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 modes depend nonlinearly on a set of time-varying parameters. RONS uses a set of ordinary differential equations (ODEs) for the parameters to optimally evolve the shape of the modes to adapt to the PDE's solution. This method has already proven extremely effective in tackling challenging problems such as advection-dominated flows and high-dimensional PDEs. 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, with the time-dependent parameters being the weights and biases of the network. The RONS equations dictate the optimal evolution of the network's parameters without requiring any training.

翻译：我们开发了用于计算降阶非线性解（RONS）的快速且可扩展的方法。RONS近期被提出作为时间相关偏微分方程（PDE）降阶建模的框架，其中模态非线性地依赖于一组时变参数。RONS通过一组关于参数的常微分方程（ODE）来最优地演化模态的形状，以适应偏微分方程的解。该方法在处理对流主导流动和高维偏微分方程等具有挑战性的问题方面已展现出极高的有效性。然而，随着参数数量的增加，RONS方程的积分甚至其构建在计算上变得不可行。本文开发了三种独立的方法来解决这些计算瓶颈：符号化RONS、配置点RONS和正则化RONS。我们通过两个示例（高维福克-普朗克方程和库拉莫托-西瓦辛斯基方程）证明了这些方法的有效性。在这两种情况下，我们观察到所提出的方法在加速比和精度上实现了数个数量级的提升。所提出的方法将RONS的适用性扩展到降阶建模之外，使其能够用于线性和非线性偏微分方程的精确数值求解。最后，作为RONS的一个特例，我们讨论了其在偏微分方程解由神经网络近似的问题中的应用，其中时变参数为网络的权重和偏置。RONS方程决定了网络参数的最优演化，而无需任何训练。