Nonlinear dynamical systems with regime transitions are typically described by ordinary differential equations with jumping parameters parameters. Traditional methods often treat change-point detection and parameter estimation as separate tasks, ignoring the inherent coupling between them. To address this, we propose residual-loss anomaly analysis of physics-informed neural networks, a unified framework that leverages dynamical consistency within the physics-informed learning paradigm. This approach jointly infers piecewise parameters and transition points under a single set of constraints. The method follows a two-stage strategy: First, local physical residuals are analyzed through overlapping subinterval decomposition. When a subinterval spans a true transition point, the residual exhibits a distinct structural elevation in noise-free conditions, which has a non-zero lower bound, enabling effective localization of potential transition intervals. Second, within our framework, change-point locations and piecewise parameters are integrated into a unified physical loss function for joint optimization, enabling simultaneous identification. Experiments on benchmark nonlinear dynamical systems, including Malthusian and logistic growth models, Van der Pol oscillator, Lotka-Volterra model and Lorenz system, demonstrate that the proposed method outperforms traditional decoupled approaches in both change-point localization and parameter estimation accuracy. This study provides an efficient, unified solution for structurally coupled inverse problems in nonlinear dynamical systems with regime switching.

翻译：具有状态切换的非线性动力系统通常由带有跳跃参数的常微分方程描述。传统方法往往将变点检测与参数估计视为独立任务，忽略了二者固有的耦合性。为解决这一问题，我们提出基于物理信息的神经网络的残差损失异常分析——一种在物理信息学习范式下利用动力学一致性的统一框架。该方法在单一约束条件下联合推断分段参数与过渡点，采用两阶段策略：首先，通过重叠子区间分解分析局部物理残差。当子区间跨越真实过渡点时，残差在无噪声条件下呈现显著的结构性抬升，其非零下界能有效定位潜在过渡区间。其次，在我们的框架中，变点位置与分段参数被整合到统一的物理损失函数中进行联合优化，实现同步辨识。针对马尔萨斯增长模型、逻辑斯蒂增长模型、范德波尔振荡器、洛特卡-沃尔泰拉模型及洛伦兹系统等基准非线性动力系统的实验表明，所提方法在变点定位与参数估计精度上均优于传统解耦方法。本研究为具有状态切换的非线性动力系统中结构耦合逆问题提供了高效统一的解决方案。