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Learning Dissipative Dynamics with Dissipativity-by-Construction Discrete-Time Neural Networks

Tuan Luong, Hyungpil Moon

arXiv:2609.27188v1cs.RO

TL;DR

Learning dissipative dynamics has often relied on computationally costly continuous-time methods, despite practical implementations operating in discrete time. This paper constructs a discrete-time neural network that guarantees incremental dissipativity through constrained parameters. It reports competitive prediction accuracy and greater computational efficiency than Neural ODE approaches, with improved noisy-training results on robotic dynamics.

  • Problem

    Many dissipative-learning approaches use continuous-time formulations that incur high computational cost, while practical implementations are discrete-time.

    Source: §1 · p. 2

  • Method

    A discrete-time neural network uses constrained parameters and a dedicated training procedure to enforce incremental dissipativity by construction.

    Source: §1 · p. 2

  • Results

    DMLP achieves competitive prediction accuracy and significantly greater computational efficiency than Neural ODE approaches; for noisy 3-DOF training (N = 100), error falls from 20.0×10−2 to 6.9×10−2.

    Source: §1 · p. 2, §4.1 · p. 9

  • Takeaways & Limitations

    The discrete-time framework is suited to digital hardware and embedded control platforms.

    Source: §1 · p. 2

    The sufficient condition used in place of Eq. (16) is more conservative.

    Source: §2.2 · p. 5

Abstract

from arXiv · show

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.

1 Introduction

Learned dynamics can be accurate yet unstable outside training data, while existing dissipative-learning approaches often use computationally costly continuous-time formulations. This work builds a discrete-time neural network whose parameters enforce incremental dissipativity by construction.

1 Introduction

  • Standard data-driven dynamics models can become unstable or physically implausible outside their training distribution, motivating explicit stability guarantees.The risk is especially relevant to long-horizon simulation, control, and generalization.Source: §1 · p. 1
  • The proposed hypothesis class enforces incremental dissipativity in discrete time, avoiding post-hoc projection and the high computational cost of Neural ODE-based methods.The paper reports competitive prediction accuracy and significantly greater computational efficiency than continuous-time Neural ODE approaches.Source: §1 · p. 2

2.1 Problem formulation

  • The problem formulation represents nonlinear input-output dynamics with a neural network and enforces dissipativity through parameter construction, without iterative LMI problems.The system assumes f(0) = 0 and g(0) = 0.Source: §2.1 · pp. 2–3

2.2 Necessary condition of a discrete-time dissipative neural network

  • A positive definite matrix P supplies a condition for the network to be (Q, S, R)-dissipative, with a more conservative sufficient alternative also stated.The formulation uses an incremental storage function to relate differences between trajectories to input differences.Source: §2.1 · p. 3, §2.2 · pp. 4–5

2.3 Training algorithm for the dissipative network

  • Training first chooses D so R1 is positive definite, then treats C, B, and ΛWu as free variables while finding P and Λ.This procedure is presented as a way to satisfy the dissipativity constraint.Source: §2.3 · p. 5

H11 H12 H21 H22

  • The construction forms H* with a block-diagonal H12 and a trainable Y sharing W1's structure, then uses a sufficient condition to satisfy the constraint.The stated condition requires ϵ2 > 0.Source: H11 H12 H21 H22 · p. 6, H11 H12 HT · p. 6

2.4 Improved training algorithm

  • The improved network replaces each deeper-layer W2 matrix with W2 + Iq to address vanishing gradients that are difficult to mitigate under the constraints.The concern arises when repeated multiplication by W2 yields small norms; layer normalization cannot be applied in this setting.Source: §2.4 · p. 6

2.5 Computational complexity of the algorithm

  • The proposed training algorithm has O((Lq)^3) per-iteration complexity, while Neural ODE approaches scale roughly as Neval O((Lq)^2), with solver evaluations often a bottleneck.Neval denotes the number of solver function evaluations, which can reach hundreds or thousands of effective layers.Source: §2.5 · p. 7

3 Case studies

The case studies evaluate the proposed network on mass–spring–damper and n-DOF manipulator systems, both described as incrementally dissipative. For the MSD example, the caption reports good accuracy and robustness for the dissipative network under noisy training, while the unconstrained MLP fails to preserve incremental dissipativity.

3 Case studies

  • Both the mass–spring–damper and n-DOF manipulator examples are described as incrementally dissipative; their storage functions establish this property but are not prior information required by the algorithm.The network's storage matrix P is learned during training.Source: §3 · p. 7

3.1 Mass-spring-damper system

  • The MSD example models position x(t) under input force f(t), with m, b, and k denoting mass, damping, and spring coefficients.Its incremental dissipativity analysis defines storage and supply-rate functions for differences between two trajectories.Source: §3.1 · p. 7

3.2 n-DOF manipulator dynamics

  • The n-DOF manipulator model takes torque τ(t) as input and joint velocity q̇(t) as output, with q(t) the joint-position vector and M, D, and K the inertia, damping, and stiffness matrices.The section describes the manipulator as incrementally dissipative and introduces storage and supply-rate functions for trajectory differences.Source: §3.2 · pp. 7–9

3.1 Mass-spring-damper system

  • In the MSD comparison, the dissipative network maintains good prediction accuracy and robustness with Gaussian training noise (σ = 0.3), while the standard MLP fails to preserve incremental dissipativity.The noise-free test uses data from 10–20 s after training on 0–10 s; an additional disturbed input tests incremental dissipativity.Source: Figure 1 · p. 8

4 Evaluation results

DMLP achieves competitive prediction accuracy with substantially lower computation than a dissipative Neural ODE, while improving low-data predictions and preserving incremental dissipativity. The evaluation also reports smoother training-loss behavior from the improved dissipative training procedure.

4 Evaluation results

  • The evaluation compares DMLP with an unconstrained supervised MLP and checks incremental dissipativity along rollout trajectories using a discrete-time inequality.The inequality uses input, output, and storage increments.Source: §4 · p. 9

4.1 Results on the prediction performance of MSD and n-DOF manipulator dynamics with and without training noises

  • On MSD rollouts, DMLP tracks more consistently than MLP and satisfies the incremental dissipativity condition, which MLP does not preserve.The condition is checked with the training input and an additional disturbed input.Source: §4.1 · p. 9
  • For noisy MSD training at N=100, DMLP reduces mean prediction error from 38.9 × 10−2 to 14.3 × 10−2.DMLP also maintains accurate predictions and incremental dissipativity under Gaussian-noise training, while the baseline MLP is more noise-sensitive.Source: §4.1 · p. 9, Table 1 · p. 10
  • At N=100, DMLP reduces mean prediction errors under noisy and clean training for both manipulator systems.For 2-DOF, errors fall from 32.80 × 10−2 to 15.3 × 10−2 with noise and from 23.2 × 10−2 to 14.7 × 10−2 without noise; for 3-DOF, they fall from 20.0 × 10−2 to 6.9 × 10−2 with noise and from 10.3 × 10−2 to 5.84 × 10−2 without noise.Source: §4.1 · p. 9, Table 1 · p. 10

4.2 Training loss update

  • The improved dissipative training procedure yields the lowest and most stable loss profile across the three evaluated systems.The loss histories compare baseline MLP, original DMLP, and improved DMLP in the noise-free setting.Source: §4.2 · p. 10, Figure 2 · p. 10

4.3 Comparison with baseline methods

  • DMLP achieves competitive prediction accuracy with substantially lower computation than the dissipative Neural ODE approach.On MSD, DMLP requires 491.7 ± 5 seconds, compared with 6.33 ± 0.2 hours for Neural ODE.Source: §4.3 · p. 11, Table 2 · p. 11
  • An appendix ablation reports that the improved training algorithm enhances DMLP performance while reducing training time across the considered systems.The ablation also compares network depths and activation functions.Source: §4.3 · p. 11

5 Conclusions

The paper concludes that its dissipative network learns dynamics while preserving the dissipative property of known dissipative nonlinear models. Its evaluation compares the method with a naive model on selected systems and input-output pairs.

  • The network is designed and trained so its weights satisfy dissipativity constraints while remaining solvable with non-constrained optimization.The method is evaluated on a mass-spring-damper system and a flexible joint model, both incrementally dissipative for particular input-output pairs.Source: §5 · p. 11
  • The reported evaluation finds good modeling performance alongside preservation of physical meaning compared with a naive model.The conclusion limits the examples to the specified systems and input-output pairs.Source: §5 · p. 11
  • Future work is to study more complex cases and extend the approach to other network types.The authors describe broader applications as an expected use, not a demonstrated result.Source: §5 · p. 11

A PROOF OF EQ. (18)

The proof uses the stated increment in v and the storage-function difference to rewrite Eq. (39).

  • The proof passage also includes a transposed k-indexed vector immediately before deriving the relation.The supplied text identifies it only as a vector ending in k and a transpose.Source: A PROOF OF EQ. (18) · p. 11
  • The proof substitutes the stated expression for ∆v_k and the storage-function difference to rewrite Eq. (39).The expression for ∆v_k combines input, z, and x increments with the corresponding weights.Source: A PROOF OF EQ. (18) · p. 12
  • The storage term is expressed as the difference between consecutive storage-function values.The difference is written using ∆x at the current and next time indices.Source: A PROOF OF EQ. (18) · p. 12

B PROOF OF EQ. (19)

The proof uses bounded activation-function slopes, with derived quantities defined from the slope bounds α and β.

C Choose D such that

This section states a matrix condition and a sufficient alternative involving Q_1. It also specifies the definition of Q_1 and dimensions for free variables.

C Choose D such that

D Incremental dissipativity of MSD.

For the MSD system, the derivation uses two trajectories and obtains a storage-function derivative bounded by the input–output difference product. The section concludes that the system is incrementally dissipative from u to y = ˙x.

D Incremental dissipativity of MSD.

E Incremental dissipativity of the n-DOF manipulator and modeling performance

The section establishes incremental dissipativity for the n-DOF manipulator and describes a comparison of baseline MLP and dissipative-model performance under disturbed inputs.

E Incremental dissipativity of the n-DOF manipulator and modeling performance

F Ablation studies

The ablations report more reliable modeling from the dissipativity-constrained DMLP than the unconstrained baseline, and improved performance with reduced training time from the improved training algorithm.

F Ablation studies

  • The ablations compare training algorithms, network depths, and activation functions across the MSD, 2-DOF, and 3-DOF systems.Source: F Ablation studies · p. 15
  • The dissipativity-constrained DMLP provides more reliable modeling than the unconstrained MLP baseline.Source: F Ablation studies · p. 15
  • Table 3 reports the ablation study with N = 100 data samples.Source: Table 3 · p. 16
  • The improved training algorithm consistently enhances DMLP performance while reducing training time across the considered systems.Source: F Ablation studies · p. 16

G Implementation settings and Hyperparameters

The simulations use an MSE-trained MLP and DMLP with Adam, while 25–50% of each available dataset is used for training. Training duration varies by system, and the two networks share hyperparameters listed in Table 4.

G Implementation settings and Hyperparameters

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