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Data-driven discrete-time deep recurrent neural network-based modeling for dissipative systems

Tuan Luong, Hyungpil Moon

arXiv:2609.27186v1cs.LGcs.RO

TL;DR

Neural-network models fitted to dynamical-system data may fail to preserve physical properties such as stability. This paper proposes DissipNet, a discrete-time recurrent network with structural dissipativity constraints and a dedicated learning algorithm. In the evaluated dissipative systems, it preserves dissipativity and achieves stronger modeling performance than the naive RNN and PINN comparisons.

  • Problem

    Data-fitted neural-network models of dynamical systems may violate physical constraints, including stability-related properties.

    Source: §1 · p. 1

  • Method

    DissipNet uses a linear matrix inequality condition and a weight-learning algorithm to enforce dissipativity in a discrete-time recurrent network.

    Source: §1 · p. 2

  • Results

    On the mass-spring-damper system and 2-DOF manipulator, DissipNet preserves dissipativity and achieves better modeling performance than the naive RNN and PINN.

    Source: §4 · p. 13, §5 · p. 14

  • Takeaways & Limitations

    The proposed network preserves the dissipative property of known dissipative nonlinear models while providing good modeling performance in the evaluated cases.

    Source: §5 · p. 14

    Future work includes modeling more complex case studies and improving training time.

    Source: §5 · p. 14

Abstract

from arXiv · show

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.

1 Introduction

The introduction motivates preserving physical properties, especially stability-related dissipativity, in neural-network models of dynamical systems. It presents DissipNet as a discrete-time approach with explicit dissipativity constraints and a learning algorithm.

1 Introduction

  • Data-fitted neural networks may violate physical constraints, motivating models that preserve properties such as stability.The introduction identifies dissipativity as relevant to stability analysis and notes its use in nonlinear systems.Source: §1 · pp. 1–2
  • Prior neural-network passivity and dissipativity methods face scaling challenges, restricted input conditions, multi-step learning, or possible accuracy loss during projection.The cited work also includes a continuous-time Neural ODE approach whose training was time-consuming.Source: §1 · p. 2
  • DissipNet uses a linear matrix inequality and a sufficient weight-learning algorithm to enforce dissipativity, with stability analyzed using Lyapunov theory.The introduction contrasts this model-level property guarantee with PINNs, which incorporate governing equations into training loss but do not guarantee internal properties such as passivity.Source: §1 · p. 2

2 Dissipative Recurrent Neural Networks

The section formulates dissipative system modeling with a recurrent network, then specifies structural conditions and a training procedure for obtaining weights that satisfy them.

2.1 Problem formulation

  • Given input-output data, the formulation represents discrete-time dynamics with states, inputs, and outputs; a nonzero direct-feedthrough matrix is necessary for dissipativity.The RNN hidden state can serve as the system state.Source: §2.1 · p. 3

2.2 Dissipative Recurrent Neural Networks

  • The model in Eq. (4) is expanded in a stated recurrence, while Assumption 1 restricts the activation function’s slope to [α = 0, β].The proposed training algorithm is restricted to α = 0; extending it to α ≠ 0 is left for future work.Source: §2.2 · pp. 3–5
  • A positive-definite matrix condition provides a sufficient test for the network to be (Q, S, R) dissipative.The section states that satisfying the condition implies the network model is dissipative.Source: §2.2 · p. 5

2.3 Training algorithm for DissipNet

  • The training procedure chooses D to make R1 positive definite, then uses a Schur-complement reformulation to handle the LMI constraint.It selects other matrices as free variables and derives the network weights from the resulting matrices.Source: §2.3 · pp. 5–6, H11 H12 H21 H22 · p. 6

H11 H12 H21 H22

  • The construction forms H12 by zeroing its off-diagonal block matrices while preserving its diagonal blocks, and defines Y as a trainable matrix shaped like W_hh.The text specifies choosing ϵ1 so the stated matrix condition is satisfied.Source: H11 H12 HT · p. 6

H11 H12 HT

  • The section identifies diag(·) as the diagonal operator and ReLU(·) as the Rectified Linear Unit function in its matrix construction.The resulting matrices are used to calculate W_hh, W_ih, and W_u, with the text stating that the dissipativity constraint is then satisfied.Source: H11 H12 HT · p. 6, H11 H12 H21 H22 · p. 6

3 Case studies

The case studies evaluate DissipNet on two dissipative systems: a mass-spring-damper model and a planar two-degree-of-freedom manipulator.

3 Case studies

  • The evaluation uses a mass-spring-damper model and a planar two-degree-of-freedom manipulator, both described as dissipative.These are the two robotic modeling examples used to evaluate DissipNet.Source: §3 · p. 7

3.1 Mass-spring-damper system

  • The mass-spring-damper model is specified by mass, damping, and spring coefficients, with force input and mass position as the system variables.The continuous system is described as dissipative; the reported parameters are m = 1 kg, k = 4.2 N/m, b = 2.5 Ns/m, with a 0.05 s sampling time.Source: §3.1 · p. 7

3.2 Two DOF manipulator

  • The manipulator model uses joint angles as its coordinates and includes mass, Coriolis and centrifugal, friction, and applied-torque terms.Its input-output pair of applied torque and joint velocity is stated to be passive; the sampling time is 0.05 s.Source: §3.2 · pp. 7–8

4 Evaluation results

DissipNet is evaluated against a naive RNN and a PINN on mass-spring-damper and 2-DOF manipulator models. Across the reported comparisons, it achieves lower prediction errors while preserving dissipativity.

Evaluation setup

  • The PINN baseline incorporates known system dynamics into its cost through a regularization coefficient, while all models are trained with MSE and Adam at a 10−3 learning rate.RNN and DRNN use two recurrent layers with hidden dimension 64; PINN uses a four-layer feed-forward network with hidden dimension 128.Source: §4 · p. 8

MSD results

  • For the MSD model, DRNN preserves dissipativity while PINN does not; under σ = 0.1 training noise, DRNN also outperforms RNN and PINN.The MSD evaluation uses data from 0–40 s for training and 40–80 s for testing.Source: Figure 1 · p. 9, Figure 2 · p. 10, §4 · p. 13

Manipulator results

  • On the 2-DOF manipulator, DRNN preserves dissipativity while PINN does not, and DRNN is less sensitive to Gaussian noise than both baselines.Training uses 0–20 s and testing uses 20–40 s; the quantitative comparison reports mean errors and standard deviations under different data lengths and noise conditions.Source: Figure 3 · p. 11, Figure 4 · p. 12, §4 · pp. 13–14

MSD results

  • For noisy MSD data with N = 200, DRNN reduces mean error from 6.53 × 10−2 for RNN to 1.12 × 10−2; for N = 400, its mean error is 1.24 × 10−2 versus 6.59 × 10−2.The comparison covers mean prediction errors and standard deviations across training noise conditions and sample counts.Source: §4 · p. 13, Table 1 · p. 13

Manipulator results

  • On the 2-DOF manipulator, DRNN reduces error relative to RNN by approximately 22% at N = 200 and 40% at N = 400 under noise.DRNN performs competitively with PINN under noisy conditions and has a clear advantage in noise-free cases as data length increases.Source: Table 2 · p. 13, §4 · p. 14

5 Conclusions

DissipNet is reported to preserve dissipativity while modeling two known dissipative nonlinear systems, outperforming a naive RNN and PINN in modeling performance and physical-meaning preservation. The authors identify more complex case studies and training time as future work.

5 Conclusions

  • On a mass-spring-damper system and a 2-dof planar manipulator, DissipNet showed superior performance to Naive RNN and PINN in modeling and preserving physical meaning.The evaluations used particular input-output pairs for systems described as dissipative.Source: §5 · p. 14
  • The network’s weights are constrained by its design and learning algorithm, with the constraints solved using non-constrained optimization.The authors state that Lyapunov theory is used to prove stability.Source: §5 · p. 14
  • Unlike the proposed network, PINNs are reported not to guarantee preservation of dissipativity and to require explicit equations describing system dynamics.Source: §5 · p. 14
  • More complex case studies and improvements to training time are identified as future work.Source: §5 · p. 14

A PROOF OF EQ. (24)

The proof uses an expression for the network quantity v_k and a Lyapunov-function increment, then gives a definition of R1 and rewrites Eq. (50).

A PROOF OF EQ. (24)

B PROOF OF EQ. (25)

The proof of Eq. (25) invokes a slope-range restriction on widely used activation functions.

B PROOF OF EQ. (25)

  • The proof assumes slopes of widely used activation functions, including ReLU, tanh, and exponential linear functions, lie within [alpha, beta].Source: B PROOF OF EQ. (25) · p. 15

B.1 Choose D such that

This subsection gives conditions for choosing D and related matrices in the dissipativity argument, including a positive-definiteness condition involving R1.

B.1 Choose D such that

  • A sufficient stated condition is R1 = R + SD + D^T S^T + D^T QD ≻ 0.Source: §B.1 · p. 15
  • The text states that Eq. (57) is satisfied if L = R + SD + D^T S^T + D^T Q1D ≻ 0.Source: §B.1 · p. 15
  • Q1 is defined as Q − ϵI, with ϵ chosen as zero when Q ≺ 0; the text also represents Q1 as −Lq^T Lq.Source: §B.1 · p. 15
  • The construction specifies forms for D, N, and M, and identifies free variables and identity-matrix dimensions in the associated conditions.The stated free-variable dimensions include Y1 ∈ R^{p×p}, Y2 ∈ R^{p×p}, and Y3 ∈ R^{(m−p)×p}.Source: §B.1 · pp. 15–16
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