Discovering Unknown Inverter Governing Equations via Physics-Informed Sparse Machine Learning

TL;DR

A Physics-Informed Sparse Machine Learning (PISML) framework that provides a unified pathway to convert structurally inaccessible devices into explicit mathematical models, enabling stability analysis of power systems with unknown inverter governing equations.

Abstract

Discovering the unknown governing equations of grid-connected inverters from external measurements holds significant attraction for analyzing modern inverter-intensive power systems. However, existing methods struggle to balance the identification of unmodeled nonlinearities with the preservation of physical consistency. To address this, this paper proposes a Physics-Informed Sparse Machine Learning (PISML) framework. The architecture integrates a sparse symbolic backbone to capture dominant model skeletons with a neural residual branch that compensates for complex nonlinear control logic. Meanwhile, a Jacobian-regularized physics-informed training mechanism is introduced to enforce multi-scale consistency including large/small-scale behaviors. Furthermore, by performing symbolic regression on the neural residual branch, PISML achieves a tractable mapping from black-box data to explicit control equations. Experimental results on a high-fidelity Hardware-in-the-Loop platform demonstrate the framework's superior performance. It not only achieves high-resolution identification by reducing error by over 340 times compared to baselines but also realizes the compression of heavy neural networks into compact explicit forms. This restores analytical tractability for rigorous stability analysis and reduces computational complexity by orders of magnitude. It also provides a unified pathway to convert structurally inaccessible devices into explicit mathematical models, enabling stability analysis of power systems with unknown inverter governing equations.

Figures (19)

• Figure 1: Problem formulation for identifying grid-connected inverter dynamics.
• Figure 2: Overview of the Physics-Informed Symbolic Machine Learning (PISML) framework. The approach combines a sparse symbolic backbone with a residual neural ODE to capture unknown dynamics. The model is trained using a composite physics-informed loss function ($\mathcal{L}_{total}$), followed by a symbolic regression step to extract an explicit, interpretable ODE representation from the neural residue.
• Figure 3: Schematic illustration of the Sparse Identification of Nonlinear Dynamics. The method constructs a library of candidate nonlinear functions $\mathbf{\Theta}(\mathbf{X})$ and employs sparse regression to select the active coefficients $\mathbf{\Xi}$ that best describe the time-series data derivatives $\dot{\mathbf{X}}$, enabling the reconstruction of the underlying governing equations.
• Figure 4: Architecture of the proposed Neural Residual ODE. This module compensates for the dynamics gap in the sparse backbone by superimposing a learnable neural vector field. The merged derivatives are integrated via a shared ODE solver, enabling direct end-to-end training using trajectory data.
• Figure 5: Multi-time-scale physics-informed training mechanism.