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.

翻译：利用非线性参数化进行模型降阶可以克服影响输运主导问题的Kolmogorov障碍。本工作基于神经伽辽金方案给出的降阶动力学，提出在二次流形上参数化相应的降阶解。我们证明了所提出的二次流形神经伽辽金降阶模型的解具有局部唯一性，并能随时间最小化残差范数，从而提升稳定性和精度。对于线性问题，二次流形神经伽辽金降阶模型实现了在线效率，即预测成本与底层全模型的状态维度无关。对于非线性问题，我们证明神经伽辽金方案允许使用独立于全模型网格点的配置点来评估残差函数，这可视为一种超降阶形式。通过对电场中平流波和带电粒子密度的数值实验表明，二次流形神经伽辽金降阶模型相比全模型可实现数量级的速度提升。