We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the reduced model depends nonlinearly on a set of time-varying parameters. RONS obtains an explicit set of ordinary differential equations (ODEs) for the parameters, which optimally evolve the shape of the approximate solution. However, as the number of parameters grow, integrating the RONS equation and even its formation become computationally prohibitive. Here, we develop three separate methods to address these computational bottlenecks: symbolic RONS, collocation RONS and regularized RONS. We demonstrate the efficacy of these methods on two examples: Fokker-Planck equation in high dimensions and the Kuramoto--Sivashinsky equation. In both cases, we observe that the proposed methods lead to several orders of magnitude in speedup and accuracy. Our proposed methods extend the applicability of RONS beyond reduced-order modeling by making it possible to use RONS for accurate numerical solution of linear and nonlinear PDEs. Finally, as a special case of RONS, we discuss its application to problems where the PDE's solution is approximated by a neural network, where the time-dependent parameters are the weights and biases of the network.

翻译：我们开发了用于计算降阶非线性解（RONS）的快速可扩展方法。RONS近期被提出作为时间相关偏微分方程（PDEs）降阶建模的框架，其中降阶模型非线性依赖于一组时变参数。RONS为这些参数导出一组显式常微分方程（ODEs），从而最优地演化近似解的形状。然而，随着参数数量增长，积分RONS方程乃至其形成过程在计算上变得难以承受。本文提出了三种独立方法来解决这些计算瓶颈：符号RONS、配置点RONS和正则化RONS。我们通过两个实例验证了这些方法的有效性：高维福克-普朗克方程和库拉莫托-西瓦辛斯基方程。在这两种情况下，观测结果表明所提出的方法在加速和精度上均提升了数个数量级。我们的方法将RONS的适用性从降阶建模扩展到线性与非线性PDE的精确数值解。最后，作为RONS的特例，我们讨论了其在PDE解由神经网络近似问题中的应用，其中时变参数即为网络的权重和偏置。