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.
T HE rapid proliferation of inverter-based resources (IBRs) is fundamentally reshaping the landscape of modern power systems [1], [2]. As conventional synchronous machines are increasingly replaced by software-controlled power electronic converters, grid dynamics are now dominated by embedded control algorithms rather than intrinsic physical properties such as mechanical or electromagnetic coupling [3], [4]. A critical challenge arises because both grid-following (GFL) and grid-forming (GFM) inverters typically operate with proprietary and closed-source control logic, essentially black boxes to grid operators [5], [6]. This structural inaccessibility jeopardizes the effectiveness of traditional stability assessment paradigms, which rely on explicit state-space equations to analyze eigenvalue trajectories and damping characteristics [7]. As a result, a crucial modeling gap is rapidly widening: while inverter penetration continues to increase, theoretical tools for understanding and predicting their dynamic behaviors lag significantly behind [8], [9].
With the deployment of advanced sensing technologies such as phasor measurement units (PMUs) and high-bandwidth impedance measurement systems, massive amounts of highresolution dynamic data have become available, offering new opportunities to uncover inverter dynamics from observations [10], [11]. However, most existing studies still focus on local linear impedance identification [12]. These methods have evolved from traditional frequency sweeping to realtime signal injection (e.g., chirp signals or pseudo-random binary sequences) and can handle complex scenarios such as parallel inverter configurations through sophisticated decoding networks [13], [14]. While these approaches are effective in fitting observed data, they inherently assume a locally linear time-invariant (LTI) physical structure and thus fail to capture the global consistency of the underlying nonlinear dynamics [15]. As a result, there are still challenges for critical analytical tasks beyond impedance analysis, such as eigenvalue-based stability assessment or transient stability analysis [16]. For grid operators seeking a comprehensive understanding of stability boundaries, relying solely on black-box impedance predictors could be insufficient when large-signal dynamics need to be considered. Consequently, there is a strong need for a data-driven approach capable of discovering explicit control equations governing the system dynamics [17].
From a broader scientific perspective, automatically discovering the governing equations of nonlinear systems from external measurements remains a major interdisciplinary challenge [18]. Early approaches such as equation-free modeling and empirical dynamic modeling established foundational ideas but often encountered difficulties in scalability and robustness when dealing with high-dimensional and noisy data [19]. With the development of symbolic regression and genetic programming, researchers began to directly search the mathematical expression space to reconstruct differential equations [20]. In the field of power and energy systems, symbolic regression has shown potential for identifying battery degradation processes [21] and extracting reduced-order grid dynamics [22]. However, symbolic regression implicitly relies on the strong assumption that a predefined function library is sufficiently complete to describe the unknown physics. This assumption often fails in high-dimensional multi-time-scale control architectures, where the search space grows exponentially and the sensitivity to noise increases significantly [23].
The emergence of Scientific Machine Learning (SciML) introduced a new paradigm by incorporating physical priors into the learning process. Graph Neural Networks (GNNs) have been adopted to capture the topological structure and component interactions within complex power electronic and power system [24]. By explicitly modeling the connectivity, these methods offer scalability to large-scale networks that is difficult to achieve with standard dense layers [25], [26]. Meanwhile, Neural Ordinary Differential Equations (Neural ODEs) and Universal Differential Equations (UDEs) represent system dynamics through differentiable neural architectures [27], [28]. These methods have been used to characterize battery thermal behavior, learn grid frequency responses, and predict dynamic features of power electronic devices [15], [29]- [31].Although these approaches are expressive in representation, purely neural methods often prioritize data fitting, rather than obey physical consistency constraints and guarantee physical validity [32]. This may result in models that are accurate in prediction but physically inconsistent in structure. Physics-Informed Neural Networks (PINNs) address this issue by integrating physical laws such as energy conservation into the loss function, thereby enhancing generalization under data-scarce conditions [33]- [35]. Howeve
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