Learning Deep Dissipative Dynamics

TL;DR

The paper addresses the challenge of guaranteeing dissipativity when learning dynamical systems with neural networks from time-series data. It derives a general solution to the nonlinear Kalman–Yakubovich–Popov (KYP) lemma and introduces differentiable dissipative projections that map arbitrary neural-network dynamics into a dissipative subspace, ensuring internal stability, input–output stability, and energy conservation. By embedding these projections into a gradient-based learning framework, the authors train models that not only fit data but also strictly satisfy dissipativity for all input sequences, improving robustness to out-of-domain inputs. The approach is demonstrated on linear and nonlinear benchmarks, including a mass–spring–damper, an $n$-link pendulum, and fluid flow around a cylinder, highlighting the practical impact for robotics, physics-based modeling, and energy-aware control. Future work will focus on selecting dissipativity hyperparameters and extending the framework to system identification and real-world deployments.

Abstract

This study challenges strictly guaranteeing ``dissipativity'' of a dynamical system represented by neural networks learned from given time-series data. Dissipativity is a crucial indicator for dynamical systems that generalizes stability and input-output stability, known to be valid across various systems including robotics, biological systems, and molecular dynamics. By analytically proving the general solution to the nonlinear Kalman-Yakubovich-Popov (KYP) lemma, which is the necessary and sufficient condition for dissipativity, we propose a differentiable projection that transforms any dynamics represented by neural networks into dissipative ones and a learning method for the transformed dynamics. Utilizing the generality of dissipativity, our method strictly guarantee stability, input-output stability, and energy conservation of trained dynamical systems. Finally, we demonstrate the robustness of our method against out-of-domain input through applications to robotic arms and fluid dynamics. Code is https://github.com/kojima-r/DeepDissipativeModel

Figures (10)

• Figure 1: Sketch of the dissipativity: The red difference in storage energy is less than the total energy supplied along the blue line, which represents the trajectory of the internal state $x(s)$.
• Figure 2: Sketch of the proposed method: The dynamics of the input-output system $(f,g,h,j)$ is projected into a space with guaranteed dissipativity using dissipative projection $\mathcal{P}_\theta$, and the output signal $y(t)$ is predicted using the projected dynamics $(f_{\bf d},g_{\bf d},h_{\bf d},j_{\bf d})$ and the input signal $u(t)$.
• Figure 3: (A) Sketch of mass-spring-damper system. (B) Prediction results for out-of-domain inputs, i.e., five long step signals with different amplitudes. The top figure is the input signal behaviors, the middle figure shows the position of the mass predicted by the model trained using the naive model, and the bottom figure shows the position predicted by the proposed dissipative model. The dashed lines are the plots of the ground truth, and each color of lines shows the results with the same input signals.
• Figure 4: (A) Sketch of $n$-link pendulum model. (B) Prediction results for out-of-domain inputs. The top figure is the input signal behaviors, the middle figure shows the angle of the first pendulum predicted by the model trained using the naive model, and the bottom figure shows the angle predicted by the proposed dissipative model. The dashed lines are the plots of the ground truth, and each color of lines shows the results with the same input signals. (C) RMSE$(t)$ related to long input step signal.
• Figure 5: (A) Sketch of the flow around cylinder model. (B) Time-average RMSE of test triangular (in-domain) waves by changing the number of training inputs$N$. (C) The predicted output flow for a test triangular (in-domain) wave with $N=100$. Each curved line represents the spatial distribution of the output flow at each time point. The blue, orange, and red lines shows the ground truth, the output flow predicted by the naive model, and the output predicted by our dissipative model, respectively. (D) The predicted output flow for a test clip (out-of-domain) wave with $N=100$. (E) RMSE$(t)$ for long time simulation with test clip waves.

Theorems & Definitions (25)

• Definition 1: Dissipativity
• Proposition 2: Nonlinear KYP lemma brogliato2020dissipative
• proof
• Definition 3: Dissipative projection
• Lemma 5
• proof
• Theorem 6: Dissipative Projection
• proof
• Corollary 7: Stable Projection
• proof