AI 中文总结
研究变分不等式求解，提出状态依赖缩放投影神经网络（SD-SPNN），其投影算子几何随状态演变，通过嵌入连续时间预处理推广经典模型，建立相关性质，统一多种投影神经网络，数值实验验证其有效性。
AI 中文摘要
基于投影的动力系统和投影神经网络提供了一种连续时间方法，通过将状态驱动到其在可行集上的投影来解决变分不等式。然而，大多数现有模型基于固定的欧几里得或恒定度量，当算子或可行集几何高度各向异性时，可能导致条件不佳和收敛缓慢。本文引入了一种状态依赖度量投影动力系统，即状态依赖缩放投影神经网络（SD-SPNN），其中投影算子的几何形状随状态平滑演变。该动力学通过将连续时间预处理直接嵌入流中，推广了经典投影动力系统和投影神经网络。在算子的标准单调性假设和度量的适度正则性条件下，我们建立了解的存在性、与基础变分不等式的精确平衡解对应关系以及基于李雅普诺夫的稳定性性质。该框架在单个连续时间模型中统一了欧几里得、恒定度量和状态依赖投影神经网络。数值实验说明了状态依赖度量如何重塑投影动力学的瞬态几何形状，同时收敛到相同的平衡。
英文摘要
Projection-based dynamical systems and projection neural networks offer a continuous-time approach to solving variational inequalities by driving the state toward its projection onto the feasible set. However, most existing models are built on a fixed Euclidean or constant metric, which can lead to poor conditioning and slow convergence when the operator or feasible set geometry is highly anisotropic. This paper introduces a state-dependent metric projection dynamical system, referred to as a state-dependent scaled projection neural network (SD-SPNN), in which the geometry of the projection operator evolves smoothly with the state. The dynamics generalize classical projected dynamical systems and projection neural networks by embedding continuous-time preconditioning directly into the flow. Under standard monotonicity assumptions on the operator and mild regularity conditions on the metric, we establish existence of solutions, an exact equilibrium-solution correspondence with the underlying variational inequality, and Lyapunov-based stability properties. The framework unifies Euclidean, constant-metric, and state-dependent projection neural network within a single continuous-time model. Numerical experiments illustrate how state-dependent metrics reshape the transient geometry of the projected dynamics while converging to the same equilibrium.
Comments28 pages, 6 figures
