APS Logo

Low-Order Modeling and Machine Learning in Fluid Dynamics: Methods II

ORAL · J10 · ID: 3583409

Presentations

  • Development of reduced-order modeling of a multiphase flow in an ejector

    ORAL

    –

    Publication: 1. Bhaduri, Sreetam, Leonard J. Peltier, David Ladd, Eckhard A. Groll, and Davide Ziviani. "Turbulence and Interface Physics of a Carbon-Dioxide Jet in an Ejector." Physics of Fluids (Under Review).<br>2. Bhaduri, Sreetam, Leonard J. Peltier, David Ladd, Eckhard A. Groll, and Davide Ziviani. "Regimes of a Decelerating Wall-bounded Multiphase Jet inside Ejectors." Physics of Fluids 37 (2025).<br>3. Bhaduri, Sreetam, Junyan Ren, Leonard J. Peltier, David Ladd, Eckhard A. Groll, and Davide Ziviani. "Flow physics of a subcritical carbon dioxide jet in a multiphase ejector." Applied Thermal Engineering 256 (2024): 124043.

    Presenters

    • Sreetam Bhaduri

      Purdue University

    Authors

    • Sreetam Bhaduri

      Purdue University

    • Leonard J Peltier

      Bechtel Nuclear, Security, & Environmental, 12011 Sunset Hills Road, Reston, 20190, Virginia, United States

    • David Ladd

      Bechtel Manufacturing & Technology, USA

    • Eckhard A Groll

      Purdue University, West Lafayette

    • Davide Ziviani

      Purdue University

    View abstract →

  • Fourier analysis of the physics of transfer learning for data-driven subgrid-scale models of ocean turbulence

    ORAL

    –

    Publication: Abernathey R, Rochanotes, Ross A, Jansen M, Ziwei Li, Poulin FJ, Constantinou NC, Anirban Sinha, Dhruv Balwada,<br>SalahKouhen, Jones S, Rocha CB, Wolfe CLP, Chuizheng Meng, Van Kemenade H, Bourbeau J, Penn J, Busecke J,<br>Bueti M and , Tobias (2022) pyqg/pyqg: v0.7.2. doi: 10.5281/ZENODO.6563667. Available at https://zenodo.org/record/<br>6563667.<br>Arakawa A and Lamb VR (1977) Computational Design of the Basic Dynamical Processes of the UCLA General Circulation<br>Model. In General Circulation Models of the Atmosphere. Elsevier, 173–265. doi: 10.1016/b978-0-12-460817-7.50009-4.<br>Available at http://dx.doi.org/10.1016/B978-0-12-460817-7.50009-4.<br>Bracco A, Brajard J, Dijkstra HA, Hassanzadeh P, Lessig C and Monteleoni C (2025) Machine learning for the physics of<br>climate. Nature Reviews Physics 7(1), 6–20.<br>Bracco A, Brajard J, Dijkstra HA, Hassanzadeh P, Lessig C and Monteleoni C (2024) Machine learning for the physics of<br>climate. Nature Reviews Physics 7(1), 6–20. issn: 2522-5820. doi: 10.1038/s42254-024-00776-3. http://dx.doi.org/10.1038/<br>s42254-024-00776-3.<br>Carati D, Ghosal S and Moin P (1995) On the representation of backscatter in dynamic localization models. Physics of Fluids<br>7(3), 606–616. issn: 1089-7666. doi: 10.1063/1.868585. http://dx.doi.org/10.1063/1.868585.<br>Chattopadhyay A, Gray M, Wu T, Lowe AB and He R (2024) OceanNet: A principled neural operator-based digital twin for<br>regional oceans. Scientific Reports 14(1), 21181.<br>Chattopadhyay A and Hassanzadeh P (2023) Long-term instabilities of deep learning-based digital twins of the climate system:<br>The cause and a solution. arXiv preprint arXiv:2304.07029.<br>Chattopadhyay A, Subel A and Hassanzadeh P (2020) Data-driven super-parameterization using deep learning: Experimen-<br>tation with multiscale Lorenz 96 systems and transfer learning. Journal of Advances in Modeling Earth Systems 12(11),<br>e2020MS002084.<br>Chen X, Gong C, Wan Q, Deng L, Wan Y, Liu Y, Chen B and Liu J (2021) Transfer learning for deep neural network-based<br>partial differential equations solving. Advances in Aerodynamics 3(1). issn: 2524-6992. doi: 10.1186/s42774-021-00094-7.<br>http://dx.doi.org/10.1186/s42774-021-00094-7.<br>Desai S, Mattheakis M, Joy H, Protopapas P and Roberts S (2021) One-Shot Transfer Learning of Physics-Informed Neural<br>Networks. doi: 10.48550/ARXIV.2110.11286. Available at https://arxiv.org/abs/2110.11286.<br>Dipankar A, Stevens B, Heinze R, Moseley C, Zängl G, Giorgetta M and Brdar S (2015) Large eddy simulation using<br>the general circulation model <scp>ICON</scp>. Journal of Advances in Modeling Earth Systems 7(3), 963–986. issn:<br>1942-2466. doi: 10.1002/2015ms000431. http://dx.doi.org/10.1002/2015MS000431.<br>Domaradzki JA, Metcalfe RW, Rogallo RS and Riley JJ (1987) Analysis of subgrid-scale eddy viscosity with use of results<br>from direct numerical simulations. Physical Review Letters 58(6), 547–550. issn: 0031-9007. doi: 10.1103/physrevlett.58.547.<br>http://dx.doi.org/10.1103/PhysRevLett.58.547.<br>Fox DG and Orszag SA (1973) Pseudospectral approximation to two-dimensional turbulence. Journal of Computational Physics<br>11(4), 612–619.<br>Fox RO (2012) Large-Eddy-Simulation Tools for Multiphase Flows. Annual Review of Fluid Mechanics 44(1), 47–76. issn:<br>1545-4479. doi: 10.1146/annurev-fluid-120710-101118. http://dx.doi.org/10.1146/annurev-fluid-120710-101118.<br>Fox-Kemper B and Menemenlis D (2008) Can large eddy simulation techniques improve mesoscale rich ocean models? In<br>Geophysical Monograph Series. American Geophysical Union, 319–337. doi: 10.1029/177gm19. Available at http://dx.doi.<br>org/10.1029/177GM19.<br>Gallet B and Ferrari R (2021) A quantitative scaling theory for meridional heat transport in planetary atmospheres and oceans.<br>AGU Advances 2(3), e2020AV000362.<br>Gao Y, Cheung KC and Ng MK (2022) SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular<br>Value Decomposition. In 2022 IEEE Symposium Series on Computational Intelligence (SSCI). IEEE. doi: 10.1109/ssci51031.<br>2022.10022281. Available at http://dx.doi.org/10.1109/SSCI51031.2022.10022281.<br>Germano M, Piomelli U, Moin P and Cabot WH (1991) A dynamic subgrid-scale eddy viscosity model. Physics of Fluids A:<br>Fluid Dynamics 3(7), 1760–1765. issn: 0899-8213. doi: 10.1063/1.857955. http://dx.doi.org/10.1063/1.857955.<br>Goswami S, Anitescu C, Chakraborty S and Rabczuk T (2020) Transfer learning enhanced physics informed neural network<br>for phase-field modeling of fracture. Theoretical and Applied Fracture Mechanics 106, 102447. issn: 0167-8442. doi:<br>10.1016/j.tafmec.2019.102447. http://dx.doi.org/10.1016/j.tafmec.2019.102447.<br>Goswami S, Kontolati K, Shields MD and Karniadakis GE (2022) Deep transfer operator learning for partial differential<br>equations under conditional shift. Nature Machine Intelligence 4(12), 1155–1164. issn: 2522-5839. doi: 10.1038/s42256-<br>022-00569-2. http://dx.doi.org/10.1038/s42256-022-00569-2.<br>Gray MA, Chattopadhyay A, Wu T, Lowe A and He R (2024) Long-term Prediction of the Gulf Stream Meander Using<br>OceanNet: a Principled Neural Operator-based Digital Twin. EGUsphere 2024, 1–23.<br>Grooms I (2023a) Backscatter in energetically-constrained Leith parameterizations. Ocean Modelling 186, 102265.<br>Grooms I (2023b) Backscatter in energetically-constrained Leith parameterizations. Ocean Modelling 186, 102265. issn: 1463-<br>5003. doi: 10.1016/j.ocemod.2023.102265. http://dx.doi.org/10.1016/j.ocemod.2023.102265.<br>Guan Y, Chattopadhyay A, Subel A and Hassanzadeh P (2022a) Stable a posteriori LES of 2D turbulence using convolutional<br>neural networks: Backscattering analysis and generalization to higher Re via transfer learning. Journal of Computational<br>Physics 458, 111090.<br>Guan Y, Hassanzadeh P, Schneider T, Dunbar O, Huang DZ, Wu J and Lopez-Gomez I (2024) Online learning of eddy-<br>viscosity and backscattering closures for geophysical turbulence using ensemble Kalman inversion. doi: 10.48550/ARXIV.<br>2409.04985. Available at https://arxiv.org/abs/2409.04985.<br>Guan Y, Subel A, Chattopadhyay A and Hassanzadeh P (2023) Learning physics-constrained subgrid-scale closures in the<br>small-data regime for stable and accurate LES. Physica D: Nonlinear Phenomena 443, 133568.<br>Guo H, Zhuang X, Chen P, Alajlan N and Rabczuk T (2022) Analysis of three-dimensional potential problems in non-<br>homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis.<br>Engineering with Computers 38(6), 5423–5444. issn: 1435-5663. doi: 10.1007/s00366-022-01633-6. http://dx.doi.org/10.<br>1007/s00366-022-01633-6.<br>Haghighat E, Raissi M, Moure A, Gomez H and Juanes R (2021) A physics-informed deep learning framework for inversion<br>and surrogate modeling in solid mechanics. Computer Methods in Applied Mechanics and Engineering 379, 113741. issn:<br>0045-7825. doi: 10.1016/j.cma.2021.113741. http://dx.doi.org/10.1016/j.cma.2021.113741.<br>Hallberg R (2013) Using a resolution function to regulate parameterizations of oceanic mesoscale eddy effects. Ocean Modelling<br>72, 92–103.<br>Hanna JM, Aguado JV, Comas-Cardona S, Askri R and Borzacchiello D (2022) Residual-based adaptivity for two-phase<br>flow simulation in porous media using Physics-informed Neural Networks. Computer Methods in Applied Mechanics and<br>Engineering 396, 115100. issn: 0045-7825. doi: 10.1016/j.cma.2022.115100. http://dx.doi.org/10.1016/j.cma.2022.115100.<br>Hornik K, Stinchcombe M and White H (1989) Multilayer feedforward networks are universal approximators. Neural Networks<br>2(5), 359–366. issn: 0893-6080. doi: 10.1016/0893-6080(89)90020-8. http://dx.doi.org/10.1016/0893-6080(89)90020-8.<br>Jakhar K, Guan Y, Mojgani R, Chattopadhyay A and Hassanzadeh P (2023) Learning Closed-form Equations for Subgrid-<br>scale Closures from High-fidelity Data: Promises and Challenges. doi: 10.48550/ARXIV.2306.05014. Available at https:<br>//arxiv.org/abs/2306.05014.<br><br>Jansen MF, Adcroft A, Khani S and Kong H (2019) Toward an Energetically Consistent, Resolution Aware Parameterization<br>of Ocean Mesoscale Eddies. Journal of Advances in Modeling Earth Systems 11(8), 2844–2860. issn: 1942-2466. doi:<br>10.1029/2019ms001750. http://dx.doi.org/10.1029/2019MS001750.<br>Jansen MF and Held IM (2014) Parameterizing subgrid-scale eddy effects using energetically consistent backscatter. Ocean<br>Modelling 80, 36–48. issn: 1463-5003. doi: 10.1016/j.ocemod.2014.06.002. http://dx.doi.org/10.1016/j.ocemod.2014.06.002.<br>Kent J, Jablonowski C, Thuburn J and Wood N (2016) An energy-conserving restoration scheme for the shallow-water<br>equations. Quarterly Journal of the Royal Meteorological Society 142(695), 1100–1110. issn: 1477-870X. doi: 10.1002/qj.<br>2713. http://dx.doi.org/10.1002/qj.2713.<br>Kerr RM, Domaradzki JA and Barbier G (1996) Small-scale properties of nonlinear interactions and subgrid-scale energy<br>transfer in isotropic turbulence. Physics of Fluids 8(1), 197–208. issn: 1089-7666. doi: 10.1063/1.868827. http://dx.doi.org/<br>10.1063/1.868827.<br>Khani S and Waite ML (2016) Backscatter in stratified turbulence. European Journal of Mechanics - B/Fluids 60, 1–12. issn:<br>0997-7546. doi: 10.1016/j.euromechflu.2016.06.012. http://dx.doi.org/10.1016/j.euromechflu.2016.06.012.<br>Kingma DP and Ba J (2014) Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980.<br>Knaepen B and Moin P (2004) Large-eddy simulation of conductive flows at low magnetic Reynolds number. Physics of Fluids<br>16(5), 1255–1261. issn: 1089-7666. doi: 10.1063/1.1651484. http://dx.doi.org/10.1063/1.1651484.<br>Krueger D, Caballero E, Jacobsen JH, Zhang A, Binas J, Zhang D, Priol RL and Courville A (2020) Out-of-Distribution<br>Generalization via Risk Extrapolation (REx). doi: 10.48550/ARXIV.2003.00688. Available at https://arxiv.org/abs/2003.<br>00688.<br>Larraondo PR, Renzullo LJ, Inza I and Lozano JA (2019) A data-driven approach to precipitation parameterizations using<br>convolutional encoder-decoder neural networks. doi: 10.48550/ARXIV.1903.10274. Available at https://arxiv.org/abs/1903.<br>10274.<br>Leslie DC and Quarini GL (1979) The application of turbulence theory to the formulation of subgrid modelling procedures.<br>Journal of Fluid Mechanics 91(01), 65. issn: 1469-7645. doi: 10.1017/s0022112079000045. http://dx.doi.org/10.1017/<br>S0022112079000045.<br>Li Z, Zheng H, Kovachki N, Jin D, Chen H, Liu B, Azizzadenesheli K and Anandkumar A (2021) Physics-Informed Neural<br>Operator for Learning Partial Differential Equations. doi: 10.48550/ARXIV.2111.03794. Available at https://arxiv.org/abs/<br>2111.03794.<br>Lund T (1997) On the use of discrete filters for large eddy simulation. Annual Research Briefs, 83–95.<br>Lupin-Jimenez L, Darman M, Hazarika S, Wu T, Gray M, He R, Wong A and Chattopadhyay A (2025) Simultane-<br>ous emulation and downscaling with physically-consistent deep learning-based regional ocean emulators. arXiv preprint<br>arXiv:2501.05058.<br>Mason PJ and Thomson DJ (1992) Stochastic backscatter in large-eddy simulations of boundary layers. Journal of Fluid<br>Mechanics 242, 51–78. issn: 1469-7645. doi: 10.1017/s0022112092002271. http://dx.doi.org/10.1017/S0022112092002271.<br>Mathieu M, Henaff M and LeCun Y (2013) Fast training of convolutional networks through ffts. arXiv preprint arXiv:1312.5851.<br>Mattheakis M, Joy H and Protopapas P (2021) Unsupervised Reservoir Computing for Solving Ordinary Differential Equations.<br>doi: 10.48550/ARXIV.2108.11417. Available at https://arxiv.org/abs/2108.11417.<br>Meneveau C and Katz J (2000) Scale-Invariance and Turbulence Models for Large-Eddy Simulation. Annual Review of Fluid<br>Mechanics 32(1), 1–32. issn: 1545-4479. doi: 10.1146/annurev.fluid.32.1.1. http://dx.doi.org/10.1146/annurev.fluid.32.1.1.<br>Orszag SA (1971) On the elimination of aliasing in finite-difference schemes by filtering high-wavenumber components. Journal<br>of Atmospheric Sciences 28(6), 1074–1074.<br>Pahlavan HA, Hassanzadeh P and Alexander MJ (2024a) Explainable Offline-Online Training of Neural Networks for<br>Parameterizations: A 1D Gravity Wave-QBO Testbed in the Small-Data Regime. Geophysical Research Letters 51(2),<br>e2023GL106324.<br>Pahlavan HA, Hassanzadeh P and Alexander MJ (2024b) Explainable Offline-Online Training of Neural Networks for<br>Parameterizations: A 1D Gravity Wave-QBO Testbed in the Small-Data Regime. Geophysical Research Letters 51(2). issn:<br>1944-8007. doi: 10.1029/2023gl106324. http://dx.doi.org/10.1029/2023GL106324.<br>Piomelli U (1999) Large-eddy simulation: achievements and challenges. Progress in Aerospace Sciences 35(4), 335–362. issn:<br>0376-0421. doi: 10.1016/s0376-0421(98)00014-1. http://dx.doi.org/10.1016/S0376-0421(98)00014-1.<br>Pope S (2000) Turbulent flows. Cambridge university press.<br>Porta Mana P and Zanna L (2014) Toward a stochastic parameterization of ocean mesoscale eddies. Ocean Modelling 79, 1–20.<br>issn: 1463-5003. doi: 10.1016/j.ocemod.2014.04.002. http://dx.doi.org/10.1016/j.ocemod.2014.04.002.<br>Pressel KG, Mishra S, Schneider T, Kaul CM and Tan Z (2017) Numerics and subgrid-scale modeling in large eddy<br>simulations of stratocumulus clouds. Journal of Advances in Modeling Earth Systems 9(2), 1342–1365. issn: 1942-2466. doi:<br>10.1002/2016ms000778. http://dx.doi.org/10.1002/2016MS000778.<br>Rasp S, Pritchard MS and Gentine P (2018) Deep learning to represent subgrid processes in climate models. Proceedings of<br>the National Academy of Sciences 115(39), 9684–9689. issn: 1091-6490. doi: 10.1073/pnas.1810286115. http://dx.doi.org/<br>10.1073/pnas.1810286115.<br>Ross A, Li Z, Perezhogin P, Fernandez-Granda C and Zanna L (2023) Benchmarking of machine learning ocean subgrid<br>parameterizations in an idealized model. Journal of Advances in Modeling Earth Systems 15(1), e2022MS003258.<br>Sagaut P (2005) Large eddy simulation for incompressible flows: an introduction. Springer Science & Business Media.<br><br>Sagaut P, Terracol M and Deck S (2013) Multiscale and multiresolution approaches in turbulence-LES, DES and Hybrid<br>RANS/LES Methods: Applications and Guidelines. World Scientific.<br>Sarlak H, Meneveau C and Sørensen J (2015) Role of subgrid-scale modeling in large eddy simulation of wind turbine wake<br>interactions. Renewable Energy 77, 386–399. issn: 0960-1481. doi: 10.1016/j.renene.2014.12.036. http://dx.doi.org/10.<br>1016/j.renene.2014.12.036.<br>Schneider T, Lan S, Stuart A and Teixeira J (2017) Earth System Modeling 2.0: A Blueprint for Models That Learn<br>From Observations and Targeted High-Resolution Simulations. Geophysical Research Letters 44(24). issn: 1944-8007. doi:<br>10.1002/2017gl076101. http://dx.doi.org/10.1002/2017GL076101.<br>Shevchenko I and Berloff P (2021) On a minimum set of equations for parameterisations in comprehensive ocean circulation<br>models. Ocean Modelling 168, 101913. issn: 1463-5003. doi: 10.1016/j.ocemod.2021.101913. http://dx.doi.org/10.1016/j.<br>ocemod.2021.101913.<br>Shinde V (2020) Proper orthogonal decomposition assisted subfilter-scale model of turbulence for large eddy simulation. Physical<br>Review Fluids 5(1). issn: 2469-990X. doi: 10.1103/physrevfluids.5.014605. http://dx.doi.org/10.1103/PhysRevFluids.5.<br>014605.<br>Smagorinsky (1963) GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS: I. THE BASIC<br>EXPERIMENT*. Monthly Weather Review 91(3), 99–164. issn: 1520-0493. doi: 10 . 1175 / 1520 - 0493(1963 ) 091<0099 :<br>gcewtp>2.3.co;2. http://dx.doi.org/10.1175/1520-0493(1963)091%3C0099:GCEWTP%3E2.3.CO;2.<br>Stevens RJ, Martínez-Tossas LA and Meneveau C (2018) Comparison of wind farm large eddy simulations using actuator disk<br>and actuator line models with wind tunnel experiments. Renewable Energy 116, 470–478. issn: 0960-1481. doi: 10.1016/j.<br>renene.2017.08.072. http://dx.doi.org/10.1016/j.renene.2017.08.072.<br>Subel A, Chattopadhyay A, Guan Y and Hassanzadeh P (2021) Data-driven subgrid-scale modeling of forced Burgers<br>turbulence using deep learning with generalization to higher Reynolds numbers via transfer learning. Physics of Fluids 33(3),<br>031702.<br>Subel A, Guan Y, Chattopadhyay A and Hassanzadeh P (2023) Explaining the physics of transfer learning in data-driven<br>turbulence modeling. PNAS Nexus, pgad015.<br>Subramanian S, Harrington P, Keutzer K, Bhimji W, Morozov D, Mahoney M and Gholami A (2023) Towards Foundation<br>Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior. doi: 10.48550/ARXIV.2306.00258.<br>https://arxiv.org/abs/2306.00258.<br>Sun YQ, Pahlavan HA, Chattopadhyay A, Hassanzadeh P, Lubis SW, Alexander MJ, Gerber E, Sheshadri A and<br>Guan Y (2023) Data Imbalance, Uncertainty Quantification, and Generalization via Transfer Learning in Data-driven<br>Parameterizations: Lessons from the Emulation of Gravity Wave Momentum Transport in WACCM. doi: 10.48550/ARXIV.<br>2311.17078. Available at https://arxiv.org/abs/2311.17078.<br>Tan Z, Schneider T, Teixeira J and Pressel KG (2017) Large-eddy simulation of subtropical cloud-topped boundary layers:<br>2. Cloud response to climate change. Journal of Advances in Modeling Earth Systems 9(1), 19–38. issn: 1942-2466. doi:<br>10.1002/2016ms000804. http://dx.doi.org/10.1002/2016MS000804.<br>Thuburn J, Kent J and Wood N (2013) Cascades, backscatter and conservation in numerical models of two-dimensional<br>turbulence. Quarterly Journal of the Royal Meteorological Society 140(679), 626–638. issn: 1477-870X. doi: 10.1002/qj.2166.<br>http://dx.doi.org/10.1002/qj.2166.<br>Xu C, Cao BT, Yuan Y and Meschke G (2023) Transfer learning based physics-informed neural networks for solving inverse prob-<br>lems in engineering structures under different loading scenarios. Computer Methods in Applied Mechanics and Engineering<br>405, 115852. issn: 0045-7825. doi: 10.1016/j.cma.2022.115852. http://dx.doi.org/10.1016/j.cma.2022.115852.<br>Xu W, Lu Y and Wang L (2022) Transfer Learning Enhanced DeepONet for Long-Time Prediction of Evolution Equations. doi:<br>10.48550/ARXIV.2212.04663. Available at https://arxiv.org/abs/2212.04663.<br>Yosinski J, Clune J, Bengio Y and Lipson H (2014) How transferable are features in deep neural networks? doi: 10.48550/<br>ARXIV.1411.1792. https://arxiv.org/abs/1411.1792.<br>Zanna L and Bolton T (2020) Data-Driven Equation Discovery of Ocean Mesoscale Closures. Geophysical Research Letters<br>47(17). issn: 1944-8007. doi: 10.1029/2020gl088376. http://dx.doi.org/10.1029/2020GL088376.<br>Zhou Y (1991) Eddy damping, backscatter, and subgrid stresses in subgrid modeling of turbulence. Physical Review A 43(12),<br>7049–7052. issn: 1094-1622. doi: 10.1103/physreva.43.7049. http://dx.doi.org/10.1103/PhysRevA.43.7049.<br>Zhuang F, Qi Z, Duan K, Xi D, Zhu Y, Zhu H, Xiong H and He Q (2021) A Comprehensive Survey on Transfer Learning.<br>Proceedings of the IEEE 109(1), 43–76. issn: 1558-2256. doi: 10.1109/jproc.2020.3004555. http://dx.doi.org/10.1109/<br>JPROC.2020.3004555.<br>

    Presenters

    • Moein Darman

      University of California, Santa Cruz

    Authors

    • Moein Darman

      University of California, Santa Cruz

    • Pedram Hassanzadeh

      University of Chicago

    • Ashesh K Chattopadhyay

      University of California, Santa Cruz

    • Laure Zanna

      Courant Institute of Mathematical Sciences

    View abstract →