Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
166 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A Multifidelity deep operator network approach to closure for multiscale systems (2303.08893v2)

Published 15 Mar 2023 in physics.comp-ph, cs.LG, cs.NA, math.NA, and physics.flu-dyn

Abstract: Projection-based reduced order models (PROMs) have shown promise in representing the behavior of multiscale systems using a small set of generalized (or latent) variables. Despite their success, PROMs can be susceptible to inaccuracies, even instabilities, due to the improper accounting of the interaction between the resolved and unresolved scales of the multiscale system (known as the closure problem). In the current work, we interpret closure as a multifidelity problem and use a multifidelity deep operator network (DeepONet) framework to address it. In addition, to enhance the stability and accuracy of the multifidelity-based closure, we employ the recently developed "in-the-loop" training approach from the literature on coupling physics and machine learning models. The resulting approach is tested on shock advection for the one-dimensional viscous Burgers equation and vortex merging using the two-dimensional Navier-Stokes equations. The numerical experiments show significant improvement of the predictive ability of the closure-corrected PROM over the un-corrected one both in the interpolative and the extrapolative regimes.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (66)
  1. Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, 2012.
  2. Reduced-order modelling for flow control, volume 528. Springer Science & Business Media, 2011.
  3. Intermodal energy transfers in a proper orthogonal decomposition–galerkin representation of a turbulent separated flow. Journal of Fluid Mechanics, 491:275–284, 2003.
  4. Stable Galerkin reduced order models for linearized compressible flow. Journal of Computational Physics, 228(6):1932–1946, 2009.
  5. Spatio-temporal symmetries and bifurcations via bi-orthogonal decompositions. Journal of Nonlinear Science, 2:183–215, 1992.
  6. Stability properties of POD–Galerkin approximations for the compressible Navier–Stokes equations. Theoretical and Computational Fluid Dynamics, 13(6):377–396, 2000.
  7. Lagrangian reduced order modeling using finite time Lyapunov exponents. Fluids, 5(4):189, 2020a.
  8. Edward N Lorenz. The predictability of a flow which possesses many scales of motion. Tellus, 21(3):289–307, 1969.
  9. The real butterfly effect. Nonlinearity, 27(9):R123, 2014.
  10. Stabilization of projection-based reduced order models of the Navier–Stokes. Nonlinear Dynamics, 70:1619–1632, 2012.
  11. Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation. Journal of Fluid Mechanics, 729:285–308, 2013.
  12. Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier–Stokes equations. Journal of Computational Physics, 321:224–241, 2016.
  13. On closures for reduced order models—a spectrum of first-principle to machine-learned avenues. Physics of Fluids, 33(9):091301, 2021.
  14. The dynamics of coherent structures in the wall region of a turbulent boundary layer. Journal of fluid Mechanics, 192:115–173, 1988.
  15. A spectral viscosity method for correcting the long-term behavior of POD models. Journal of Computational Physics, 194(1):92–116, 2004.
  16. Artificial viscosity proper orthogonal decomposition. Mathematical and Computer Modelling, 53(1-2):269–279, 2011.
  17. Problem reduction, renormalization, and memory. Communications in Applied Mathematics and Computational Science, 1(1):1–27, 2007.
  18. Approximate deconvolution reduced order modeling. Computer Methods in Applied Mechanics and Engineering, 313:512–534, 2017.
  19. An ensemble-proper orthogonal decomposition method for the nonstationary Navier–Stokes equations. SIAM Journal on Numerical Analysis, 55(1):286–304, 2017.
  20. Neural network closures for nonlinear model order reduction. Advances in Computational Mathematics, 44:1717–1750, 2018.
  21. Data-driven discovery of closure models. SIAM Journal on Applied Dynamical Systems, 17(4):2381–2413, 2018.
  22. A deep learning enabler for nonintrusive reduced order modeling of fluid flows. Physics of Fluids, 31(8):085101, 2019.
  23. Neural closure models for dynamical systems. Proceedings of the Royal Society A, 477(2252):20201004, 2021.
  24. Neural Galerkin scheme with active learning for high-dimensional evolution equations. arXiv preprint arXiv:2203.01360, 2022.
  25. Stabilized neural ordinary differential equations for long-time forecasting of dynamical systems. Journal of Computational Physics, 474:111838, 2023.
  26. Closure learning for nonlinear model reduction using deep residual neural network. Fluids, 5(1):39, 2020b.
  27. Stephan Rasp. Coupled online learning as a way to tackle instabilities and biases in neural network parameterizations: general algorithms and Lorenz 96 case study (v1. 0). Geoscientific Model Development, 13(5):2185–2196, 2020.
  28. Interpreting and stabilizing machine-learning parametrizations of convection. Journal of the Atmospheric Sciences, 77(12):4357–4375, 2020.
  29. Opportunities and challenges for machine learning in weather and climate modelling: hard, medium and soft AI. Philosophical Transactions of the Royal Society A, 379(2194):20200083, 2021.
  30. A toy model to investigate stability of ai-based dynamical systems. Geophysical Research Letters, 48(8):e2020GL092133, 2021.
  31. A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence. Theoretical and Computational Fluid Dynamics, 34:429–455, 2020.
  32. Stable a posteriori les of 2d turbulence using convolutional neural networks: Backscattering analysis and generalization to higher re via transfer learning. Journal of Computational Physics, 458:111090, 2022a.
  33. Learning physics-constrained subgrid-scale closures in the small-data regime for stable and accurate LES. Physica D: Nonlinear Phenomena, page 133568, 2022b.
  34. Frame invariant neural network closures for Kraichnan turbulence. Physica A: Statistical Mechanics and its Applications, 609:128327, 2023.
  35. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature machine intelligence, 3(3):218–229, 2021.
  36. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020.
  37. A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data. Computer Methods in Applied Mechanics and Engineering, 393:114778, 2022a.
  38. Multifidelity deep operator networks. arXiv preprint arXiv:2204.09157, 2022.
  39. Solver-in-the-loop: Learning from differentiable physics to interact with iterative PDE-solvers. Advances in Neural Information Processing Systems, 33:6111–6122, 2020.
  40. Machine learning–accelerated computational fluid dynamics. Proceedings of the National Academy of Sciences, 118(21):e2101784118, 2021.
  41. Physics-based Deep Learning. WWW, 2021. URL https://physicsbaseddeeplearning.org.
  42. Learned turbulence modelling with differentiable fluid solvers: physics-based loss functions and optimisation horizons. Journal of Fluid Mechanics, 949:A25, 2022.
  43. Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Transactions on Neural Networks, 6(4):911–917, 1995.
  44. Improved architectures and training algorithms for deep operator networks. Journal of Scientific Computing, 92(2):35, 2022.
  45. Learning the solution operator of parametric partial differential equations with physics-informed DeepONets. Science advances, 7(40):eabi8605, 2021.
  46. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378:686–707, 2019.
  47. Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport. Physical Review Research, 4(2):023210, 2022b.
  48. Bi-fidelity modeling of uncertain and partially unknown systems using DeepONets. arXiv preprint arXiv:2204.00997, 2022.
  49. On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flows. Journal of Computational Physics, 419:109681, 2020.
  50. Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations. International Journal for numerical methods in engineering, 86(2):155–181, 2011.
  51. Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction. Journal of Computational Physics, 330:693–734, 2017.
  52. End-to-end differentiable physics for learning and control. Advances in Neural Information Processing Systems, 31, 2018.
  53. Carlos A Michelén Ströfer and Heng Xiao. End-to-end differentiable learning of turbulence models from indirect observations. Theoretical and Applied Mechanics Letters, 11(4):100280, 2021.
  54. Windowed least-squares model reduction for dynamical systems. Journal of Computational Physics, 426:109939, 2021.
  55. Windowed space–time least-squares Petrov–Galerkin model order reduction for nonlinear dynamical systems. Computer Methods in Applied Mechanics and Engineering, 386:114050, 2021.
  56. Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations. International Journal for Numerical Methods in Fluids, 76(8):497–521, 2014.
  57. Comparative numerical analysis using reduced-order modeling strategies for nonlinear large-scale systems. Journal of Computational and Applied Mathematics, 310:32–43, 2017.
  58. The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. Journal of Computational Physics, 242:623–647, 2013.
  59. On the stochastic stability of deep markov models. Advances in Neural Information Processing Systems, 34:24033–24047, 2021.
  60. Data-driven variational multiscale reduced order models. Computer Methods in Applied Mechanics and Engineering, 373:113470, 2021.
  61. Error estimates for DeepONets: A deep learning framework in infinite dimensions. Transactions of Mathematics and Its Applications, 6(1):tnac001, 2022.
  62. Benjamin Peherstorfer. Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling. SIAM Journal on Scientific Computing, 42(5):A2803–A2836, 2020.
  63. Predictive reduced order modeling of chaotic multi-scale problems using adaptively sampled projections. arXiv preprint arXiv:2301.09006, 2023.
  64. Real-time reduced-order modeling of stochastic partial differential equations via time-dependent subspaces. Journal of Computational Physics, 415:109511, 2020.
  65. On-the-fly reduced order modeling of passive and reactive species via time-dependent manifolds. Computer Methods in Applied Mechanics and Engineering, 382:113882, 2021.
  66. Machine-learning-based spectral methods for partial differential equations. Scientific Reports, 13(1):1739, 2023.
Citations (12)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now