Greedy Orthogonal Polynomial Manifolds with Lipschitz Control for Dimensionality Reduction and Reduced Modeling
Authors: Tomoki Koike, Benjamin Peherstorfer, Elizabeth Qian
Organizations: School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA · Institute of Mathematics, EPFL, Lausanne, Switzerland · School of Aerospace Engineering and School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, GA, USA
Polynomial manifolds provide nonlinear low-dimensional representations through polynomial decoders. We propose to represent the polynomial terms of the decoder using features that are orthonormal on the training data. This empirical orthogonality enables a greedy algorithm, Greedy-OPM, which jointly selects encoder modes and polynomial features while minimizing the reconstruction error under a prescribed bound on the decoder's Lipschitz constant. Numerical experiments show that this Lipschitz constraint improves generalization away from the training data while retaining high reconstruction accuracy.
Figures & tables
Figure 1: Linear advection-diffusion. Relative reconstruction error ( 36 ) against the reduced dimension n on the test parameters within ( w=2.81×10−3 ) and outside ( w=0.5×10−3 ) the training range.
Figure 2: Linear advection-diffusion. Active nonlinear features against n , with the Greedy-OPM (p=3) bars split into degree-2 and degree-3 contributions.
Figure 3: Linear advection-diffusion. Validation objective Eval and number of active nonlinear features against the Lipschitz budget P at n=10 . Dotted lines mark the selected P⋆ of each variant.
Figure 4: Linear advection-diffusion. Space-time field u(x,t) of the test parameter outside the training range ( w=0.5×10−3 ) reconstructed at n=10 , with the absolute error below each reconstruction.
Figure 5: Two-dimensional viscous Burgers. Relative reconstruction error ( 36 ) against n on the test parameters within ( Re=264.7 ) and outside ( Re=580 ) the training range.
Figure 6: Two-dimensional viscous Burgers. Active nonlinear features against n , degree-split for Greedy-OPM (p=3) .
Figure 7: Two-dimensional viscous Burgers. Validation objective Eval and number of active nonlinear features against the Lipschitz budget P at n=10 , with the selected P⋆ marked.
Figure 8: Two-dimensional viscous Burgers. Speed u2+v2 of the snapshot least captured by the linear POD subspace at the test parameter outside the training range ( Re=580 ), reconstructed at n=10 , with the absolute error below each reconstruction.
Figure 9: Boussinesq rising thermal plume. Relative reconstruction error ( 36 ) against n on the test parameters within ( Re=561.6 ) and outside ( Re=1250 ) the training range.
Figure 10: Boussinesq rising thermal plume. Active nonlinear features against n , degree-split for Greedy-OPM (p=3) .
Figure 11: Boussinesq rising thermal plume. Validation objective Eval and number of active nonlinear features against the Lipschitz budget P at n=10 , with the selected P⋆ marked.
Figure 12: Boussinesq rising thermal plume. Speed u2+v2 at t=10 for the test parameter outside the training range ( Re=1250 ), reconstructed at n=10 , with the absolute error below each reconstruction.