In this work, we present reduced-order models (ROMs) for a nonlinear cross-diffusion problem from population dynamics, the Shigesada-Kawasaki-Teramoto (SKT) equation with Lotka-Volterra kinetics. The formation of the patterns of the SKT equation consists of a fast transient phase and a long stationary phase. Reduced order solutions are computed by separating the time into two time-intervals. In numerical experiments, we show for one- and two-dimensional SKT equations with pattern formation, the reduced-order solutions obtained in the time-windowed form, i.e., principal decomposition framework, are more accurate than the global proper orthogonal decomposition solutions obtained in the whole time interval. The finite-difference discretization of the SKT equation in space results in a system of linear-quadratic ordinary differential equations. The ROMs have the same linear-quadratic structure as the full order model. Using the linear-quadratic structure of the ROMs, the computation of the reduced-order solutions is further accelerated by the use of proper orthogonal decomposition in a tensorial framework so that the computations in the reduced system are independent of the full-order solutions. Furthermore, the prediction capabilities of the ROMs are illustrated for one- and two-dimensional patterns. Finally, we show that the entropy is decreasing by the reduced solutions, which is important for the global existence of solutions to the nonlinear cross-diffusion equations such as the SKT equation.

在这项工作中，我们针对种群动力学，具有Lotka-Volterra动力学的Shigesada-Kawasaki-Teramoto（SKT）方程，提出了一种基于种群动力学的非线性交叉扩散问题的降阶模型（ROM）。SKT方程模式的形成包括一个快速的瞬态阶段和一个长的静止阶段。通过将时间分成两个时间间隔来计算降阶解。在数值实验中，我们证明了对于具有模式形成的一维和二维SKT方程，以时间窗形式获得的降阶解（即主分解框架）比从中获得的全局固有正交分解解更准确。整个时间间隔。空间中SKT方程的有限差分离散化导致了一个线性二次方常微分方程组。ROM具有与全阶模型相同的线性二次结构。使用ROM的线性二次结构，通过在张量框架中使用适当的正交分解，进一步简化了降阶解的计算，从而使简化系统中的计算与全阶解无关。此外，针对一维和二维模式示出了ROM的预测能力。最后，我们表明，熵通过减少的解而减小，这对于非线性交叉扩散方程（例如SKT方程）的全局存在性很重要。通过在张量框架中使用适当的正交分解，进一步简化了降阶解的计算，从而使降阶系统中的计算独立于全阶解。此外，针对一维和二维模式示出了ROM的预测能力。最后，我们表明，熵通过减少的解而减小，这对于非线性交叉扩散方程（例如SKT方程）的全局存在性很重要。通过在张量框架中使用适当的正交分解，进一步简化了降阶解的计算，从而使降阶系统中的计算独立于全阶解。此外，针对一维和二维模式示出了ROM的预测能力。最后，我们表明，熵通过减少的解而减小，这对于非线性交叉扩散方程（例如SKT方程）的全局存在性很重要。