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

Shivanshu Tripathi; Maziar Raissi

eess.SYOct 24, 2026

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

Authors: Shivanshu Tripathi, Maziar Raissi

Organizations: Department of Electrical and Computer Engineering, University of California, Riverside, Riverside, CA 92521, USA. · Department of Mathematics, University of California, Riverside, Riverside, CA 92521, USA.

Abstract

Hybrid dynamical systems flow continuously but jump at discrete events. Physics-informed neural networks (PINNs) fit data while enforcing known dynamics, but a single network is smooth in time and cannot model these jumps. We show that the more closely a PINN follows a jump, the larger its physics loss must be. We propose hybrid-time PINNs (HT-PINNs), which assign one network to each flow interval of the hybrid time domain and train the networks jointly with the jump times and the unknown parameters. The number of jumps and the sequence of modes are found by enumeration. HT-PINNs recover the number of jumps and the restitution coefficient of a bouncing ball. In a MuJoCo simulation of robotic disassembly, where a robot pulls a press-fitted pin out of a housing through a spring, HT-PINNs find the number of jumps and the sequence of modes, and estimate the static and kinetic friction coefficients with small errors of 0.1% and 1.7%.

Figures & tables

Fig. 1: Solution of

\mathcal{H}

flows within a mode and switches modes at guards.

Panel of Figure 2 — Fig. 2: Neural network for each region (black), and resets when guards become zero (red).

Fig. 3: Bouncing ball with height over time (left), unmeasured velocity over time (middle), smallest loss for number of jumps

J

(right). The first

J

below the tolerance

\varepsilon

J=4

Panel of Figure 4 — Fig. 4: Extraction of a press-fitted pin. (a) The MuJoCo cell: the gantry robot pulls the pin out of the housing through the spring of its series-elastic puller. (b) The hybrid system ( 12 ): the flow of each mode, with $F=k\delta+c\,(\dot{U}-v)$ , and the jumps with the conditions that trigger them. The resets are the identity, and the measured output is $y=\delta$ .

Fig. 5: Extraction of a press-fitted pin in MuJoCo. Left: measured spring force

k\delta

; the shading shows the modes of the HT-PINN, stick (green), slip (orange) and released (purple). Middle: unmeasured velocity of the pin during the last slip and the exit. Right: smallest loss for each

J

, shaded below the tolerance

\varepsilon

Fig. 6: Force and displacement during the extraction of the press-fitted pin in MuJoCo, colored by the state of the pin: stuck (

\mathsf{S}

), slipping (

\mathsf{K}

) and released (

\mathsf{R}

). (a) Spring force of the puller; the dashed lines are the static and kinetic friction levels, offset by the weight of the pin. (b) Displacement of the hook, which the robot raises at a constant

20

mm/s, and of the pin.

	$\tau_{1}$ ( $\mathsf{K}$ )	$\tau_{2}$ ( $\mathsf{S}$ )	$\tau_{3}$ ( $\mathsf{K}$ )	$\tau_{4}$ ( $\mathsf{S}$ )	$\tau_{5}$ ( $\mathsf{K}$ )	$\tau_{6}$ ( $\mathsf{S}$ )	$\tau_{7}$ ( $\mathsf{K}$ )	$\tau_{8}$ ( $\mathsf{R}$ )	$\mu_{s}$	$\mu_{k}$
MuJoCo	0.5975	0.6145	0.8840	0.9010	1.1705	1.1875	1.4570	1.4640	0.4500	0.3000
HT-PINN	0.5967	0.6141	0.8845	0.9008	1.1697	1.1864	1.4568	1.4644	0.4495	0.3052

TABLE I: The table shows the event times and friction coefficients (

\mu_{s},\mu_{k}

) for MuJoCo simulator (true values) and HT-PINNs (our approach). Each

\tau_{j}

is marked with the mode it enters:

\mathsf{S}

(stick),

\mathsf{K}

(slip),

\mathsf{R}

(released).

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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.

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