Greedy construction of quadratic manifolds for nonlinear dimensionality reduction and nonlinear model reduction (2403.06732v2)
Abstract: Dimensionality reduction on quadratic manifolds augments linear approximations with quadratic correction terms. Previous works rely on linear approximations given by projections onto the first few leading principal components of the training data; however, linear approximations in subspaces spanned by the leading principal components alone can miss information that are necessary for the quadratic correction terms to be efficient. In this work, we propose a greedy method that constructs subspaces from leading as well as later principal components so that the corresponding linear approximations can be corrected most efficiently with quadratic terms. Properties of the greedily constructed manifolds allow applying linear algebra reformulations so that the greedy method scales to data points with millions of dimensions. Numerical experiments demonstrate that an orders of magnitude higher accuracy is achieved with the greedily constructed quadratic manifolds compared to manifolds that are based on the leading principal components alone.
- Interpolatory Methods for Model Reduction. SIAM, Philadelphia, 2020.
- J. Barnett and C. Farhat. Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction. J. Comput. Phys., 464:111348, 2022.
- Neural-network-augmented projection-based model order reduction for mitigating the Kolmogorov barrier to reducibility. J. Comput. Phys., 492:112420, 2023.
- M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6):1373 – 1396, 2003. Cited by: 5995; All Open Access, Green Open Access.
- P. Benner and T. Breiten. Two-sided projection methods for nonlinear model order reduction. SIAM Journal on Scientific Computing, 37(2):B239–B260, 2015.
- ℋ2subscriptℋ2\mathcal{H}_{2}caligraphic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-quasi-optimal model order reduction for quadratic-bilinear control systems. SIAM Journal on Matrix Analysis and Applications, 39(2):983–1032, 2018.
- A quadratic decoder approach to nonintrusive reduced-order modeling of nonlinear dynamical systems. PAMM, 23(1):e202200049, 2023.
- A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev., 57(4):483–531, 2015.
- AMR-Wind: Adaptive mesh-refinement for atmospheric-boundary-layer wind energy simulations. In APS Division of Fluid Dynamics Meeting Abstracts, APS Meeting Abstracts, page T29.007, 2021.
- Robust principal component analysis? J. ACM, 58(3), jun 2011.
- The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math. Program., 155(1–2):57–79, 2014.
- Nonlinear compressive reduced basis approximation for PDE’s. working paper or preprint, 2023.
- Intermodal energy transfers in a proper orthogonal decomposition–Galerkin representation of a turbulent separated flow. Journal of Fluid Mechanics, 491:275–284, 2003.
- D. L. Donoho and C. Grimes. Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. Proceedings of the National Academy of Sciences, 100(10):5591–5596, 2003.
- Turbulence modeling in the age of data. Annual Review of Fluid Mechanics, 51(1):357–377, 2019.
- Learning latent representations in high-dimensional state spaces using polynomial manifold constructions, 2023.
- Learning physics-based reduced-order models from data using nonlinear manifolds, 2023.
- Operator inference for non-intrusive model reduction with quadratic manifolds. Comput. Methods Appl. Mech. Engrg., 403:115717, 2023.
- A priori estimation of memory effects in reduced-order models of nonlinear systems using the Mori-Zwanzig formalism. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2205):20170385, 2017.
- P. Goyal and P. Benner. Generalized quadratic embeddings for nonlinear dynamics using deep learning. arXiv, 2211.00357, 2024.
- Guaranteed stable quadratic models and their applications in SINDy and operator inference. arXiv, 2308.13819, 2024.
- C. Gu. Qlmor: A projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlinear systems. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 30(9):1307–1320, 2011.
- Reducing the dimensionality of data with neural networks. Science, 313(5786):504–507, 2006.
- A quadratic manifold for model order reduction of nonlinear structural dynamics. Computers & Structures, 188:80–94, 2017.
- Promoting global stability in data-driven models of quadratic nonlinear dynamics. Phys. Rev. Fluids, 6:094401, Sep 2021.
- Learning nonlinear reduced models from data with operator inference. Annual Review of Fluid Mechanics, 56(1):521–548, 2024.
- Dynamic mode decomposition: data-driven modeling of complex systems. SIAM, 2016.
- S. Pan and K. Duraisamy. Data-driven discovery of closure models. SIAM Journal on Applied Dynamical Systems, 17(4):2381–2413, 2018.
- B. Peherstorfer. Breaking the Kolmogorov barrier with nonlinear model reduction. Notices of the American Mathematical Society, 69(5):725–733, 2022.
- B. Peherstorfer and K. Willcox. Data-driven operator inference for nonintrusive projection-based model reduction. Computer Methods in Applied Mechanics and Engineering, 306:196–215, 2016.
- Lift & learn: Physics-informed machine learning for large-scale nonlinear dynamical systems. Physica D: Nonlinear Phenomena, 406:132401, 2020.
- An explicit nonlinear mapping for manifold learning. IEEE Transactions on Cybernetics, 43(1):51 – 63, 2013. Cited by: 74; All Open Access, Green Open Access.
- Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323–2326, 2000.
- Spectral analysis of nonlinear flows. Journal of Fluid Mechanics, 641:115–127, 2009.
- Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng., 15(3):229–275, 2008.
- Generalization of quadratic manifolds for reduced order modeling of nonlinear structural dynamics. Computers & Structures, 192:196–209, 2017.
- P. Sagaut. Large Eddy Simulation for Incompressible Flows: An Introduction. Springer-Verlag, 2006.
- Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference. Computer Methods in Applied Mechanics and Engineering, 404:115836, 2023.
- Robust principal component analysis for modal decomposition of corrupt fluid flows. Phys. Rev. Fluids, 5:054401, May 2020.
- M. Schlegel and B. R. Noack. On long-term boundedness of Galerkin models. Journal of Fluid Mechanics, 765:325–352, 2015.
- P. J. Schmid. Dynamic mode decomposition of numerical and experimental data. Fluid Mech., 656:5–28, 2010.
- Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10(5):1299 – 1319, 1998. Cited by: 6579; All Open Access, Green Open Access.
- Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds. Computer Methods in Applied Mechanics and Engineering, 417:116402, 2023.
- A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500):2319–2323, 2000.
- On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 1(2):391–421, 2014.
- W. I. T. Uy and B. Peherstorfer. Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectories. Journal of Scientific Computing, 88(3):91, Aug 2021.
- L. van der Maaten and G. Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, 9(86):2579–2605, 2008.
- Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison. Computer Methods in Applied Mechanics and Engineering, 237-240:10–26, 2012.
- Data-driven filtered reduced order modeling of fluid flows. SIAM Journal on Scientific Computing, 40(3):B834–B857, 2018.
- Data-driven identification of quadratic representations for nonlinear Hamiltonian systems using weakly symplectic liftings. arXiv, 2308.01084, 2024.
- L. Zanna and T. Bolton. Data-driven equation discovery of ocean mesoscale closures. Geophysical Research Letters, 47(17):e2020GL088376, 2020.