Hybrid-time PINNs: Discontinuity-Aware Physics-Informed Neural Networks for Learning Hybrid Dynamical Systems

A multi-domain eXtended Physics-Informed Neural Network (XPINN) framework is developed for nonsmooth Delay Differential Equations (DDEs). This is the first implementation to demonstrate the efficacy of partitioning the temporal domain into subdomains of integer multiples of the characteristic time delay and progressively training the associated subnetworks while freezing previously learned parameters. The efficacy of the proposed framework is demonstrated using a machining dynamics model that incorporates both regenerative and nonsmooth frictional effects. Results demonstrate that the proposed multi-domain XPINN framework leads to better solution reconstruction in DDEs and improved parameter estimation compared to a generic PINN (SPINN) formulation. The proposed method works particularly well for extended temporal domains and non-constant history functions. The robustness of inverse XPINN (I-XPINN) is also assessed using reference data contaminated with Gaussian measurement noise. Results indicate that I-XPINN remains resilient to measurement noise and the physics-informed constraints guide the network toward accurately recovering the underlying dynamics. This demonstrates, for the first time, the potential of the proposed framework for reliable parameter identification in DDEs characterised by nonsmoothness and large time delays.

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.

Physics-Informed Neural Networks (PINNs) are a common class of machine learning-based partial differential equation (PDE) solvers which train a network to represent a solution by minimizing a residual loss that encodes the PDE. Despite their successes, they are known to fail on certain simple equations, converging to an incorrect solution despite low loss. These failure modes have garnered significant attention in the literature over the past several years, motivating both architectural and optimization based solutions. By directly visualizing the residual, we show that failure modes are the result of overfitting: the loss is minimized on the collocation points, but not elsewhere. Applying regularization causes the failure modes to vanish. Finally, we extend double backpropagation over the full set of residuals, and use it to achieve state-of-the-art performance on four standard failure mode equations with up to $23\times$ fewer collocation points and a vanilla architecture.