System Identification

Extrapolative prediction of complex nonlinear dynamics remains a central challenge in engineering. This study proposes a one-shot learning method to identify global frequency-response curves from a single excitation time history by learning governing equations. We introduce MEv-SINDy (Multi-frequency Evolutionary Sparse Identification of Nonlinear Dynamics) to infer the governing equations of non-autonomous and multi-frequency systems. The methodology leverages the Generalized Harmonic Balance (GHB) method to decompose complex forced responses into a set of slow-varying evolution equations. We validated the capabilities of MEv-SINDy on two critical Micro-Electro-Mechanical Systems (MEMS). These applications include a nonlinear beam resonator and a MEMS micromirror. Our results show that the model trained on a single point accurately predicts softening/hardening effects and jump phenomena across a wide range of excitation levels. This approach significantly reduces the data acquisition burden for the characterization and design of nonlinear microsystems.

Recent advances in Deep Learning (DL) have strengthened data-driven System Identification (SysID), with Neural and Fuzzy Ordinary Differential Equation (NODE/FODE) models achieving high accuracy in nonlinear dynamic modeling. Yet, system states in these frameworks are often reconstructed without clear physical meaning, and input contributions to the state derivatives remain difficult to interpret. To address these limitations, we propose Explainable FODE (xFODE), an interpretable SysID framework with integrated DL-based training. In xFODE, we define states in an incremental form to provide them with physical meanings. We employ fuzzy additive models to approximate the state derivative, thereby enhancing interpretability per input. To provide further interpretability, Partitioning Strategies (PSs) are developed, enabling the training of fuzzy additive models with explainability. By structuring the antecedent space during training so that only two consecutive rules are activated for any given input, PSs not only yield lower complexity for local inference but also enhance the interpretability of the antecedent space. To train xFODE, we present a DL framework with parameterized membership function learning that supports end-to-end optimization. Across benchmark SysID datasets, xFODE matches the accuracy of NODE, FODE, and NLARX models while providing interpretable insights.

Recent advances in Deep Learning (DL) have boosted data-driven System Identification (SysID), but reliable use requires Uncertainty Quantification (UQ) alongside accurate predictions. Although UQ-capable models such as Fuzzy ODE (FODE) can produce Prediction Intervals (PIs), they offer limited interpretability. We introduce Explainable Type-2 Fuzzy Additive ODEs for UQ (xFODE+), an interpretable SysID model which produces PIs alongside point predictions while retaining physically meaningful incremental states. xFODE+ implements each fuzzy additive model with Interval Type-2 Fuzzy Logic Systems (IT2-FLSs) and constraints membership functions to the activation of two neighboring rules, limiting overlap and keeping inference locally transparent. The type-reduced sets produced by the IT2-FLSs are aggregated to construct the state update together with the PIs. The model is trained in a DL framework via a composite loss that jointly optimizes prediction accuracy and PI quality. Results on benchmark SysID datasets show that xFODE+ matches FODE in PI quality and achieves comparable accuracy, while providing interpretability.

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.

Power System State Estimation (PSSE) converts geographically distributed measurements into the voltage magnitudes and phase angles needed for grid monitoring and control. Learned estimators can perform this mapping rapidly, but model-aware False Data Injection Attacks (FDIAs) may corrupt their inputs while retaining AC plausibility and residual-based stealth. Physics-Informed Neural Networks (PINNs) limit candidate states through power-flow consistency; however, their robustness depends on balancing supervised and physics losses whose scales and gradient contributions evolve during training. This paper proposes a PINN that jointly learns homoscedastic uncertainty parameters and uses them to adapt the two objectives. The formulation assigns trainable log-uncertainties to the active-power, reactive-power, voltage, and angle losses while safeguarding against an underweighted physics objective. The estimator is trained only on clean steady-state data and is evaluated, without adversarial retraining, under baseline state-distortion and stricter residual-profile-matching regimes on the IEEE~118-bus system. Accuracy is measured against the uncompromised system state. Relative to a fixed-weight PINN, dynamic weighting reduces overall Mean Absolute Error (MAE) by $55\%$ while also improving voltage- and angle-estimation accuracy. The results show that learning the physics/data balance from clean data improves robustness to unseen FDIAs.

This paper studies finite-sample set-membership identification for discrete-time bilinear systems under bounded symmetric log-concave disturbances. Our analysis considers trajectory-dependent regressors and allows marginally stable dynamics with polynomial mean-square state growth. We prove that the diameter of the feasible parameter set shrinks with sample complexity $\widetilde{\mathcal O}(1/ε)$ where $ε$ is the estimation error. Simulation supports the theory and illustrates the advantage of the proposed estimator for uncertainty quantification.

Is it possible to understand the intricacies of a dynamical system not solely from its input/output pattern, but also by observing the behavior of other systems within the same class? This central question drives the study presented in this paper. In response to this query, we introduce a novel paradigm for system identification, addressing two primary tasks: one-step-ahead prediction and multi-step simulation. Unlike conventional methods, we do not directly estimate a model for the specific system. Instead, we learn a meta model that represents a class of dynamical systems. This meta model is trained on a potentially infinite stream of synthetic data, generated by simulators whose settings are randomly extracted from a probability distribution. When provided with a context from a new system-specifically, an input/output sequence-the meta model implicitly discerns its dynamics, enabling predictions of its behavior. The proposed approach harnesses the power of Transformers, renowned for their \emph{in-context learning} capabilities. For one-step prediction, a GPT-like decoder-only architecture is utilized, whereas the simulation problem employs an encoder-decoder structure. Initial experimental results affirmatively answer our foundational question, opening doors to fresh research avenues in system identification.