Residual-loss Anomaly Analysis of Physics-Informed Neural Networks: An Inverse Method for Change-point Detection in Nonlinear Dynamical Systems with Regime Switching

Nonlinear system identification must balance physical interpretability with model flexibility. Classical methods yield structured, control-relevant models but rely on rigid parametric forms that often miss complex nonlinearities, whereas Neural ODEs are expressive yet largely black-box. Physics-Informed Neural Networks (PINNs) sit between these extremes, but inverse PINNs typically assume a known governing equation with fixed coefficients, leading to identifiability failures when the true dynamics are unknown or state-dependent. We propose \textbf{SOLIS}, which models unknown dynamics via a \emph{state-conditioned second-order surrogate model} and recasts identification as learning a Quasi-Linear Parameter-Varying (Quasi-LPV) representation, recovering interpretable natural frequency, damping, and gain without presupposing a global equation. SOLIS decouples trajectory reconstruction from parameter estimation and stabilizes training with a cyclic curriculum and \textbf{Local Physics Hints} windowed ridge-regression anchors that mitigate optimization collapse. Experiments on benchmarks show accurate parameter-manifold recovery and coherent physical rollouts from sparse data, including regimes where standard inverse methods fail.

This work proposes a two-stage physics-informed deep learning framework that combines neural-network-based sampling with statistical inference and constrained parameter refinement. In the first stage, a dual-network physics-informed architecture is used, where a main-network approximates the PDE solution and an auxiliary coefficient sub-network provides a relaxed continuous soft approximation of the true discontinuous coefficient field. A gradient-adaptive weighting strategy is incorporated to improve residual training and enhance sampling reliability near possible discontinuity regions. The sampled coefficient values are then analyzed using Bayesian learning for Gaussian mixture models and birth-death Markov chain model selection, which identify the number of coefficient regimes and provide candidate intervals for coefficient values and transition regions. In the second stage, the inverse problem is reformulated as a constrained physics-informed estimator, in which the coefficient is replaced by a form-consistent hard approximation explicitly represented as a piecewise-constant function over the spatiotemporal domain. Comprehensive numerical experiments on PDEs with jump-discontinuous coefficients demonstrate that the proposed framework achieves adaptability and accurate parameter identification with acceptable computational costs compared to existing methods. Applications to solution reconstruction further illustrate its practical potential. This work provides a generalizable computational approach for inverse problems governed by PDEs with discontinuous parameter structures, particularly in non-stationary and heterogeneous systems.

Statistical change point (CP) detection methods typically rely on likelihood-based inference and ignore contextual information about plausible CP locations beyond the observed sequence. Although informative priors provide a natural way to incorporate such information, general and computationally efficient methods for doing so are lacking, especially for multiple CP detection. To address this gap, we propose a prior-informed CP detection algorithm (Pi-Change) that incorporates prior information on CP locations through a time-varying penalty term. We prove that the proposed penalty can be embedded in the Pruned Exact Linear Time framework while preserving the dynamic programming recursion and pruning rule required for efficient multiple CP detection. Across simulation studies and three time-series applications, Pi-Change discourages spurious CPs unsupported by prior information, remains robust to prior misspecification, and improves detection accuracy. More broadly, Pi-Change extends multiple CP detection beyond purely data-driven fitting by incorporating partial prior knowledge in a computationally efficient and interpretable way. It is particularly useful when CPs arise from heterogeneous mechanisms or are associated with known external events, helping quantify the delay between an event and the resulting structural change.