Nonlinear model reduction with Neural Galerkin schemes on quadratic manifolds

TL;DR

The paper advances nonlinear model reduction by integrating Neural Galerkin schemes with quadratic manifold parametrizations to produce QMNG reduced models. These models exhibit locally unique solutions and minimal residual norms, providing stable, accurate dynamics and enabling hyper-reduction via separate collocation points from full-model grids. The framework offers online efficiency for linear full models and demonstrates substantial speedups across transport-dominated problems, including Vlasov-type and Hamiltonian systems, with robust stability when residual minimization is enforced. The work further clarifies the theoretical structure of Jacobians for quadratic decoders and provides practical offline/online algorithms, validated by comprehensive numerical experiments.

Abstract

Leveraging nonlinear parametrizations for model reduction can overcome the Kolmogorov barrier that affects transport-dominated problems. In this work, we build on the reduced dynamics given by Neural Galerkin schemes and propose to parametrize the corresponding reduced solutions on quadratic manifolds. We show that the solutions of the proposed quadratic-manifold Neural Galerkin reduced models are locally unique and minimize the residual norm over time, which promotes stability and accuracy. For linear problems, quadratic-manifold Neural Galerkin reduced models achieve online efficiency in the sense that the costs of predictions scale independently of the state dimension of the underlying full model. For nonlinear problems, we show that Neural Galerkin schemes allow using separate collocation points for evaluating the residual function from the full-model grid points, which can be seen as a form of hyper-reduction. Numerical experiments with advecting waves and densities of charged particles in an electric field show that quadratic-manifold Neural Galerkin reduced models lead to orders of magnitude speedups compared to full models.

Figures (9)

• Figure 1: Acoustic waves: Density field snapshots from the full model of the Hamiltonian wave problem (first row), their approximations on the quadratic manifold (second row) and the corresponding approximation obtained with the QMNG reduced model of dimension $n=30$ (third row). Results are shown for the test parameter $\mu_3^{\text{(test)}} = 0.5$.
• Figure 2: Acoustic waves: Averaged relative test error of QMNG reduced models for different reduced dimensions $n$ and the corresponding online speedups. The QMNG reduced model achieves an error that is close to the reconstruction error of the data on the quadratic manifold, which shows that QMNG leverages the expressivity of the quadratic manifold.
• Figure 3: Acoustic waves: Reduced models that use constant approximations of the Jacobian matrix are unstable for quadratic manifolds that are fitted well to the data (low regularization parameter $\gamma$). In contrast, QMNG reduced models use the actual Jacobian matrix without an approximation and so minimize the residual norm, which leads to stable predictions also when quadratic manifolds are trained with small regularization parameters.
• Figure 4: Acoustic waves: The reduced dynamics corresponding to constant approximations of the Jacobian matrix are unstable in our experiments and require strongly regularized quadratic manifolds to be stable (here $\gamma = 10^3$). At the same time, using constant test spaces avoids having to assemble the Jacobian matrix as in QMNG reduced models, which leads to higher speedups compared to QMNG.
• Figure 5: Charged particles: Snapshots from the full model of the Vlasov problem, their reconstruction after projection onto the quadratic manifold and the corresponding reduced solutions from the QMMG reduced model of size $n = 30$.

Theorems & Definitions (4)

• Lemma 4.1
• proof
• Proposition 4.2
• proof