):1026-1030. DOI: 10.1126/science.aaw4741.

Shin Y, J Darbon, and GE Karniadakis. 2020. "On the convergence and generalization of physics informed neural networks." arXiv preprint arXiv:2004.01806.

Shin Y and GE Karniadakis. 2020. "Trainability of ReLU Networks and data-dependent initialization." Journal of Machine Learning for Modeling and Computing 1, 1. DOI: 10.1615/2020034126.

Stinis P. 2020. "Enforcing Constraints for Time Series Prediction in Supervised, Unsupervised and Reinforcement Learning" in Proceedings of the AAAI 2020 Spring Symposium on Combining Artificial Intelligence and Machine Learning with Physical Sciences, CEUR Workshop Proceedings, Vol. 2587.

Suzuki JL, Y Zhou, M D'Elia, and M Zayernouri. 2020. "A Thermodynamically Consistent Fractional Visco-Elasto-Plastic Model with Memory-Dependent Damage for Anomalous Materials." Computer Methods in Applied Mechanics and Engineering (CMAME), arXiv:1911.07114.

Trask N, A Huang, and X Hu. 2020. "Enforcing exact physics in scientific machine learning: a data-driven exterior calculus on graphs." arXiv preprint arXiv: 2012.11799.

Trask N, P Bochev, and M Perego. 2020. "A conservative, consistent, and scalable meshfree mimetic method." Journal of Computational Physics 409, 109187. DOI: 10.1016/j.jcp.2019.109187.

Trask N and P Kuberry. 2020. "Compatible meshfree discretization of surface PDEs." Computational Particle Mechanics 7:271-277. DOI: 10.1007/s40571-019-00251-2.

Trask N, R Patel, P Atzberger, and B Gross. 2020. “GMLS-Nets: A machine learning framework for unstructured data" in Proceedings of the AAAI 2020 Spring Symposium on Combining Artificial Intelligence and Machine Learning with Physical Sciences, March 23-25, 2020, Stanford, CA.

Wang Y, Z Li, J Ouyang, and GE Karniadakis. 2020. "Controlled release of entrapped nanoparticles from thermoresponsive hydrogels with tunable network characteristics." Soft Matter 16:4756-4766. DOI: 10.1039/D0SM00207K.

Wu S, H Zhang, G Valiant, and C Re. 2020. "On the Generalization Effects of Linear Transformations in Data Augmentation." International Conference on Machine Learning (ICML). arXiv:2005.00695.

Xu K and E Darve. 2020. "ADCME: Learning Spatially-varying Physical Fields using Deep Neural Networks" in 3rd Workshop on Machine Learning and the Physical Sciences, Workshop at the 34th Conference on Neural Information Processing Systems.

Xu K and E Darve. 2020. "Calibrating multivariate Lévy processes with neural networks" in Proceedings of The First Mathematical and Scientific Machine Learning Conference, PMLR 107:207-220. 

Xu K and E Darve. 2020. "Data-Driven Inverse Modeling with Incomplete Observations" in AAAI 2020 Spring Symposium on Combining Artificial Intelligence and Machine Learning with Physical Sciences.

Xu K and E Darve. 2020. "Isogeometric collocation method for the fractional Laplacian in the 2D bounded domain." Computer Methods in Applied Mechanics and Engineering, 364:112936. DOI: 10.1016/j.cma.2020.112936.

Xu K and E Darve. 2020. "Physics constrained learning for data-driven inverse modeling from sparse observations." arXiv preprint arXiv:2002.10521.

Xu K, DZ Huang, and E Darve. 2020. "Learning constitutive relations using symmetric positive definite neural networks." Journal of Computational Physics, 428,110072. DOI: 10.1016/j.jcp.2020.110072.

Xu K, AM Tartakovsky, J Burghardt, and E Darve. 2020. "Inverse Modeling of Viscoelasticity Materials using Physics Constrained Learning." arXiv preprint arXiv:2005.04384.

Xu K, W Zhu, and E Darve. 2020. "Distributed machine learning for computational engineering using MPI." arXiv preprint arXiv: 2011.01349.

Yang L, C Daskalakis, and GE Karniadakis. 2020. "Generative Ensemble-Regression: Learning Stochastic Dynamics from Discrete Particle Ensemble Observations." arXiv preprint arXiv: 2008.01915.

Yang L, D Zhang, and GE Karniadakis. 2020. "Physics-informed generative adversarial networks for stochastic differential equations." SIAM Journal on Scientific Computing 42, A292-A317. DOI: 10.1137/18M1225409.

You H, X Lu, N Trask, and Y Yu. 2020. "An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems." ESAIM: Mathematical Modelling and Numerical Analysis 54, 1373-1413 (2020) 39. arXiv:1908.03853.

You H, Y Yu, N Trask, M Gulian, and M D'Elia. 2020. "Data-driven learning of robust nonlocal physics from high-fidelity synthetic data.” Computer Methods in Applied Mechanics and Engineering 374: 113553. DOI: 10.1016/j.cma.2929.113553.

Zhang D, L Guo, and GE Karniadakis. 2020. “Learning in modal space: Solving time-dependent stochastic PDEs using physics-informed neural networks.” SIAM Journal on Scientific Computing 42, A639-A665. DOI: 10.1137/19M1260141.

Zhao L, Z Li, Z Wang, B Caswell, J Ouyang, and GE Karniadakis. 2020. “Active- and transfer-learning applied to microscale-macroscale coupling to simulate viscoelastic flows.” arXiv preprint arXiv:2005.04382.

Zheng Q, L Zeng, and GE Karniadakis. 2020. “Physics-informed semantic inpainting: Application to geostatistical modeling.” Journal of Computational Physics 419:109676. DOI: 10.1016/j.jcp.2020.109676.

Zhu W, K Xu, E Darve, and GC Beroza. 2020. “A General Approach to Seismic Inversion with Automatic Differentiation.” arXiv:2003.06027.

2019

Ainsworth M and Z Mao. 2019. “Phase field crystal based prediction of temperature and density dependence of elastic constants through a structural phase transition.” Physical Review B 100, 104101. DOI: 10.1103/PhysRevB.100.104101.

Blumers AL, Z Li, and GE Karniadakis. 2019. "Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: Application to hydrodynamics." Journal of Computational Physics 393(2019):214-228. DOI: 10.1016/j.jcp.2019.05.016.

Chen X, J Duan, and GE Karniadakis. 2019. "Learning and Meta-Learning of Stochastic Advection-Diffusion-Reaction Systems from Sparse Measurements." European Journal of Applied Mathematics 1-24. DOI: 10.1017/S0956792520000169.

Chen Y, L Lu, GE Karniadakis, and LD Negro. 2019. "Physics-informed neural networks for inverse problems in nano-optics and metamaterials." Optics Express 28(8):11618-11633. DOI: 10.1364/OE.384875.

Daskalakis C, N Dikkala, and G Kamath. 2019. "Testing Ising Models." In IEEE Transactions on Information Theory 65(11):6829-6852. DOI: 10.1109/TIT.2019.2932255.

GaoP, X Yang, and A Tartakovsky. 2019. "A New Approach For Learning Coarse-Grained Potentials with Application to Immiscible Fluids." arXiv preprint arXiv:1907.06144.

Gao P and A Tartakovsky. 2019. "MARTINI-based Coarse-grained Model for Poly (alpha-peptoid)s." arXiv preprint arXiv:1903.01975.

Ghorbanidehno H, J Lee, M Farthing, T Hesser, PK Kitanidis, and EF Darve. 2019. "Novel data assimilation algorithm for nearshore bathymetry." Journal of Atmospheric and Oceanic Technology 36(4):699-715. DOI: 10.1175/JTECH-D-18-0067.1.

Gross BJ, N Trask, P Kuberry, and PJ Atzberger. 2019. "Meshfree Methods on Manifolds for Hydrodynamic Flows on Curved Surfaces: A Generalized Moving Least-Squares (GMLS) Approach." arXiv preprint arXiv:1904.02137.

Huang DZ, K Xu, C Farhat, and E Darve. 2019. "Learning Constitutive Relations from Indirect Observations Using Deep Neural Networks." arXiv preprint arXiv:1905.12530.

Jin P, L Lu, Y Tang, and GE Karniadakis. 2019. "Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness." arXiv preprint arXiv:1905.11427.

Kharazmi E, Z Zhang, and GE Karniadakis. 2019. "Variational Physics-Informed Neural Networks For Solving Partial Differential Equations." arXiv preprint arXiv:1912.00873.

Li D, K Xu, JM Harris, and E Darve. 2019. "Time-lapse Full Waveform Inversion for Subsurface Flow Problems with Intelligent Automatic Differentiation." arXiv preprint arXiv:1912.07552.

Li J and AM Tartakovsky. 2019. "Gaussian Process Regression and Conditional Polynomial Chaos for Parameter Estimation." arXiv preprint arXiv:1908.00424.

Lu L, P Jin, and GE Karniadakis. 2019. "DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators." arXiv preprint arXiv:1910.03193.

Lu L, X Meng, Z Mao, and GE Karniadakis. 2019. "DeepXDE: A deep learning library for solving differential equations." arXiv preprint arXiv:1907.04502.

Meng X, Z Li, D Zhang, and GE Karniadakis. 2019. "PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs." arXiv preprint arXiv:1909.10145.

Pang G, L Lu, and GE Karniadakis. 2019. "fPINNs: Fractional physics-informed neural networks." SIAM Journal on Scientific Computing 41, A2603-A2626. arXiv preprint arXiv:1811.08967.

Pazner W, N Trask, and PJ Atzberger. 2019. "Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance." Results in Applied Mathematics 4:100068. DOI: 10.1016/j.rinam.2019.100068.

Rower D, M Padidar, and PJ Atzberger. 2019. "Surface Fluctuating Hydrodynamics Methods for the Drift-Diffusion Dynamics of Particles and Microstructures within Curved Fluid Interfaces." arXiv prerpint arXiv:1906.01146.

Shin Y and GE Karniadakis. 2019. "Trainability and Data-dependent Initialization of Over-parameterized ReLU Neural Networks." arXiv preprint arXiv:1907.09696

Stinis P, T Hagge, AM Tartakovsky, and E Yeung. 2019. "Enforcing constraints for interpolation and extrapolation in Generative Adversarial Networks." Journal of Computational Physics 397: 108844. DOI:10.1016/j.jcp.2019.07.042.

Trask N, P Bochev, and M Perego. 2019. "A conservative, consistent, and scalable meshfree mimetic method." arXiv preprint arXiv: 1903.04621.

Trask N, RG Patel, BJ Gross, and PJ Atzberger. 2019. "GMLS-Nets: A Framework for Learning from Unstructured Data." arXiv preprint arXiv:1909.05371.

Wang Y, Z Li, J Xu, C Yang, and GE Karniadakis. 2019. "Concurrent coupling of atomistic simulation and mesoscopic hydrodynamics for flows over soft multi-functional surfaces." Soft Matter 15(2019):1747-1757. DOI: 10.1039/C8SM02170H.

Xu K and E Darve. 2019. "Adversarial Numerical Analysis for Inverse Problems." arXiv preprint arXiv:1910.06936.

Xu K and E Darve. 2019. "The neural network approach to inverse problems in differential equations." arXiv preprint arXiv:1901.07758.

Xu K and E Darve. 2019. "Physics Constrained Learning for Data-driven Inverse Modeling from Sparse Observations." arXiv preprint arXiv:2002.10521.

Xu K, D Li, E Darve, and JM Harris. 2019. "Learning Hidden Dynamics using Intelligent Automatic Differentiation." arXiv preprint arXiv:1912.07547.

Xu K and E Darve. 2019. "The Neural Network Approach to Inverse Problems in Differential Equations." arXiv preprint arXiv:1901.07758.

Yang L and GE Karniadakis. 2019. "Potential Flow Generator with L2 Optimal Transport Regularity for Generative Models." arXiv preprint arXiv:1908.11462.

Zhang K, Z Li, M Maxey, S Chen, and GE Karniadakis. 2019. "Self-cleaning of hydrophobic rough surfaces by coalescence-induced wetting transition." Langmuir 35, 6: 2431-2442. DOI: 10.1021/acs.langmuir.8b03664.

Zheng Q, L Zeng, Z Cao, and GE Karniadakis. 2019. "Physics-informed semantic inpainting: Application to geostatistical modeling." arXiv preprint arXiv:1909.09459 X.

2018

Atzberger PJ. 2018. "Importance of the Mathematical Foundations of Machine Learning Methods for Scientific and Engineering Applications." Position paper presented at SciML2018 Workshop, U.S. Department of Energy. arXiv preprint arXiv:1808.02213.

Daskalakis C. 2018. "Equilibria, Fixed Points, and Computational Complexity - Nevanlinna Prize Lecture." In Proceedings of the International Congress of Mathematicians (ICM), Vol 1, pp. 147-210, 2018. DOI: 10.1142/9789813272880_0009.

Mao Z, Z Li, and GE Karniadakis. 2018. "Nonlocal flocking dynamics: Learning the fractional order of PDEs from particle simulations.” Communications on Applied Mathematics and Computation." 1:597-619. arXiv preprint arXiv:1810.11596.

Pang G, L Lu, and GE Karniadakis. 2018. "fPINNs: Fractional Physics-Informed Neural Networks." arXiv preprint arXiv:1811.08967.

Tartakovsky AM, CM Ortiz Marrero, P Perdikaris, GD Tartakovsky, and DA Barajas-Solano. 2018. “Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Network.” Abstract submitted to Computational Science and Engineering (CSE), Spokane, Washington. PNNL-SA-140297.

Xu K and E Darve. 2018. "Calibrating Lévy Process from Observations Based on Neural Networks and Automatic Differentiation with Convergence Proofs." arXiv preprint arXiv:1812.08883.

Yang L, D Zhang, and GE Karniadakis. 2018. "Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations.” SIAM Journal on Scientific Computing 42(1):A292-A317. DOI: 10.1137/18M1225409.