Dissipativity is a fundamental system-theoretic property closely related to stability, passivity, and input--output stability, and is particularly important in robotics, where learned dynamics models are often embedded within feedback control loops. However, most existing approaches for learning dissipative dynamics are based on continuous-time formulations, which require ODE solvers during training or inference and can therefore be computationally expensive. Moreover, because practical implementations are inherently discrete-time, direct discretization of a continuous-time passive system does not necessarily preserve passivity, motivating the need for explicit discrete-time guarantees. This study proposes a method for learning incrementally dissipative dynamics from input--output time-series data using a deep multilayer perceptron formulated directly in discrete time. Through a constrained parameterization and a dedicated training procedure, the proposed model guarantees incremental dissipativity by construction rather than through regularization. Lyapunov-based analysis establishes the corresponding dissipativity and stability guarantees, while simulations on robotic dynamical systems demonstrate competitive prediction accuracy, computational efficiency, and consistent preservation of incremental dissipativity compared with baseline methods.
Figures & tables
Figure 1: Performance comparison for the mass–spring–damper (MSD) system. (a) Noise-free case, where data from 0 – 10s are used for training and data from 10 – 20s for testing; an additional disturbed input is used to verify incremental dissipativity. (b) Noisy case with Gaussian training noise (σ=0.3) , showing that the proposed dissipative network maintains good prediction accuracy and robustness. (c) The standard MLP without dissipativity constraints fails to preserve incremental dissipativity, evaluated using Eq. ( 3 ) with known MSD parameters.
System
N
Noise
MLP
DMLP
MSD
100
Yes
38.9±0.02
14.3±0.02
No
11.1±0.03
8.21±0.01
200
Yes
5.02±0.01
4.42±0.00
No
2.53±0.02
3.06±0.01
2-DOF
100
Yes
32.80±0.02
15.3±0.01
No
23.2±0.01
14.7±0.02
Table 1: Mean prediction errors ( ± std) of MLP and DMLP on the MSD, 2-DOF, and 3-DOF systems for different training sizes N and noise settings.
Figure 2: Training loss histories of the baseline model, the original dissipative network, and the improved dissipative network, demonstrating the effectiveness of the improved training algorithm. (a) MSD system. (b) 2-DOF manipulator. (c) 3-DOF manipulator.
Model
Performance
MLP
DMLP
Dissipative Neural ODE
MSD (10000 epochs)
Mean error
2.53±0.02(×10−2)
3.06±0.01(×10−2)
0.108±0.03
Computation time
403±10 s
491.7±5 s
6.33±0.2 h
2-DOF manipulator (5000 epochs)
Mean error
6.96±0.02(×10−2)
11±0.03(×10−2)
0.15±0.02
Computation time
262±11 s
442±8 s
4.15±0.5 h
3-DOF manipulator (5000 epochs)
Mean error
2.44±0.01(×10−2)
5.79±0.02(×10−2)
0.10±0.01
Computation time
302±15 s
525±10 s
7.22±0.7 hour
Table 2: Performance comparison between MLP, DMLP, and Dissipative Neural ODE.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: Comparison of the performance between baseline MLP model and the dissipative model. (a) 2-DOF manipulator without training noise, where data from 0s to 20s were used for training and data from 20s to 40s were used for testing. (b) 3-DOF manipulator without training noise, where data from 0s to 10s were used for training and data from 10s to 20s were used for testing. In both cases, besides the input used in training, a disturbed input signal was also used to check the incremental dissipativity condition of the proposed network.
Ablation cases
MSD (10000 epochs)
2-DOF manipulator (5000 epochs)
3-DOF manipulator (5000 epochs)
Mean error (×10−2)
Training time
Mean error (×10−2)
Training time
Mean error (×10−2)
Training time
Dissipativity constraints
No (MLP)
2.53±0.02
403±10 s
6.96±0.02
262±11 s
2.44±0.01
302±15 s
Yes (DMLP)
3.06±0.01
491.7±5 s
11±0.03
423±10 s
5.79±0.02
525±10 s
Training algorithm (DMLP)
Improved
3.06±0.01
491.7±13 s
11±0.03
423±10 s
5.79±0.02
525±10 s
Original
4.98±0.02
625±12 s
13.4±0.02
472±12 s
6.02±0.03
578±11 s
Number of layers (DMLP)
1
5.02±0.01
500±15 s
15.5±0.01
412±13 s
7.33±0.02
502±13 s
Appendix
Table 3: Ablation study with the number of data samples N=100 .
Physical AI has gained increasing attention for its role in developing AI systems that better understand, predict, and control real-world dynamics. Achieving this requires AI models that not only achieve high prediction accuracy but also preserve fundamental physical properties of dynamical systems. In this paper, we propose a deep discrete-time dissipative recurrent neural network (DissipNet) that explicitly enforces dissipativity, a key property related to stability and energy dissipation, through structural weight constraints and a dedicated training algorithm. By construction, the proposed network is capable of learning dissipative dynamics while preserving their inherent stability, which is formally analyzed using Lyapunov theory. In contrast to Physics-Informed Neural Networks (PINNs), which incorporate governing equations into the training loss but do not guarantee preservation of internal analytical properties such as dissipativity or passivity, our approach provides explicit guarantees on stability at the model level. We demonstrate the effectiveness of the proposed method through several modeling applications, and compare its performance with a naive recurrent neural network (RNN) and a PINN-based model.
Tuan Luong, Hyungpil Moon
Department of Mechanical Engineering, Sungkyunkwan University, Suwon 16419, South Korea.
In this paper, we propose the Dissipative Lagrangian Neural Network (dLNN), a continuous-time system identification technique that separately learns the conservative and dissipative dynamics of a mechanical system directly from data. A key contribution is a novel parametrisation of dissipative forces that (i) is structurally constrained to be dissipative, enforcing passivity of the learned model, (ii) cannot absorb conservative mechanics, and (iii) retains sufficient expressivity to represent a broad family of state-dependent damping and regularised friction forces. The model is validated on prediction tasks on two real-world mechanical systems: trajectory prediction of a chaotic double pendulum and inverse-dynamics modelling of a six-degree-of-freedom industrial robot. Across these tasks, the dLNN demonstrates improved predictive performance over existing Lagrangian architectures while remaining competitive with a black-box neural model. The imposed physical structure also improves generalisation to unseen states, with the benefits becoming particularly pronounced when training data is limited. Finally, we introduce a symbolic distillation procedure that extracts closed-form expressions from the learned physical components. Applied to the industrial robot, this produces interpretable symbolic models that retain high predictive accuracy while reducing the computational overhead associated with evaluating neural Lagrangian models.
Rob Seabourne, Alice Cicirello
Department of Engineering, University of Cambridge Trumpington Street, Cambridge CB2 1PZ, UK
Learning dynamical systems through purely data-driven methods is challenging as they do not learn the underlying conservation laws that enable them to correctly generalize. Existing port-Hamiltonian neural network methods have recently been successfully applied for modeling mechanical systems. However, even though these methods are designed on power-balance principles, they usually do not consider power-preserving discretizations and often rely on Runge-Kutta numerical methods. In this work, we propose to use a second-order discrete gradient method embedded in the learning of dynamical systems with port-Hamiltonian neural networks. Numerical results are provided for three systems deliberately selected to span different ranges of dynamical behavior under control: a baseline harmonic oscillator with quadratic energy storage; a Duffing oscillator, with a non-quadratic Hamiltonian offering amplitude-dependent effects; and a self-sustained oscillator, which can stabilize in a controlled limit cycle through the incorporation of a nonlinear dissipation. We show how the use of this discrete gradient method outperforms the performance of a Runge-Kutta method of the same order. Experiments are also carried out to compare two theoretically equivalent port-Hamiltonian systems formulations and to analyze the impact of regularizing the Jacobian of port-Hamiltonian neural networks during training.
Maximino Linares, Guillaume Doras, Thomas Hélie
IRCAM, 1, place Igor-Stravinsky, 75004, Paris, France