Physics-informed learning of interpretable surrogates for modeling and control

Abstract

This thesis presents a physics-informed framework for learning interpretable surrogate models of nonlinear dynamical systems from sparse trajectory data and for using these learned models in controller synthesis. The main motivation of the study is the long-standing trade-off between flexibility and interpretability in nonlinear system identification. Classical identification methods provide structured models that are suitable for analysis and control design, but they may become restrictive when the system exhibits strong nonlinearities or state-dependent effects. In contrast, neural network based models can represent complex dynamics with high flexibility, yet they often lack a transparent structure that can be directly used for physical interpretation or controller design. To address this issue, the thesis proposes a structured learning framework based on a second-order Quasi-LPV surrogate hypothesis. Instead of assuming a fixed global parametric model, the unknown nonlinear system is locally approximated by a second-order surrogate whose coefficients vary with the state. These coefficients correspond to effective stiffness, damping, and input gain, which preserve the semantics of classical second-order systems while allowing nonlinear behavior to be represented through state dependence. In this way, the identification problem is reformulated as the learning of a state-conditioned coefficient field rather than the estimation of a single global parameter vector. The proposed method employs a dual-network architecture. A Solution Network is used to reconstruct smooth continuous-time trajectories from sparse measurements, while a Parameter Network predicts the local surrogate coefficients along the reconstructed trajectory. These two components are coupled through a physics-informed residual derived from the surrogate dynamics. In order to reduce the optimization difficulties commonly observed in inverse physics-informed learning, a curriculum-based training strategy is adopted. The training alternates between trajectory reconstruction and parameter identification stages. In the parameter learning stage, local ridge-regression based physics hints are used as weak supervision in order to guide the optimization and reduce convergence to trivial or physically inconsistent solutions. The framework is evaluated on three benchmark systems with different nonlinear characteristics: the Van der Pol oscillator, the Duffing oscillator, and a real two-tank process. The performance analysis is carried out from two main identification perspectives. First, the trajectory reconstruction ability of the framework is examined in order to assess how accurately the measured system responses can be recovered under the second-order surrogate assumption. Second, the quality of the learned surrogate model is evaluated through open-loop prediction and phase portrait analysis, in order to determine how well the local coefficient field captures the underlying nonlinear dynamics. The results show that the proposed framework is able to reconstruct trajectories accurately and to produce coherent local surrogate models that remain physically interpretable. Compared with baseline inverse PINN formulations based on fixed global parameters, the proposed state-dependent surrogate approach provides better consistency in representing nonlinear dynamics and better open-loop predictive behavior on the considered benchmark systems. These findings support the main premise of the thesis, namely that nonlinear system identification can be formulated in a way that preserves both data-driven flexibility and control-oriented structure. In addition to identification, the thesis also considers the use of the learned surrogate model in controller design. After the offline learning stage, the identified local second-order model is used to construct a gain-scheduled state-feedback controller through algebraic polynomial matching. By using the learned local stiffness, damping, and input gain, the controller gains are updated according to the current operating point. This provides a simple and interpretable way to connect the learned surrogate model with real-time control. The closed-loop control study is conducted on the Duffing oscillator as a case study. The results indicate that the learned surrogate can support scheduled controller design and can be used not only for analysis and prediction, but also for control-oriented applications. Overall, the thesis proposes a unified perspective that connects nonlinear system identification, interpretable surrogate modeling, and gain-scheduled control. Its main contribution is to show that physics-informed learning can be used not only to fit trajectories, but also to recover local dynamical structure that is meaningful from a control point of view. The presented framework provides a basis for future studies on higher-dimensional systems, uncertainty-aware surrogate modeling, and more advanced control strategies built on learned local models.