Residual Physics-Informed Neural Networks for High-Fidelity BLDC Motor Modeling

Accurate dynamical modeling is essential for simulation and control of embodied systems, yet first-principles models of electromechanical systems often fail to capture complex dissipative effects such as joint friction, stray losses, and structural damping. While residual-learning physics-informed neural networks (PINNs) can effectively augment imperfect first-principles models with data-driven components, the residual terms are typically implemented as unconstrained multilayer perceptrons (MLPs), which may inadvertently inject artificial energy into the system. To more faithfully model the dissipative dynamics, we propose DiLaR-PINN, a dissipative latent residual PINN designed to learn unmodeled dissipative effects in a physically consistent manner. Structurally, the residual network operates only on unmeasurable (latent) state components and is parameterized in a skew-dissipative form that guarantees non-increasing energy for any choice of network parameters. To enable stable and data-efficient training under partial measurability of the state, we further develop a recurrent rollout scheme with a curriculum-based sequence length extension strategy. We validate DiLaR-PINN on a real-world helicopter system and compare it against four baselines: a pure physical model (without a residual network), an unstructured residual MLP, a DiLaR variant with a soft dissipativity constraint, and a black-box LSTM. The results demonstrate that DiLaR-PINN more accurately captures dissipative effects and achieves superior long-horizon extrapolation performance.

Modeling concentric tube robots (CTRs) involves complex nonlinear continuum mechanics, and despite recent progress, physics-based models often lack an accurate representation of the experimental setups. To overcome these limitations, deep neural network-based models have been explored as alternatives with superior accuracy; however, they often overlook known mechanics, require large training datasets, and typically discard shape estimation of the robot. We present a physics-informed neural network (PINN) for kinematic modeling of a 6-DoF CTR with three pre-curved tubes that embeds the Cosserat rod differential equations and learns from few-shot observational data, balancing physics priors with data-driven fitting. PINN enables full-state estimation of shape, twist angle, torsional strain, bending moment, and orientation. Benchmark tests show a mean shape error below 1% of the robot length and accurately recovered other kinematic states, outperforming a purely physics-based Cosserat rod model baseline while using a minimal training set. The resulting model is also computationally efficient and robust, making it well-suited for real-time control applications.

The integration of machine learning with domain-specific physics is transforming the design, monitoring, and control of electricity systems, where data scarcity, limited interpretability, and the need to enforce physical laws constrain purely data-driven models. Physics-informed machine learning (PIML) addresses these limitations by embedding governing equations directly into the learning process, yielding accurate, efficient, and scalable solutions for Industry 4.0 applications. This article reviews hybrid PIML architectures for electricity systems, including physics-informed neural networks (PINNs), Deep Operator Networks (DeepONets), Fourier Neural Operators, Extreme Learning Machine-enhanced PINNs, graph-based PINNs (PIGNNs), and domain-decomposition PINNs. Each approach is examined through case studies spanning field analysis, fault detection, digital twins, surrogate modeling, and control optimization. The review shows that embedding Maxwell's equations and other first-principles constraints substantially improves predictive accuracy under sparse and noisy data, reduces simulation time by orders of magnitude relative to finite element methods, and enhances generalization across operating regimes. Hybrid frameworks consistently outperform purely data-driven baselines on parameter sensitivity, dynamic behavior, and robustness, while supporting real-time digital-twin calibration and uncertainty quantification. Persistent challenges include training instability for stiff multi-scale problems, computational cost of high-fidelity models, and the absence of standardized benchmarks. The findings demonstrate that PIML enables a paradigm shift from black-box data-driven methods to transparent, physics-informed strategies, positioning the field for sustained innovation in resilient and intelligent electricity systems.