Title: Input-to-State Stable Coupled Oscillator Networks for Closed-form Model-based Control in Latent Space
Venue: NeurIPS 2024
Accepted: yes
Keywords: Dynamical Systems

Control Theory

Robotics

Decision and Control

Deep Autoencoders
Abstract: Even though a variety of methods have been proposed in the literature, efficient and effective latent-space control (i.e., control in a learned low-dimensional space) of physical systems remains an open challenge.
We argue that a promising avenue is to leverage powerful and well-understood closed-form strategies from control theory literature in combination with learned dynamics, such as potential-energy shaping.
We identify three fundamental shortcomings in existing latent-space models that have so far prevented this powerful combination: (i) they lack the mathematical structure of a physical system, (ii) they do not inherently conserve the stability properties of the real systems, (iii) these methods do not have an invertible mapping between input and latent-space forcing.
This work proposes a novel Coupled Oscillator Network (CON) model that simultaneously tackles all these issues. 
More specifically, (i) we show analytically that CON is a Lagrangian system - i.e., it possesses well-defined potential and kinetic energy terms. Then, (ii) we provide formal proof of global Input-to-State stability using Lyapunov arguments.
Moving to the experimental side, we demonstrate that CON reaches SoA performance when learning complex nonlinear dynamics of mechanical systems directly from images.
An additional methodological innovation contributing to achieving this third goal is an approximated closed-form solution for efficient integration of network dynamics, which eases efficient training.
We tackle (iii) by approximating the forcing-to-input mapping with a decoder that is trained to reconstruct the input based on the encoded latent space force.
Finally, we leverage these three properties and show that they enable latent-space control. We use an integral-saturated PID with potential force compensation and demonstrate high-quality performance on a soft robot using raw pixels as the only feedback information.

Input-to-State Stable Coupled Oscillator Networks for Closed-form Model-based Control in Latent Space

Abstract

Even though a variety of methods have been proposed in the literature, efficient and effective latent-space control (i.e., control in a learned low-dimensional space) of physical systems remains an open challenge. We argue that a promising avenue is to leverage powerful and well-understood closed-form strategies from control theory literature in combination with learned dynamics, such as potential-energy shaping. We identify three fundamental shortcomings in existing latent-space models that have so far prevented this powerful combination: (i) they lack the mathematical structure of a physical system, (ii) they do not inherently conserve the stability properties of the real systems, (iii) these methods do not have an invertible mapping between input and latent-space forcing. This work proposes a novel Coupled Oscillator Network (CON) model that simultaneously tackles all these issues. More specifically, (i) we show analytically that CON is a Lagrangian system - i.e., it possesses well-defined potential and kinetic energy terms. Then, (ii) we provide formal proof of global Input-to-State stability using Lyapunov arguments. Moving to the experimental side, we demonstrate that CON reaches SoA performance when learning complex nonlinear dynamics of mechanical systems directly from images. An additional methodological innovation contributing to achieving this third goal is an approximated closed-form solution for efficient integration of network dynamics, which eases efficient training. We tackle (iii) by approximating the forcing-to-input mapping with a decoder that is trained to reconstruct the input based on the encoded latent space force. Finally, we leverage these three properties and show that they enable latent-space control. We use an integral-saturated PID with potential force compensation and demonstrate high-quality performance on a soft robot using raw pixels as the only feedback information.

1 Introduction

Learning how the environment evolves around us from high-dimensional observations (i.e., world models [1]) is essential for achieving both artificial and physical intelligence [2]. For example, world models are required for effectively planning an artificial/robotic agent's actions in complex and unstructured environments [3]. However, learning such dynamics directly in high-dimensional observation space is usually intractable. Seminal works have shown that we can leverage autoencoders to compress the state information into a low-dimensional latent space [4, 5] in which it is much more feasible to learn the dynamics [6, 7, 8, 9, 10]. However, strong limitations still persist when it comes to using these learned models to generate low-level intelligence.

One outstanding challenge is how to perform closed-loop control in the learned latent space i.e., how to generate control inputs based on a high dimensional sensory input such that a desired movement is generated. Prior works have explored, among other approaches, Reinforcement Learning (RL) [11, 12, 13, 14], Model Predictive Control (MPC) [7, 15, 16, 17], Linear-quadratic Regulators (LQRs) [18, 19, 20] and gradient-based optimization [21] for planning and control towards a target evolution that is given in observation space. However, all existing latent-space control strategies have

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shortcomings, such as a limited planning horizon and slow control rates (MPC and gradient-based approaches), sample inefficiency (RL), or they pose a requirement for learning linear dynamics [6] (LQR), which is not even possible for systems that are inherently non-linearizable [22]. One interesting avenue is to leverage model-based control approaches, such as potential shaping [23, 24, 10], for effective and computationally efficient control in latent space [25]. For these techniques to be feasible, the dynamical model needs to fulfill four characteristics: (i) the dynamics need to have the mathematical structure of physical systems, (ii) conserve the stability properties of real systems, (iii) the latent state needs to be relatively low-dimensional, and (iv) there needs to exist a well-defined, invertible mapping between the input and the forcing in latent space. However, existing model structures that are used for learning latent dynamics [26] do not meet all of these criteria. Relevant examples are Multilayer Perceptrons (MLPs), Neural ODEs (NODEs) [27, 28], many variants of Recurrent Neural Networks (RNNs) (e.g., LSTMs [29], Gated Recurrent Units (GRUs) [30], etc.), and physics-informed neural networks (e.g., Lagrangian Neural Networks (LNNs) [31, 32, 10], Hamiltonian Neural Networks (HNNs)) [33]. For example, MLPs do not have a physical interpretation and do not provide an invertible mapping of the forcing generated by the input, NODEs are usually not easily stabilizable [34], most RNNs require a relatively high-dimensional latent space (i.e., many hidden states), and energy-shaping control approaches based on LNNs [10] do not come with any formal stability guarantees.

In recent years, oscillatory networks [35, 36, 37, 38, 39] have been shown to exhibit state-of-the-art performance on time sequence modeling tasks while being parameter-efficient, thus fulfilling our requirement (iii). Consequently, we believe that they are a promising option for control-oriented dynamics learning in latent space. Still, these models do not fulfill the remaining requirements that we have listed above. Despite being an interpretable combination of harmonic oscillators, they do not have the structure of a physical system - i.e., they do not possess a well-defined energy function. Moreover, only local stability [35, 37] has been shown, with sufficient conditions that appear to be very stringent. Finally, in addition to training an encoder that maps inputs to latent-space forcing, we propose also training a decoder that learns to reconstruct inputs based on latent-space forcing. This enables us to easily switch between inputs and forcing, which is essential when implementing control strategies.

We resolve all the above-mentioned challenges by proposing Coupled Oscillator Networks (CONs), a new formulation of a coupled oscillator network that is inherently Input-to-State Stability (ISS) stable, for learning the dynamics of physical systems and subsequently exploiting its structure for model-based control in latent space. The network consists of damped, harmonic oscillators connected through elastic springs, damping elements, and a neuron-like coupling force and can be excited by a nonlinear actuation term. We identify a transformation into a set of coordinates from which we can derive the networks' kinetic and potential energy. This allows us to leverage Lyapunov arguments [40] for proving the global asymptotic stability of the unforced system and ISS stability for the forced system under relatively mild assumptions on the network parameters. Even though we constrain the dynamics to a very specific structure, we demonstrate (a) the CON network achieves similar performance as NODEs when learning the dynamics of unactuated, mechanical systems with two

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orders of magnitude fewer parameters and (b) that the proposed model achieves, for the complex task of learning the actuated, highly nonlinear dynamics of continuum soft robots [41, 42] directly from pixels, a 60 % lower prediction error than Coupled Oscillatory Recurrent Neural Network (coRNN) [35] and reaches the SoA performance across all techniques that we tested. Finally, we show some initial results that the proposed CON model is also able to learn the latent dynamics of Partial Differential Equations (PDEs), in this case containing reaction-diffusion [9, 43] dynamics.

Subsequently, we derive an approximate closed-form solution, that is, in parameter regimes in which the linear, decoupled dynamics dominate transient, more accurate than numerical integrators with comparable computational requirements and which increases training speed by 2x with a small decrease in prediction accuracy. Finally, as we can derive the system's potential energy, we can leverage potential shaping [23, 24] to derive a controller that combines an integral-saturated PID controller with a feedforward term compensating potential forces. As the feedback acts on a wellshaped potential field, tuning the feedback gains becomes very simple and out-of-the-box, and the controller exhibits a faster response time and a 26 % lower trajectory tracking Root Mean Squared Error (RMSE) than a pure feedback controller based on a latent NODE [27] model.

The proposed methodology is particularly well-suited for learning the latent dynamics of mechanical systems with continuous dynamics, dissipation, and a single, attractive equilibrium point. Examples of such systems include many soft robots, deformable objects with dominant elastic behavior, Lagrangian systems immersed in a dominant potential field, or locally other mechanical systems such as robotic manipulators, legged robots, etc. For these systems, we can fully leverage the structural prior of the proposed latent dynamics, including the integrated stability guarantees. If the system is actuated, the learned dynamics can be subsequently exploited for model-based control, as demonstrated in Sec. 5.

The code associated with this paper is available on GitHub 1 .

2 Input-to-State Stable (ISS) Coupled Oscillator Networks (CONs)

Formulation. The integral component to (coupled) oscillatory RNNs [35, 36, 37, 38] are onedimensional, potentially damped, harmonic oscillators, which are described by their state y i = [ x i ˙ x i ] T ∈ R 2 , where x i and ˙ x i are the position and velocity of the oscillator, respectively. Then, the oscillator's dynamics are defined by the following Equation of Motion (EOM)

Here, m i is the mass, κ i is the stiffness, and d i is the damping coefficient of the damped harmonic oscillator. F i ( t ) ∈ R is a (possibly time-dependent) external forcing term acting on the mass.

ω n ,i = √ κ i m i and ζ i = d i 2 √ κ i m i let us determine the natural frequency and the damping factor, respectively and allow us to design the transient behavior. For example, ω n ,i lets us isolate a spectrum of the input signal F i ( t ) [37] and ζ i determines the damping regime: underdamped ( ω n ,i < 1 ), critically damped ( ω n ,i = 1 ), overdamped ( ω n ,i > 1 ). Furthermore, as (damped) harmonic oscillators are omnipresent in nature (and especially in physical systems), they have been intensively studied and are well understood (e.g., characteristics, closed-form solutions, etc.). In this work, we will exploit some of these properties and knowledge to learn stable (latent) dynamics efficiently.

Even though the state is extremely low dimensional and the number of parameters is small, this single, damped harmonic oscillator can already exhibit a variety of (designable) behaviors: The expressions

By intercoupling damped harmonic oscillators, we can drastically increase the expressiveness of the dynamical system [35, 37, 38] while preserving some of the intuition and understanding we have for these systems. In this work, we propose a ISS-stable CON consisting of n damped harmonic oscillators that are coupled through both linear and nonlinear terms. The networks' state is defined as y = [ x T ˙ x T ] T ∈ R 2 n and its dynamics can be formulated as a 2 nd -order Ordinary Differential Equation (ODE)

where K,D ∈ R n × n are the linear stiffness and damping matrices, respectively. The neuron-inspired term tanh( Wx ( t ) + b ) with W ∈ R n × n , b ∈ R n provides nonlinear coupling between the harmonic oscillators. The network is excited by the time-dependent input u ( t ) ∈ R m through the possibly nonlinear mapping g : R m → R n . Specifically, we consider in this work a formulation where an input-dependent matrix

B ( u ) ∈ R n × m projects the input u ( t ) to a time-dependent forcing on the oscillators: τ = g ( u ) = B ( u ) u . Here B ( u ) could, for example, be parametrized by a MLP.

We specifically designed the network architecture such that (i) the system exhibits a unique and isolated equilibrium and (ii) we can derive expressions for the kinetic and potential energies. These two features allow us to (a) prove Global Asymptotic Stability (GAS) and ISS stability using an established procedure based on strict Lyapunov arguments [44, 45], and (b) implement model-based controller based on potential shaping.

One key insight of this work is that in the coordinates x ( t ) , ˙ x ( t ) , we cannot derive a potential as the hyperbolic force tanh( Wx ( t )+ b ) is not symmetric. Therefore, we propose a coordinate transformation into W -coordinates:

y w ( t ) = [ x w ( t ) ˙ x w ( t ) ] = [ Wx ( t ) W ˙ x ( t ) ] ∈ R 2 n . The coordinate transformation is valid if its Jacobian is full-rank, which is the case if rank( W ) = n . In W -coordinates, the dynamics can be rewritten as

with K w = KW , D w = DW and M w = W -1 .

A difference of this formulation compared to prior work [35, 36, 37, 38] is that (i) the forcing produced by the input term τ = g ( u ) is fully separated from the forcing produced by the elastic coupling terms K w , and (ii) the generalized force is symmetric, which we prove in Appendix A.1, allowing us to define a potential energy expression, which we can later on leverage for stability analysis and control.

The equilibria ¯ y w = [ ¯ x T w 0 T ] T ∈ R 2 n of the unforced network are given by the roots of the characteristic equation tanh(¯ x w + b ) + K w ¯ x w = 0 .

Lemma 1. Let K w ≻ 0 . Then, the dynamics defined in (3) have a single, isolated equilibrium ¯ y w = [ ¯ x T w 0 T ] T .

Proof. The proof is straightforward and provided in Appendix A.2.

Next, we introduce a mapping into the ˜ tilde coordinates ˜ y w = y w -¯ y w . The residual dynamics (w.r.t. the equilibrium ¯ y w ) can now be stated as

In the following, we will write ∥ A ∥ to denote the induced norm of matrix A and λ m ( A ) , λ M ( A ) to refer to its minimum and maximum Eigenvalue respectively.

Global Asymptotic Stability (GAS) for the unforced system. We first consider the unforced system with τ = g ( u ) = 0 , ∀ t ∈ [ t 0 , t ∞ ) and strive to prove global asymptotic stability [40] for the attractor ¯ x . We propose a strict Lyapunov candidate with skewed level sets [45]

Lemma 2. The scalar function V µ (˜ y w ) defined in (5) is continuously differentiable and verifies the condition V µ (0) = 0 . Furthermore, let M w , K w ≻ 0 . Now, if we choose 0 < µ < √ λ m ( M w ) λ m ( K w ) ∥ M w ∥ := µ V , then V µ (˜ y w ) > 0 ∀ ˜ y w ∈ R 2 n \{ 0 } . Additionally, then V µ (˜ y w ) is radially unbounded as ∥ ˜ y w ∥ → ∞ ⇒ V µ (˜ y w ) → ∞ .

Proof. We provide the proof in Appendix A.3 and demonstrate that the bounds on µ are required for the Lyapunov candidate to be positive-definite.

Theorem 1. Let M w , K w and D w be positive definite and suppose the system be unforced: g ( u ( t )) = 0 . Then, ˜ y w = 0 is globally asymptotically stable for the system dynamics defined (4) such that ˙ V µ (˜ y w ) < 0 , ∀ ˜ y w ∈ R 2 n \ { 0 } .

Proof. First, we show that ˜ y w = 0 is an equilibrium of (4):

Lemma 2 states that we can always choose µ such that (5) is strict Lyapunov function (e.g., Lipschitz continuous, zero-valued at ˜ y w = 0 , positive definite, and radially unbounded) [40, 45]. We now evaluate the time-derivative of ˙ V µ (˜ y w ) in the case of an unforced system (i.e., g ( u ( t )) = 0 ):

where we exploited the force balance at equilibrium K w ¯ x w = -tanh(¯ x w + b ) and Lemma 7 for defining the upper bound on ˙ V µ (˜ y w ) . Lemma 8 states that P ˙ V ≻ 0 for 0 < µ < µ ˙ V . Similarly, Lemma 2 requests that 0 < µ < µ V . Indeed, both conditions can always be fulfilled by choosing µ ∈ (0 , min { µ V , µ ˙ V } ) . With P ˙ V ≻ 0 ⇔ λ m ( P ˙ V ) > 0 [46], we can state that ˙ V µ (˜ y w ) < 0 ∀ ˜ y w ∈ R 2 n \ { 0 } and conclude that the unforced system is globally asymptotically stable around ˜ y w = 0 .

Global Input-to-State Stability (ISS) for the forced system. We now take the forcing g ( u ) into account again and demonstrate that the system states remain proportionally bounded to the initial conditions and as a function of the supremum of the input forcing.

Theorem 2. Suppose M w , K w , D w ≻ 0 , 0 < θ < 1 , and that we choose 0 < µ < min { µ V , µ ˙ V }} . Then, (4) is globally Input-to-State Stable (ISS) such that the solution ˜ y w ( t ) verifies

Proof. The proof is provided in Appendix A.5. We first demonstrate that the ISS-Lyapunov function is bounded from both sides by K ∞ functions. Subsequently, we derive the energy dissipation of the forced system and establish attracting regions as a function of the norm of the forcing. Outside these regions, it is ensured that the system has a minimal rate of decay and, therefore, converges exponentially fast into the attracting region.

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3 An approximate closed-form solution for the rollout of CON

To predict future system states, we need to integrate the ODE in Eq. (2), with the solution given by y ( t k +1 ) = y t k + ∫ t k +1 t k f ( y ( t ′ ) , u ( t ′ )) d t ′ . Unfortunately, a closed-form solution for the nonlinear dynamics f ( y, u ) does not (yet) exist. Therefore, we traditionally need to revert to (high-order) numerical ODE solvers that are computationally very expensive and introduce additional memory overhead [47]. This considerably increases the training time of models involving such continuous-time dynamics. While the computational time can be reduced by increasing the (minimum) time step of the integrator, this comes at the expense of an integration error, and we lose (part of) the theoretical guarantees and practical characteristics that the nominal ODE provides. In this work, we take an alternative approach by splitting the problem into (i) decoupled linear dynamics that can be cheaply and precisely integrated using a closed-form solution and (ii) the residual, coupled nonlinear dynamics, which we integrate numerically at a slower time scale:

where κ = diag( K 11 . . . K nn ) , d = ( D 11 . . . D nn ) are the diagonal components of the stiffness and damping matrices, respectively, and F ∈ R n is a constant, external forcing term on the oscillators.

For a short-time-interval δt , we now approximate (9) as

For a scalar 2 nd -order, linear ODE of form ˙ y i = f ld ,i ( x ( t ′ ) , F ( t k )) , a well-known, closed-form solution [48] exists. We exploit this characteristic by formulating the approximate solution as

and denote f CFA -CON : R n × R m → R n as the Closed-Form Approximation of the Coupled Oscillator Network (CFA-CON) model. The implicit assumption behind (10) is that f ¨ x, ld ( y ) ≻ f ¨ x, nld ( y, u ) (i.e., the linear, decoupled dynamics dominate the nonlinear, coupled, time-varying dynamics). We refer the interested reader to Appendix B for derivation and implementation details, where we summarize the integration procedure in Algorithm 1. We also provide qualitative results for the integration accuracy in Fig. 2 and quantitative results for the integration accuracy and computational speed-up w.r.t. numerical integrators in Appendix B.

4 Learning control-oriented latent dynamics from pixels

We now move towards learning latent dynamical models based on CON and CFA-CON. CONs are an ideal fit for learning latent dynamics as they guarantee that the latent states stay bounded.

We assume to have access to observations in the form of images o ∈ R h o × w o × c o , where c o denotes the number of channels. Please note that this could also be other high-dimensional observations such as LiDAR scans, point clouds, etc. We now leverage an encoder-decoder architecture to map these high-dimensional observations into a compressed latent space: The encoder Φ : R h o × w o × c o → R n z with n z ≪ h o w o identifies a lowdimensional latent representation z ∈ R n z of the images. The decoder Ψ : R n z → R h o × w o × c o approximates the inverse operation by reconstructing an image ˆ o ∈ R h o × w o × c o based on the latent representation. To promote the learning of a smooth and monotonic mapping into latent space, we specifically choose to implement the autoencoder here as a β -Variational Autoencoder (VAE) [5, 49]. Instead of just statically reconstructing the image ˆ o ( t k ) , we are interested in predicting future observations ˆ o ( t k + l ) , where l ∈ 1 . . . N . For this, we train a 2 nd -order dynamical model that is, when integrated, able to predict future latent representations z ( t k + l ) . This requires us to define a latent state ξ ( t ) = [ z T ( t ) ˙ z T ( t ) ] T ∈ R 2 n z consisting of the latent representation and latent velocity ˙ z ( t ) ∈ R n z .

We now rely on CON with n = n z oscillators to provide us with the latent state derivative ˙ ξ = f w ( y w ( t ) , u ( t )) , where we defined ξ = y w , and z = x w . To ensure stability, we make use of the Cholvesky decomposition to ensure that M w , K w and D w always remain positive definite (see Theorem 2). It is important to note that we train the encoder, decoder, and dynamical model all jointly. Please refer to Appendix C for more implementation details.

Training. It is important to remember that because we are using a β -VAE [5, 49], the image encoding becomes stochastic, and the encoder neural network actually outputs µ z ( o ) , 2 log( σ z )( o ) ∈ R n z . After executing the reparametrization trick as z ( t k ) ∼ N ( µ z ( t k ) , σ 2 z ( t k )) , we formulate the loss function, evaluated on each trajectory consisting of N time-steps, as

where ˆ z ( t k ) is predicted by ˆ ξ ( t k ) = ∫ t k t = t 0 f ξ ( ξ ( t ′ ) , u ( t ′ )) d t ′ , and ξ ( t 0 ) = [ z T ( t 0 ) ˙ z T ( t 0 ) ] T . Here, z ( t 0 ) is given by the encoder, and ˙ z ( t 0 ) is approximated using finite differences in image-space (see Appendix C.5 for more details). β, λ ⃗ o , λ z ∈ R are loss weights.

Models. We train the CON with the input-to-forcing mapping g ( u ) = B ( u ) u , where B ( u ) is parametrized by a MLP with a hyperbolic tangent activation function applied in between layers. We report results for two variants of the CON model: for the medium-sized CON-M and small-sized CON-S , the MLP consists of five and two layers with a hidden dimension of 30 and 12 , respectively. The model CFA-CON uses the same architecture as CON-M . We compare against several popular latent space model architectures: The NODE model uses a MLP with an hyperbolic activation functions and predicts ˙ ξ ( t ) = f NODE ( ξ ( t ) , u ( t )) . To make the comparison fair, we parametrize the NODE's MLP in the same fashion as for CON-M . The MECH-NODE integrates prior knowledge towards learning 2 nd -order mechanical ODEs and, therefore, predicts ¨ z ( t ) = f MECH -NODE ( ξ ( t ) , u ( t )) . Furthermore, we consider multiple autoregressive models: RNN, GRU, and coRNN and let them parameterize the following transition function: ξ ( t k +1 ) = f ar ( ξ ( t k ) , u ( t k )) As common in the relevant literature [26], we allow the autoregressive models to perform multiple time step transitions before predicting the next sample. For the autoencoder, we use a vanilla Convolutional Neural Network (CNN). More details can be found in Appendix C.

Datasets. We consider in total six datasets that are based on simulations of unactuated mechanical systems, and actuated continuum soft robots. The first three, mechanical dataset are based on the work of Botev et al. [26] and contain video sequences of a mass-spring system with friction ( M-SP+F ), a single pendulum with friction ( S-P+F ), and a double pendulum with friction ( D-P+F ). Continuum soft robots have theoretically infinite Degree of Freedom (DOF), evolve with highly nonlinear and often time-dependent dynamical behaviors, and are notoriously difficult to model from first principles [50]. For that reason, it is a very interesting proposition if we could learn latent-space dynamical models directly from video [51] and later leverage them for control [52]. Therefore, we generate three datasets based on the Piecewise Constant Strain (PCS) soft robot model. CS considers one segment with constant strain and is modeled using three configuration variables. PCC-NS-2 and PCC-NS-3 only consider bending deformations and contain soft robots with two and three segments, respectively. For all datasets, we render images with a resolution of 32x32px of the system's state. More information on the datasets can be found in Appendix C.1.

We tune all hyperparameters for each model and dataset separately using Optuna [53].

Results. Unactuated mechanical datasets: The results in Tab. 1 show that the NODE model slightly outperforms the CON network on the M-SP+F and S-P+F datasets. However, as the datasets do not consider system inputs, we can remove the input mapping from all models (e.g., RNN , GRU , coRNN , CON , and CFACON ). With that adjustment, the CON network has the fewest parameters among all models, particularly two orders of magnitude less than the NODE model. Therefore, we find it very impressive that the CON network is roughly on par with the NODE model. For the D-P+F dataset, we can conclude that the CFA-CON model offers the best performance across all methods. Finally, most of the time, the CON & CFA-CON networks outperform the other baseline methods that have more trainable parameters. Actuated continuum soft robot datasets: The results in Tab. 1 show that CON-M matches the performance of the state-of-the-art methods

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The results for a dataset based on the 1 st -order reaction-diffusion PDE [9] are presented in Apx. D.1.

5 Exploiting the dynamic structure for latent-space control

We consider the problem setting of guiding the system towards a desired observation o d by providing a sequence of inputs u ( t k ) such that at time t N , the actual observation o ( t N ) matches o d . A relatively simple way would be to encode the desired observation into latent space z d = Φ( o d ) and then to design a feedback controller (e.g., PID) in latent space: g ( u ) = PID( z d -z ( t ) , -˙ z ) . Unfortunately, several challenges appear: first, it is not clear how the latent-space force τ = g ( u ) can be mapped back to an actual input u ( t ) as the inverse input-to-forcing mapping g -1 is generally not known. Furthermore, relying entirely on a PID controller has several well-known drawbacks, such as poor and slow transient behavior, steady-state errors (in case the integral gain is chosen to be zero), and instability for high proportional and integral gains. We take inspiration from potential shaping strategies [23, 24], which are widely used for effectively controlling (elastic) robots, and, therefore, combine a feedforward term compensating the latent-space potential forces with an integral-saturated, PID-like feedback term. For mapping the desired forcing τ back to an input u ( t ) , we train an forcing decoder η : R n → R m that approximates g -1 . Specifically, we consider here the structure u = η ( τ ) = E ( τ ) τ , where E ∈ R m × n is parameterized by an MLP.

The latent-space control law is given by

where K p , K i , K d ∈ R n × n are the proportional, integral, and derivative control gains, respectively. As integral terms can often lead to instability when applied to nonlinear systems [54], we adopt an integral term saturation [55] with the associated dimensionless gain υ ∈ R , which ensures that the integral error added at each time step is bounded to the interval ( -1 , 1) . Subsequently, τ is decoded to the input as u ( t ) = η ( τ ) = E ( τ ) τ . For training this decoder, we add a reconstruction loss to (12): L u ( t k ) = λ u MSE( u ( t k ) , ˆ u ( t k )) = λ u MSE( u ( t k ) , η ( g ( u ( t k )))) .

Experimental setup. We train a CON model with two latent variables ( n z = 2 ) on the PCC-NS-2 dataset. Analog to the input encoder mapping B ( u ) , the forcing decoder mapping E ( τ ) is parametrized by an MLP consisting of five layers with hidden dimension 30 and a hyperbolic tangent activation function. The CON model achieves an RMSE of 0 . 1628 on the test set. We benchmark two controllers on the simulated continuum soft robot consisting of two segments: (i) a pure P-satI-D controller (i.e., the feedback term in (13)) that leverages the smooth mapping into the latent representation enabled by the CON dynamic model and the β -VAE, and (ii) a P-satI-D+FF (i.e., (13)) that exploits the structure of the CON dynamics by compensating for the potential forces. The stable closed-loop system dynamics made the control gain tuning very easy, and we selected

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macroscale phenomena help us to develop RD processes for controlling the fate of cells and treating diseases? (Although some progress has been made towards this latter goal, we are still at a very early stage  before  we  can  treat  diseases 109-111 .)  Fi ft h,  multiscale  aspects: although there is an observable, phenomenological link between a variety of nanoscale processes and microscale phenomena, how can we rationalize how large-scale behaviour emerges from an understanding of nanoscale processes and interactions? We hope that this short Review will stimulate scientists to develop new, nanoscale, synthetic systems for a wide range of functions and 6 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 BZ Reduced [Ru( ii )] Oxidized [Ru( iii )] ~ 500% volume change Figure 5 | Pattern formation, communication and energy conversion produced by chemical oscillations. a , Pattern formation in BZ reactions occurring in a microemulsion of the surfactant sodium bis(2-ethylhexyl) We train all models (e.g., MECH-NODE , CON ) on three different random seed and choose the best model instance. Please refer to Appendix E for more details on the model selection. We generate a trajectory of 7 setpoints, where q d ( t j ) ∼ U ( -5 π, 5 π ) rad / m ∈ R 2 is a sampled configuration of the soft robot. Then, we render an image o d ( t j ) that represents the target observation for the controller and encode it into latent space to retrieve z d ∈ R 2 . At time step k , we render an image o ( t k ) of the robot's current configuration q ( t k ) and encode the image. Subsequently, we evaluate the control law and apply the decoder u ( t k ) = η ( τ ( t k ))) , which is finally passed to the simulator that integrates the ground-truth dynamics to the next time-step t k +1 considering the actuation u ( t k ) . The controller runs at 100Hz , and we simulate the ground-truth dynamics with a Dopri5 ODE integrator at a time-step of 1e -5 s .

Jumping waves

Time

Space

Turing patterns

Standing waves

2

3

Segmented spirals

Memory reader towards a set of speci

fi c problems that merit attention from

ples to develop new types of chemical oscillator for controlling RD

processes 106,107 or regulating RD processes in living systems? Second, communication, computation and memory: we know that BZ oscillators can communicate and exhibit certain features of computation 108 and memory systems, so how can we develop synthetic systems, with intermediate complexity between chemical oscillators and the brain, but not based on silicon electronics, that can perform such tasks? Th ird, motion: nature has evolved ATP motors by integrating selfassembly, self-organization and RD processes, so what do we need to construct an analogous e ffi cient chemomechanical engine? Fourth, periodicity: how can nanoscale oscillations that result in micro- and Packet waves Antispirals Localized waves BZ c K p = 1 , K i = 2 , K d = 0 . 02 , υ = 1 for the P-satI-D controller and K p = 0 , K i = 2 , K d = 0 . 05 , υ = 1 for the P-satI-D+FF controller, respectively. Furthermore, we compare the control performance of our model-based controllers with a baseline control strategy based on the MECH-NODE ( n z = 2 ) that achieves an error of 0 . 1104 on the test set. First, we utilize the same P-satI-D feedback controller as for the CON model to generate the control action τ ( t ) in latent space. As the MECH-NODE uses an MLP to parameterize the function ˙ ξ = f ξ ( ξ, u ) , we cannot easily map τ ( t ) into an input u ( t ) . Therefore, we linearize the latent space dynamics w.r.t to the input as f ξ, ac ( ξ, u ) = f ξ ( ξ, 0) + A ( ξ ) u , where A ( ξ ) = ∂f ξ ∂u ( ξ, 0) is computed using autodiff. Then, u ( t ) = A T ( ξ ) τ ( t ) . After tuning the control gains, we choose K p = 0 . 001 , K i = 0 . 02 , K d = 1e -5 , υ = 1 .

inspire them to shi ft from studying single molecules (or objects) to multiple entities, from homotypic to heterotypic molecular interactions, from single to multiple events, from static to dynamic patterns, and from structure to function. Of course, it is unlikely that the integration of self-assembly, self-organization and RD processes at the nanoscale will produce systems that capture all the features of living systems. But exploration along these lines will undoubtedly not only lead to a new paradigm of doing nanoscience and to the development of new nanotechnologies, but may also help address fundamental questions, such as how life evolves from the coupling of chemical reactions 1,112-114 and other types of nonlinear behaviour 115,116 . sulfosuccinate. Jumping waves are presented as snapshots (left) and as a space-time plot (right, obtained across the white line shown in the left snapshot). Three types of Turing pattern are shown: left, 'black dots' (honeycomb); centre, 'frozen waves'; and right, 'white dots' (distorted hexagons). Standing waves are shown as two antiphase snapshots. 'Memory' shows persistence of image briefly projected onto a photosensitive medium that exhibits localized subcritical Turing structures. Localized waves are obtained in the same manner using a striped mask. Other patterns are spiral in chaotic waves, bubble waves, oscillion, segmented sprials, packet waves, and antispirals. b , Pattern generation in macroemulsions produced by microfluidics. Schematic of two-dimensional ' π -S' pattern in which each triangle of drops is composed Results. As an evaluation metric, we consider the RMSE between the actual and the reference trajectory. The P-satI-D applied to the MECH-NODE model (baseline) achieves an RMSE of 2 . 88rad / m w.r.t. to the desired configuration q d (but unknown to the algorithm). The P-satI-D CON controller, which does not exploit the learned latent dynamics for control, exhibits an RMSE of 4 . 08rad / m w.r.t. to the desired configuration q d . The P-satI-D+FF controller exhibits an RMSE of 2 . 12rad / m w.r.t. to the desired configuration q d . We also visualize the closed-loop trajectories in Fig. 5 and as sequences of stills in Apx. E. We conclude that the nicely structured latent space generated by the β -VAE allows the P-satI-D controller to effectively regulate the system towards the setpoint, although the response time is rather slow. The P-satI-D+FF controller is able to exploit the structure of the CON model through its potential shaping feedforward term. With that, CON P-satI-D+FF exhibits a faster response time and a 26% lower RMSE than the MECH-NODE P-satI-D baseline. We provide results for the control of an actuated damped harmonic oscillator in Apx. D.

of two drops that are π radians out of phase with one another, and the third is stationary: grey circles represent stationary (non-oscillatory) droplets; 6 Conclusion and Limitations

published online 5 April 2016

Received 1 September 2015; accepted 18 February 2016;

Future perspective Whereas  evolution  has  integrated  self-assembly,  self-organization and periodic RD processes at the nanoscale to produce beautiful nanomachinery, there is clearly a lot to do to combine these three References 1. Epstein, I. R. Th e consequences of imperfect mixing in autocatalytic chemical and biological systems. Nature 374, 321-327 (1995). Th is paper shows that autocatalysis and heterogeneity in chemical reactions can generate complex behaviours in both biological and abiological systems. 2. Bánsagi, T. Jr, Vanag, V . K. & Epstein, I. R. Tomography of reaction-di ff usion microemulsions reveals three-dimensional Turing patterns. Science 331, 1309-1312 (2011). Th is paper demonstrates that reaction-di ff usion in nanoemulsions can generate 3D Turing patterns. blue and red circles show oscillatory droplets 180 ° out of phase with each other. c , Chemomechanical oscillation of active gels with flow of aqueous BZ solution that achieve ~ 500% volume change. Figure adapted with permission from: a , ref. 6, American Chemical Society; b , ref. 92, American Chemical Society; c , ref. 96, American Chemical Society. Conclusion. In this work, we propose a new formulation for a coupled oscillator that is inherently input-tostate stable. Additionally, we identify a closed-form approximation, that is able to simulate the network dynamics more accurately compared to numerical ODE integrators with similar computational costs. When learning latent dynamics with CON, we observe that the performance is on par or slightly better compared to SoA methods such as RNNs, NODEs, etc., even though we constrained the solution space to a ISS-stable coupled oscillator structure. Furthermore, we point out that the performance of the CON models is more consistent across latent dimensions compared to the baselines and improved when not specifically tuned for a given dimension. Furthermore, as seen in Tab. 8, the closed-form approximation achieves, with the same number of model parameters, similar accuracies and double the training speed w.r.t. to the continuous-time model. Finally, we demonstrate that even a simple PID-like latent-space controller can effectively regulate the system to a setpoint. By exploiting the advances outlined in this Review have laid the foundation for inves-

processes  in  nanoscale  abiological  systems.  We  believe  that  the tigating  self-assembly,  self-organization  and  RD  at  the  nano-  and

Recent studies of the prototypical BZ oscillating chemical reac- microscales in a holistic manner.

tion in nano- and microdroplets have provided considerable insight into how emergent properties arise from RD processes. Moreover,

Vanag, V.

K. & Epstein, I.

R. Segmented spiral waves in a reaction-di system.

ff

100,

14635-14638 (2003).

Proc. Natl Acad. Sci. USA

Nature Nanotech.

Mattia, E. & Otto, S. Supramolecular systems chemistry.

10,

111-119 (2015).

Mann, S. Self-assembly and transformation of hybrid nano-objects and nanostructures under equilibrium and non-equilibrium conditions.

Epstein, I.

8,

781-792 (2009).

Chemical oscillators in structured media.

R.

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0

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t

[s]

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d

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(e) Configuration for P-satI-D+FF with CON.

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network structure and compensating for potential forces, regulation performance can be greatly improved, and response time decreased by more than 55% .

Limitations. While we think our proposed method shows great potential and opens interesting avenues for future research, there exist certain limitations. For example, the proposed method of learning (latent) dynamics implicitly assumes that the underlying system adheres to the Markov property (e.g., the full state of the system is observable), that a system with mechanical structure can approximate it, and that it has an isolated, globally asymptotically stable equilibrium. This is, for example, the case for many mechanical systems (e.g., some continuum soft robots, deformable objects, and elastic structures) with continuous dynamics, convex elastic behavior, dissipation, and whose time-dependent effects (e.g., viscoelasticity, hysteresis) are negligible. Even if these conditions are not met globally, the method can be applied to model the local behavior around an asymptotic equilibrium point of the system (e.g., robotic manipulators, legged robots) with added stability benefits for out-of-distribution samples. Alternatively, the method could be extended to relax some of these assumptions, e.g., by allowing for multiple equilibria, zero damping, or by incorporating additional terms to capture discontinuous dynamics (e.g., stick-slip models) or period motions (e.g., limit cycles such as the Van der Pol oscillator). The proposed method might not be suitable for some physical systems, such as nonholonomic systems, partially observable systems, or systems with non-Markovian properties. Examples of such systems include mobile robots and systems with hidden states or delayed observations.

Furthermore, the approximated closed-form solution shows the best integration for situations where linear, decoupled dynamics dominate the transient. For dominant nonlinear, coupled forces, the performance of CFACON degrades, and it might be better to revert to numerical integration of the CON ODE. Finally, the control works exceptionally well in the setting where the latent dimension equals the input dimension. We hypothesize that this enables the method to identify a diffeomorphism between the input and the latent-space forcing. Still not investigated is how the performance could degrade if n z > m (or n z < m for that matter).

Acknowledgments and Disclosure of Funding

This work was supported under the European Union's Horizon Europe Program from Project EMERGE - Grant Agreement No. 101070918.

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A Appendix on proof of Input-to-State Stability (ISS)

In the following, we will write ∥ A ∥ to denote the induced norm of matrix A and λ m ( A ) , λ M ( A ) to refer to its minimum and maximum Eigenvalue respectively.

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We introduce some expressions often used throughout this section: The gradient of V µ (˜ y w ) w.r.t. the residual coordinate ˜ y w is given by

Next, the Hessian of the Lyapunov candidate can be derived as

where

Furthermore, the Schur complement of P V is given by

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Lemma 3. Let ˜ x w ∈ R n be generalized coordinates and ¯ x w , b ∈ R n constants. Then, the potential force of system (4)

stems from the potential

Proof. First, we take the derivative of U (˜ x w ) :

The Hessian of the potential is given by

As K w ≻ 0 ⇒ K w = K T w , we can easily show that the potential force is symmetric:

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A.2 Proof of Lemma 1: single & isolated equilibrium

Lemma 1 restated. Let K w ≻ 0 . Then, the dynamics defined in (3) have a single, isolated equilibrium ¯ y w = [ ¯ x T w 0 T ] T .

Proof. We regard the characteristic equation as a function: h eq ( x w ) = tanh( x w + b ) + K w x w . For there to exist multiple equilibria, h eq (¯ x w ) = 0 would need to be true for multiple ¯ x . However, we take the partial derivative of h eq ( x w ) w.r.t. x w and see that

as S sech ( x w ) ≻ 0 ∀ x w ∈ R n and K w ≻ 0 . Therefore, h eq ( x w ) is continuously increasing and can only cross the zero line once.

A.3 Proof of Lemma 2: Validity of strict Lyapunov candidate V µ (˜ y w )

Lemma 4. Suppose M w ≻ 0 , K w ≻ 0 and 0 < µ < √ λ m ( M w ) λ m ( K w ) ∥ M w ∥ := µ V . Then S P V , as defined in (17) , is positive definite.

Proof. The minimum Eigenvalue of S P V is bounded by

Based on the assumption M w ≻ 0 , K w ≻ 0 , we can state ∥ M w ∥ λ w ( K w ) > 0 . Therefore, the critical case for the lower bound on λ m ( S P V ) is µ = √ λ m ( M w ) λ m ( K w ) ∥ M w ∥ := µ V . Hence,

Consequently, the Eigenvalue sensitivity theorem [46] demands that S P V ≻ 0 .

Lemma 5. Let M w ≻ 0 , K w ≻ 0 , and 0 < µ < √ λ m ( M w ) λ m ( K w ) ∥ M w ∥ := µ V . Then, it follows that P V ≻ 0 and H V (˜ x w ) ≻ 0 ∀ ˜ x w ∈ R n .

Proof. By inspecting the expressions for P V ≻ 0 and H V (˜ x w ) ≻ 0 ∀ ˜ x w ∈ R n in Equations (5) and (15), respectively, it can be easily seen that H V (˜ x w ) ⪰ P V ∀ ˜ x w ∈ R . As Lemma 4 states that the Schur complement of P V is positive definite, it follows that H V (˜ x w ) ⪰ P V ≻ 0 .

Lemma 6. Suppose ¯ x w , ˜ x w , b ∈ R n and n ∈ N + . Then,

is a positive semi-definite function.

Proof. Proving h V , th (˜ x w ) ≥ 0 is equivalent to showing that the scalar function ˘ h V , th ( r ) = ∫ r 0 tanh( σ + a ) d σ -∫ r 0 tanh( a ) d σ ≥ 0 ∀ r, a ∈ R , where we set r = ˜ x w ,i and a = ¯ x w ,i + b i .

We strive to find the critical points (i.e., minimas and maximas) ¯ r of ˘ h V , th ( r ) and, for this, analyze where the first derivative of ˘ h V , th ( r ) is zero

which is the case only for ¯ r = 0 . Next, we compute the second derivative at ¯ r as

Thus, ˘ h V , th ( r ) is convex and its global minimum at ¯ r = 0 takes the value h V , th (0) = 0 . As a result, h V , th (˜ x w ) is also positive semi-definite.

A.3.1 Proof of Lemma 2:

Lemma 2 restated. The scalar function V µ (˜ y w ) defined in (5) is continuously differentiable and verifies the condition V µ (0) = 0 . Furthermore, let M w , K w ≻ 0 . Now, if we choose 0 < µ < √ λ m ( M w ) λ m ( K w ) ∥ M w ∥ := µ V , then V µ (˜ y w ) > 0 ∀ ˜ y w ∈ R 2 n \ { 0 } . Additionally, then V µ (˜ y w ) is radially unbounded as ∥ ˜ y w ∥ → ∞ ⇒ V µ (˜ y w ) →∞ .

Proof. Step 1: It can be easily seen that V µ (˜ y w ) in (5) is smooth and continuously differentiable.

Step 2: Proof that V µ (0) = 0 .

Step 3: Proof that the Lyapunov candidate is positive definite; i.e., V µ (˜ y w ) > 0 ∀ ˜ y w ∈ R n \ { 0 } . As the gradient of the Lyapunov candidate, as defined in (14), is zero for ˜ y w = 0 :

˜ y w = 0 is a critical point of V µ (˜ y w ) . According to Lemma 5, the Hessian in (15) is positive-definite [56]: H V (˜ y w ) ≻ 0 ∀ ˜ y w ∈ R 2 n . With that, (5) is convex and its global minimum is at ˜ y w = 0 , where V µ (0) = 0 . In summary, we state V µ (˜ y w ) > 0 ∀ ˜ y w ∈ R n \ { 0 } .

Step 4: Proof that the Lyapunov candidate is radially unbounded: i.e., ∥ ˜ y w ∥ → ∞ ⇒ V µ (˜ y w ) →∞ . Lemma 6 is exploited for identifying a lower bound on V µ (˜ y w ) :

Lemma 5 tells us that P V ≻ 0 and with that λ m ( P V ) > 0 . Therefore, if ∥ ˜ y w ∥ → ∞ , it also follows that V µ (˜ y w ) →∞ .

A.4 Proof of Theorem 1: Global asymptotic stability of unforced network

The Lemmas introduced below are used to prove Theorem 1.

Lemma 7. Suppose ¯ x w , ˜ x w , b ∈ R n and n ∈ N + . Then, the function h ˙ V , th (˜ x w ) defined as

is positive semi-definite.

Proof. Proving h ˙ V , th (˜ x w ) ≥ 0 is equivalent to proving that the scalar function ˘ h ˙ V , th ( r ) = (tanh( r + a ) -tanh( a )) r ≥ 0 ∀ r, a ∈ R , where we set r = ˜ x w ,i and a = ¯ x w ,i + b i . Now, we expand the hyperbolic tangent:

as the denominator e 2 r +4 a + e 2 r +2 a + e 2 a +1 > 0 ∀ r ∈ R and as sign ( 2 e 2 a ( e 2 r -1 )) = sign( r ) . For example, e 2 r -1 ≥ 0 ∀ r ≥ 0 . Analog, e 2 r -1 < 0 ∀ r < 0 .

Lemma 8. Let M w ≻ 0 , K w ≻ 0 , and D w ≻ 0 . Also, let µ ∈ R + be chosen such that 0 < µ < λ m ( D w ) λ m ( M w )+ ∥ D w ∥ 2 4 λ m( K w) := µ ˙ V . Then, the matrix P ˙ V = [ µK w 1 2 µD w 1 2 µD T w D w -µM w ] ∈ R n is positive definite.

Proof. The Schur complement of P ˙ V is given by

The lower bound on the smallest Eigenvalue of S P ˙ V can be identified as

As K w , D w ≻ 0 , we know that ∥ D w ∥ 2 λ m ( K w ) > 0 . Therefore, the case µ = λ m ( D w ) λ m ( M w )+ ∥ D w ∥ 2 4 λ m( K w) := µ ˙ V determines the lower bound on λ m ( S P ˙ V ) :

We conclude, based on the Eigenvalue sensitivity theorem of symmetric matrices [46], that S P ˙ V ≻ 0 and with that P ˙ V ≻ 0 [56].

A.5 Proof of Theorem 2: Proof of Input-to-State Stability (ISS)

Lemma 9. Let ¯ x w , ˜ x w , b ∈ R n . Then,

Proof. Proving h V ,th (˜ x w ) ≤ 2 | ˜ x w | is equivalent to proving that the scalar function ˘ h V , th ( r ) = ∫ r 0 tanh( σ + a ) d σ -∫ r 0 tanh( a ) d σ ≤ 2 | r | ∀ r, a ∈ R , where we set r = ˜ x w ,i and a = ¯ x w ,i + b i . We perform the integration contained in ˘ h V , th ( r ) :

Next, we demonstrate that the slope of 2 | r | is always larger than the magnitude of the slope of ˘ h V , th ( r ) :

Additionally, ˘ h V , th (0) = 2 | 0 | = 0 . We conclude that ˘ h V , th ( r ) ≤ 2 | r | ∀ r ∈ R and with that, h V ,th (˜ x w ) ≤ 2 | ˜ x w | ∀ ˜ x w ∈ R n .

Lemma 10. Let M w ≻ 0 and K w ≻ 0 . Then, (5) is bounded by the two scalar, class K ∞ functions α 1 ( r ) = 1 2 λ m ( P V ) r 2 and α 2 ( r ) = 1 2 λ M ( P V ) r 2 +2 √ nr : α 1 ( ∥ ˜ y w ∥ 2 2 ) ≤ V µ (˜ y w ) ≤ α 2 ( ∥ ˜ y w ∥ ) 2 2 .

Proof. With Lemma 2, we already showed that V µ (˜ y w ) is a Lyapunov candidate. Now, we additionally also verify the conditions for ISS-Lyapunov candidates [40].

Step 1: Establishing bounds on V µ (˜ y w ) . We first identify the lower bound of V µ (˜ y w ) by leveraging Lemma 6:

Similarly, we derive an upper bound for V µ (˜ y w ) exploiting Lemma 9.

Step 2: Proof that α 1 ( r ) , α 2 ( r ) belong to class K ∞ . According to Lemma 5, P V ≻ 0 and with that λ m ( P V ) > 0 . First, we analyze the behavior of α 1 ( r ) : as it is strictly increasing and α 1 (0) = 0 , it belongs to class K . Furthermore, we can evaluate lim r →∞ α 1 ( r ) = ∞ . Therefore, α 1 ( r ) ∈ K ∞ [40]. α 2 ( r ) is also strictly increasing for r ∈ [0 , ∞ ) , α 2 (0) = 0 , and it is radially unbounded as lim r →∞ α 2 ( r ) = ∞ . For that reason, α 2 ( r ) ∈ K ∞ as well.

A.5.1 Proof of Theorem 2

Theorem 2 restated. Suppose M w , K w , D w ≻ 0 , 0 < θ < 1 , and that we choose 0 < µ < min { µ V , µ ˙ V }} . Then, (4) is globally Input-to-State Stable (ISS) such that the solution ˜ y w ( t ) verifies

Proof. Step 1: Bounds on ISS-Lyapunov candidate.

Lemma 10 provides the K ∞ functions α 1 ( r ) = 1 2 λ m ( P V ) r 2 and α 2 ( r ) = 1 2 λ M ( P V ) r 2 +2 √ nr such that α 1 ( ∥ ˜ y w ∥ 2 2 ) ≤ V µ (˜ y w ) ≤ α 2 ( ∥ ˜ y w ∥ ) 2 2 .

Step 2: Minimum energy dissipation.

Let 0 < µ < min { µ V , µ ˙ V }} as in the proof of Theorem 1. We compute the input-dependent time-derivative of the ISS Lyapunov candidate. We do not repeat the derivations already made as part of (7) (e.g., exploiting Lemmas 7 and 8).

where we leveraged Hölder's inequality. We choose θ such that 0 < θ < 1 . As a consequence,

We define

Lemma 8 shows that λ m ( P ˙ V ) > 0 . Therefore, α 3 ( r ) is a continuous positive function. Furthermore, as µ > 0 , ρ ( r ) is a strictly increasing for r ∈ [0 , ∞ ) . Additionally with ρ (0) = 0 verified, it can be stated that ρ ( r ) belongs to class K [40]. We conclude that

Step 3: Conclusions.

As a result of Steps 1 and 2, the system is input-to-state stable, and with that, the solution ˜ y t satisfies [40]

with

Indeed, based on Theorem 1 and the associated proof, we can easily verify that γ ( r ) is strictly increasing for r ∈ [0 , ∞ ) and that γ (0) = 0 . As a consequence, γ ( r ) ∈ K [40].

B Appendix on an approximate closed-form solution for coupled oscillator networks

(b) Time evolution of a harmonic oscillator

(c) Damping regimes of a harmonic oscillator

B.1 Closed-form solution to a forced harmonic oscillator

As introduced in (1), we consider the linear dynamics of a 1D forced harmonic oscillator with state y i = [ x i ˙ x i ] ∈ R 2

where F i ( t ) ∈ R is the externally applied force acting on the oscillator.

The characteristic equation for the unforced dynamics (i.e., F i ( t ) = 0) can be stated as [48]

where ω n ,i = √ κ i and ζ i = d i 2 √ κ i are the natural frequency and the damping factor of the i th homogeneous oscillator, respectively. This harmonic oscillator exhibits three regimes: underdamped ( ζ i < 1 ), critically damped ( ζ i = 1 ), and overdamped regime ( ζ i > 1 ).

̸

We approximate the forcing using the Heavyside function H ( t ) : F i ( t ) = F i ( t k ) H ( t ) , where F i ( t k ) is the constant external forcing as computed by (10). The solution for ζ i = 1 is given by [48]

where δt = t k +1 -t k , α i = ζ i ω n ,i , and β i = ω n ,i √ 1 -ζ 2 i . After enforcing the initial conditions x i ( t k ) , x i ( t k ) , the integration constants

can be identified with ∆ λ i = λ i, 2 -λ i, 1 = -2 β i j , where j is the imaginary value. While we could derive the solution for the critically damped case d i < 2 √ κ i separately, we instead approximate it in our network dynamics with (51) by setting ∆ λ i ∼ = { sign( -2 β i j ) ϵ | 2 β i j | < ϵ -2 β i j | 2 β i j | ≥ ϵ , where ϵ ∈ R + ≪ 1 is a small, positive value.

B.2 Algorithmic implementation

We can now leverage the closed form solution to the evolution of a single, decoupled damped harmonic oscillator of (51) to solve the integral in (11)

with

We summarize the approach of integrating/rolling out the CFA-CON dynamics in Algorithm 1.

Algorithm 1 Rollout of CFA-CON.

Inputs: initial state y ( t 0 ) , input sequence { u ( t 0 ) , . . . u ( t k ) , . . . u ( t N ) } Outputs: state sequence { y ( t 0 ) , . . . y ( t k ) , . . . y ( t N ) }

2: while k ≤ N do

1: k ← 0

3: ( x ( t k ) , ˙ x ( t k )) ← y ( t k )

5: ω n , ζ ← √ κ, d 2 √ κ ▷ Compute the characteristics of the decoupled harmonic oscillators.

7: c x ( t ) k ▷ Compute integration constants using initial conditions.

k

+1

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10:

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▷

▷

Set time step.

Update state with approximated closed-form solution.

13: k ← k +1

14: end while

B.3 Approximation bounds for CFA-CON

Lemma 11 demonstrates how, for the particular case of no external input and linearly decoupled oscillators (which we are always free to choose), we can establish bounds on the approximation error when using the closed-form solution instead of the ground-truth coupled oscillator dynamics.

Lemma 11. Suppose that the network is unforced with g ( u ( t )) = 0 and that K = diag( κ 1 , . . . , κ n ) , D = diag( d 1 , . . . , d n ) such that the oscillators are not linearly coupled. Then, given any t ≥ 0 , and the initial state y (0) ∈ R 2 n , the error between the continuous dynamics ¨ x ( t ) of (2) and the approximated dynamics ˆ ˙ x ( t ) in (10) , is bounded by ∥ ¨ x ( t ) -ˆ ¨ x ( t ) ∥ ≤ 2 .

Proof. (2), (10) and F = -f ¨ x, nld ( y (0) , 0) give us

▷

Update time index.

B.4 Empirical evaluation of approximation error

In Table 3, we present a comparison of CFA-CON with several other strategies for integrating nonlinear dynamics, such as CON. Following the implicit assumption made in Section B, we consider the case of g ( u ) = 0 , K = diag( κ 1 , . . . , κ n ) and D = diag( d 1 , . . . , d n ) but with the hyperbolic coupling between the oscillators active (i.e., a full W matrix). Integrating the dynamics at a very small time step (i.e., δt = 5e -5 s ) with a high-order ODE solver would give us a very accurate solution, but this is computationally infeasible in practice. We, therefore, regard this as the upper bound on the accuracy of the solution. A feasible solution would be to implement either a high-order solver such as Tsit5 at a larger integration time-step, e.g., δt = 1e -1 s ) or a low-order solver with a slightly smaller integration time step, e.g., δt = 5e -2 s ). Therefore, we also benchmark these options. We also benchmark an implementation specialized on the underdamped case (i.e., ζ i < 1 ): Closed-Form Approximation of the Underdamped Coupled Oscillator Network (CFA-UDCON). This specialized implementation allows us to avoid using complex numbers in the algorithm and reduces the number of computations necessary for calculating the approximated solution. As a result, we see a considerable increase in the sim-time to real-time factor.

B.4.1 Integration error

We perform the integration error benchmark over 100 different network configurations, all consisting of 50 oscillators ( n = 50 ): First, we sample the natural frequency of the i th oscillator from a uniform distribution as ω n ,i ∼ U (0 . 05 Hz , 0 . 5 Hz) , then we sample κ i ∼ U (0 . 2 N / m , 2 N / m) such that K = diag( κ 1 , . . . , κ n ) ≻ 0 , which lets us determine each mass m i = κ i ω 2 n ,i . Next, the damping ratio is determined as ζ i ∼ U (0 . 1 , 0 . 9) and

ζ i ∼ U (0 . 1 , 2 . 0) for the underdamped and general case, respectively. As a result, D = diag( d 1 , . . . , d n ) ≻ 0 with d i = 2 ζ i √ m i κ i given. Finally, by leveraging the Cholesky decomposition, we sample a W ≻ 0 and b i ∼ U ( -1 , 1) . We compute the estimation error of all integrated trajectories with respect to the high-precision solution (i.e., Tsitouras' 5/4 method (Tsit5) at δt = 5e -4 s ). For this, we compute the RMSE for each 60s trajectory and then take the mean and standard deviation across the 100 different network configurations.

B.4.2 Simulation-time to real-time factor

The simulation vs. real-time factor is computed as the simulated rollout duration per second of computational time. For this, we let each method simulate a 60s trajectory for 100 times and record the minimum run time on an Intel Core i7-10870H CPU (single core) over 10 trials. Because of computational constraints, we simulated with the high-precision Tsit5 solver the trajectory only 5 times.

B.4.3 Results

The results in Table 3 show that CFA-CON is 30% more accurate than the Euler integrator at half of the speed. Compared against the Tsit5 integrator, CFA-CON exhibits a 1.56x speed increase while being significantly less accurate. For the underdamped case with ζ < 1 , the specialized implementation CFA-UDCON is 14 . 8% faster and at the same time 32% more accurate than the Euler integrator. Furthermore, CFA-UDCON is 3.7x faster and significantly less accurate than the Tsit5 integrator. We can conclude that in the pure rollout setting (i.e., no backpropagation involved) for a generic CON, the CFA-CON does not show clear advantages to an appropriately tuned Euler or Tsit5 solver. However, the specialized version CFA-UDCON demonstrates a 2.4x speed-up at no reduction of accuracy vs. CFA-CON for underdamped oscillator networks.

C Appendix on experimental setup and datasets

C.1 Datasets

For all datasets, we generate images of size 32 × 32px and subsequently normalize the pixels to the interval [ -1 , 1] .

C.1.1 Unactuated mechanical datasets

We consider multiple mechanical datasets based on a standard implementation included in the Toy Physics category of the NeurIPS 2021 Track on Datasets and Benchmarks publication by Botev et al. [26]: mass-spring with friction (M-SP+F), a single pendulum with friction (S-P+F), and a double pendulum with friction (D-P+F). All datasets contain 5000 system trajectories in the training set and 1000 trajectories each in the validation and test set. Each trajectory is generated by first randomly initializing the system, then rolling it out for 3 s using an Euler integrator with a time step size of 5 ms . Samples are recorded at a rate of 20Hz (i.e., a time step of 0 . 05s ). As a result, each trajectory contains 60 images of the system's state. As all of these datasets are unactuated, we can deactivate the input-to-forcing mapping component from all models (e.g., set g ( u ) = 0 for the CON model).

The M-SP+F dataset contains motion samples of a damped harmonic oscillator with a mass of 0 . 5 kg , a spring stiffness of 2N / m , and a damping coefficient of 0 . 05Ns / m . For each trajectory, the initial condition of the massspring is randomly sampled by combining a random sign( q ) with a uniformly sampled | q | ∼ U (0 . 1 m , 1 m) . The position of the mass is rendered with a filled circle in a grayscale image.

The SP+F and DP+F datasets include the evolutions of a single-link pendulum and double-link pendulum, respectively, with a mass of 0 . 5kg attached to the end of each link, which has a length of 1m . The dataset considers a gravitational acceleration of 3 m / s 2 . A rotational damper with coefficient 0 . 05Nms / rad provides the friction. Similarly to the M-SP+F dataset, both the sign and the absolute value of the initial configuration are randomly sampled, where | q (0) | ∼ U (1 . 3 rad , 2 . 3 rad) . The position of each mass is rendered with a filled circle. For the single-link pendulum, this is done in grayscale, and for the double pendulum, each mass is rendered with a different color (i.e., blue and red).

C.1.2 Actuated continuum soft robot datasets

The shape of slender and deformable rods can be approximated by considering the deformations along the 1D curve of the backbone [57]. While this curve is still infinite-dimensional, it is possible to discretize the backbone into (many) segments with piecewise constant strain [58, 57]. Accordingly, we describe the kinematics of a planar continuum soft robot consisting of n b segments with the PCS model [58]. We assume each segment has a length of 100mm and a diameter of 20mm . The PCS model assumes each segment to have constant strain. In the planar case, this means that the shape of the i th segment can be parametrized by ξ i = [ κ be ,i σ sh ,i σ ax ,i ] T ∈ R 3 where κ be ,i is the bending strain (i.e., the curvature) in the unit rad / m , σ sh ,i is the shear strain (dimensionless), and σ ax ,i is the axial elongation strain (dimensionless). The robot's configuration is then defined as q = [ ξ T 1 · · · ξ T i · · · ξ T n b ] T . In the case of Piecewise Constant Curvature (PCC), only the bending strain is active as shear strains and axial strains are neglected, and the configuration is now q ∈ R n b . The PCS model generates EOM in the form of [59]

where B ( q ) ≻ 0 and C ( q, ˙ q ) are the inertia and Corioli matrices, respectively. G ( q ) collects the gravitational forces, K q ≻ 0 is the stiffness matrix, and D q ≻ 0 contains the damping coefficients. u ( t ) ∈ R n b is an external force acting on the generalized coordinates, and now m = n b .

We derive the corresponding dynamics for a continuum soft robot of material density 600 kg / m 3 , elastic modulus of 20 000 Pa , shear modulus of 10 000 Pa , and damping coefficients of 0 . 000 01 Nm 2 s for bending strains, 0 . 01Ns for shear strains, and 0 . 01Ns for axial strains, respectively. Gravity is pointing downwards. The implementation of the dynamics in JAX [60] is based on the JSRM library [54, 61], and we simulate the robot using a constant integration time step of 0 . 1ms . We render grayscale images of the robot with a size of 32 × 32px at a rate of 50Hz using OpenCV [62]. We generate 10000 trajectories, each of duration 2 . 0s and a sampling time-step of 0 . 02 s . We use 60% training, 20% validation, and 20% test split. For each trajectory, we randomly sample a constant actuation/input u ∼ U ( -u max , u max ) . We choose the maximum actuation magnitude to be equal to the sum of the contribution of the potential forces (i.e., elastic and gravitational forces): u max = G ( q max ) + Kq max with q max ,i = [5 π rad / m , 0 . 2 , 0 . 2] T .

We generate three datasets based on this continuum soft robot model: in the CS dataset, we consider one segment with all three planar strains active (i.e., bending, shear, and elongation). This results in three DOF and six-state variables in the dynamical model. In the case of the PCC-NS-2 and PCC-NS-3 datasets, we base the dataset on a simulated system consisting of two planar bending segments, respectively. Each segment is parametrized using Constant Curvature (CC) [63, 64], which results in two configuration variables and a state dimension of four.

C.1.3 Unactuated PDE reaction-diffusion dataset

We consider the 1 st -order Reaction-diffusion ( R-D ) PDE on which (Champion et al, 2019). [9] evaluated their SINDy Autoencoder on. The PDE of the high-dimensional lambda-omega reaction-diffusion system is defined as

where u ( t, q ) : R × R 2 → R and v ( t, q ) : R × R 2 → R are time-dependent two vector fields defined over the spatial domain q ∈ R 2 . We choose the same system parameters and initial condition as Champion et al. [9]: d 1 , d 2 = 0 . 1 , and β = 1 and

After discretizing the spatial domain into 32 points along each dimension, we solve the PDE with a MATLAB ODE45 solver the solution of u ( t, q ) and v ( t, q ) at each time step and grid point. Subsequently, the solution is multiplied with a Gaussian centered at the origin [9]

We integrate the system from the specified initial condition for 500s and store samples at a time step of 0 . 05s . We divide the entire sequence into 99 subsequences each containing 101 samples. We train the models to predict these subsequences that have a horizon of 5 . 0 s each.

We stack the solution of ¯ u ( t, q ) and ¯ v ( t, q ) contained in the two grids o u ( t ) , o v ( t ) ∈ R 32 × 32 , respectively, to gather the images o ( t ) ∈ R 32 × 32 × 2 containing two channels. A sample sequence of the generated images is presented in Apx. 17. We use 60% of the subsequences (i.e., 59 ) as our training set, and employ 20% (i.e., 19 ) for the validation and test sets, respectively.

C.2 Autoencoder architecture

For the encoder and decoder, we rely on a vanilla CNNs implemented as a β -VAE [5].

Encoder. The encoder consists of two convolutional layers with kernel size (3 , 3) and stride (1 , 1) mapping to 16 , 32 , respectively. The features are flattened and then passed to two linear layers with hidden dimension 256 and n z . Each layer (except for the last) is followed by a layer norm [65] and a LeakyReLU nonlinearity.

Decoder. The decoder first uses two linear layers to map to hidden dimensions of 256 and 32768 , respectively. We then apply two 2D transposed convolutions [66] reducing the number of channels first to 16 , and then to 1 . Each layer (except for the last linear and last convolutional) is followed by a layer norm [65] and a LeakyReLU nonlinearity. Finally, we apply a sigmoid function to clip the output into the range [ -1 , 1] .

C.3 Latent dynamic models

In the following section, we provide implementation details for the latent dynamic models that we evaluated as part of this work.

C.3.1 Coupled Oscillator Network (CON)

We leverage the CON in W -coordinates given by (3) for learning latent space dynamics. Specifically, we consider the input-to-force mapping g ( u )) = B ( u ) u ( t ) , where B ( u ) ∈ R n × m is parametrized by few-layer MLP. We report results for two different sizes of the MLP: one medium-sized variant consisting of five layers with a hidden dimension of 30 and a small variant with two layers and a hidden dimension of 12 . In both cases, we use a hyperbolic tangent as a nonlinearity.

When training the model, we jointly optimize M -1 w , K w , D w , b and g ( u ) . However, we also need to make sure that we adhere to the stability constraints M -1 w , K w , D w ≻ 0 . For this, we leverage the Cholvesky decomposition [67]. Instead of directly learning the full matrix A ∈ R n z × n z , we designate the elements of an upper triangular matrix U ∈ R n z × n z as the trainable parameters. The Cholesky decomposition demands that diag( U 11 , . . . , U n z n z ) > 0 . Therefore, we apply the operation

where ˘ U is the learned upper triangular matrix, and ϵ 1 = 1e -6 and ϵ 2 = 2e -6 are two small, positive values. The positive-definite matrix A is now given by A = U T U ≻ 0 .

C.3.2 Neural ODEs

We consider two kinds of Neural ODEs [27]: the vanilla f NODE : ξ ( t ) × u ( t ) ↦→ ˙ ξ ( t ) maps latent state and system actuation directly into a time derivative of the latent state. In contrast, for the MECH-NODE , we enforce the latent dynamics to have a mechanical structure

We represent both f NODE and f MECH -NODE as MLPs consisting of 5 layers, a hidden dimension of 30 , and a hyperbolic tangent nonlinearity.

C.3.3 Autoregressive models

For the below stated autoregressive models, we divide the integration between two (latent) samples ξ ( t k ) and ξ ( t k +1 ) into N int integration steps ξ ( t k + δt ) , . . . , ξ ( t k + k ′ δt ) , . . . , ξ ( t k + N int δt ) where δt is the integration step size and t k +1 = t k + N int δt . The autoregressive model now describes the transition ξ ( t k ′ +1 ) = f ar ( ξ ( t k ′ ) , u ( t k ))) ∀ k ′ ∈ 1 , . . . , N int .

RNN. We implement a standard, single-layer Elman RNN with tanh nonlinearity. The hidden state captures the latent state of the system. The latent state transition functions are given by

GRU. We implement a standard, single-layer GRU [30] with sigmoid activation function where we interpret the latent state of the system as the hidden state of the cell. The latent state transition functions are given by

where σ is the sigmoid function, ⊙ the Hadamard product, W hr , W hp , W hn ∈ R 2 n z × 2 n z , W ir , W ip , W in ∈ R 2 n z × m , and b hr , b ir , b ip , b in ∈ R 2 n z .

coRNN. A time-discrete coRNN is defined by the transition function

where γ, ε ∈ R + are positive, scalar hyperparameters representing the stiffness and damping coefficients, respectively. The term tanh ( Wξ ( t k ′ ) + V u ( t k ) + b ) with W ∈ R 2 n z × 2 n z , V ∈ R n z × m , and b ∈ R n z contributes nonlinear state-to-state connections. It is implemented with a linear layer operating on ( ξ ( t k ′ ) , u ( t k )) followed by a hyperbolic tangent nonlinearity.

CFA-CON. We adapt the Alg. 1 for predicting the time evolution in latent-space

where f CFA -CON describe the autoregressive state transition by the CFA-CON model as introduced in Eq. 11.

C.4 First-order variants of dynamical models

For learning (latent) dynamics of 1 st -order systems (e.g., the reaction-diffusion dataset R-D ), it might be beneficial also to formulate the dynamical model to be of 1 st -order. While this is straightforward for some dynamics that do not explicitly take the order into account (e.g., RNN, GRU, NODE), for other models such as coRNN, CON, and CFA-CON more adjustments are necessary. Namely, we substitute the d z d t component of the ODE with the expression for d˙ z d t . Furthermore, we remove any terms that depend on the velocity ˙ z (e.g., damping effects). Below, we report in detail the adapted, 1 st -order formulations for the coRNN, CON, and CFA-CON models.

CON. In the 1 st -order version, we adapt the standard, 2 nd -order ODE of the CON network as defined in (3) to

coRNN. In the 1 st -order version, we define the transition function as

CFA-CON. We adapt a 1 st -order version of Alg. 1 for predicting the time evolution in latent-space

where the closed-form solution for the integral is given by

C.5 Estimation of the initial latent velocity

For 2 nd -order systems and when integrating the evolution of the latent state ξ ( t ) = [ z T ( t ) ˙ z T ( t ) ] T in time, we need to have access to an initial latent velocity ˙ z ( t 0 ) such that we can roll out the latent state ξ ( t ) in time. A naive approach to estimating such an initial latent velocity would be to encode multiple (at least two) images of the system at the start of the trajectory into latent space and then perform numerical differentiation (e.g., finite differences) in latent space. However, we found the resulting ˙ z ( t k ) to be relatively noisy and susceptible to small encoding errors. Instead, we propose to perform numerical differentiation in image space and then map this velocity into latent space using the encoder's Jacobian. First, we estimate the image-space velocity at t k using finite differences: ˙ o ( t k ) ≈ o ( t k +1 ) -o ( t k -1 ) t k +1 -t k -1 . The latent velocity is then estimated as ˙ z ( t k ) = ∂ Φ ∂o ( o ( t k )) ˙ o ( t k ) , where ∂ Φ ∂o is obtained with forward-mode automatic differentiation.

C.6 Training

We implement the network dynamics and the neural networks (e.g., encoder, decoder, and MLPs) in JAX [60] and Flax [68], respectively. When training or inferring time-continuous dynamical models (e.g., NeuralODE, CON), we rely on Diffrax [47] for numerical integration of the ODE using the Dormand-Prince's 5/4 method [69] (Dopri5). For the numerical integration of both the time-continuous and the time-discrete models (e.g., RNN, coRNN, CFA-CON), we use an integration time-step δt of 0 . 025s and 0 . 01 s for the Toy Physics [26] and soft robotic datasets, respectively.

Because of the GPU memory constraints, we limit ourselves to a batch size of 30 and 80 trajectories for the Toy Physics [26] and soft robotic datasets, respectively. We implement a learning rate schedule consisting of a warm-up (5 epochs) and a cosine annealing [70] period (remaining epochs). We employ an AdamW optimizer [71, 72] with β 1 = 0 . 9 , β 2 = 0 . 999 for updating both neural network weights (e.g., encoder, decoder) and parameters of the dynamical model (e.g., K,D w , M w , etc.).

Before training, we conduct a hyperparameter selection study using Optuna [53]. For this, we leverage a Tree-Structured Parzen Estimator [73] for identifying hyperparameters such as the base learning rate, the weight decay, the loss function weights, and model-specific hyperparameters such as the number of MLP layers, the hidden dimension of the MLP layers, the γ and ϵ values for the coRNN model etc. that minimize the RMSE of the predicted images. To reduce computational requirements, we employ the Asynchronous Successive Halving Algorithm [74] to stop unpromising trials early.

C.7 Evaluation metrics

Similar to other publications in the field [75, 76, 77], we state the RMSE, the Peak Signal-to-Noise Ratio (PSNR) and the Structural Similarity Index Measure (SSIM) [78] between the ground-truth image o ∈ R h o × w o and the predicted image image ˆ o ∈ R h o × w o . We use the separated test set for all evaluation results.

C.7.1 Root Mean-Square Error

The RMSE between the two images is given by

C.7.2 Peak Signal-to-Noise Ratio

The PSNR is a function of the total Mean Squared Error (MSE) loss and the maximum dynamic range of the image L .

As we work with normalized images with pixels in the interval [ -1 , 1] , the dynamic range is L = 2 .

C.7.3 Structural Similarity Index Measure

As simple pixel-by-pixel metrics such as RMSE or PSNR tend to average out any encountered errors, this could lead to a situation in which a significant reconstruction error in a part of the image is not seen in the RMSE metric but has a huge impact on the visual appearance of the reconstruction. SSIM [78] incorporates not just the absolute errors , but also the strong inter-dependencies between pixels, especially when they are spatially close. The SSIM metric between two observations o and ˆ o is given by

where

We use the constants C 1 = ( k 1 L ) 2 , C 2 = ( k 2 L ) 2 and C 3 = C 2 / 2 , where L signifies the dynamic range as previously used for the PSNR metric, and k 1 = 0 . 01 and k 2 = 0 . 03 . The average µ and the variance σ 2 is computed with a Gaussian filter with a 1D kernel of size 11 and sigma 1 . 5 . We set the weight exponents α , β , and γ for the luminance, contrast, and structure comparisons all to one. We rely on the PIX library [79] for efficiently computing the SSIM metric.

C.8 Compute resources

We trained the models on several desktop workstations for a total duration of roughly 150h . In total, we relied on 10x RTX 3090/4090 GPUs, each with 24 GB of VRAM, training the models in parallel. Each workstation contained between 64 and 128 GB of RAM, and we used roughly 100 GB of total storage. Training each model on one random seed took between 45min and 4h depending on the model type, the integration time constant, and the number of trainable parameters. The hyperparameter tuning we conducted beforehand (only on one random seed) took roughly the same time and computational resources as generating the final results.

For the control experiments, we additionally used a laptop with a 16-core Intel Core i7-10870H CPU and 32 GB RAM. We did not need to use a GPU for evaluating the model during closed-loop control.

D Appendix on learning latent dynamics

We report the full set of quantitative results, including the additional evaluation metrics PSNR and SSIM, in Tables 7 to 9. For PCC-NS-2 in Tab. 8, we additionally also recorded the training steps per second on an Nvidia RTX 3090 GPU with a batch size of 80 (which leads to 8080 images per batch). We plot the results of a sweep across the latent dimensions for the additional evaluation metrics PSNR and SSIM in Fig. 8. Correspondingly, we visualize the number of trainable parameters of each model vs. the latent dimension in Fig. 9. In Figs. 10- 16, we present sequences of stills for the rollout of the trained CON (-M) models on the various datasets.

D.1 Results for Reaction-diffusion dataset

As all previous examples exampled ODEs, we strive to test the proposed approach also on a system that is governed by PDEs. Specifically, we consider the Reaction-diffusion ( R-D ) dataset as introduced in Apx. C.1.3. To address the unactuated nature of the dataset, we remove, analog to the M-SP+F , S-P+F , and D-P+F datasets, the input-to-state mapping parameters of the dynamical models (e.g., the B ( u ) and E ( τ ) MLPs for the CON models). Furthermore, the PDE describing the system dynamics is of 1 st -order. Therefore, we leverage the 1 st -order versions of the latent dynamics as specified in Apx. C.4.

We report the metrics of the test set evaluations in Tab. 10. Furthermore, we also present a sequence of stills of the rollout of a trained latent dynamics CON model in Fig. 17. We find it impressive that CON with its strong stability guarantees can accurately model the dynamics of a high-dimensional PDE system.

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E Appendix on latent-space control

E.1 Latent-space control of a damped harmonic oscillator

We consider an actuated version of the M-SP+F dataset (i.e., a damped harmonic oscillator) and denote it with M-SP+F+A . All system, trajectory sampling and rendering parameters remain the same, except that for each trajectory in the dataset we randomly sample a forcing u ∼ U ( -1 N , 1 N) .

We train a CON model with latent dimension n z = 1 over three random seeds on the M-SP+F+A dataset. This means that the network consists of a single oscillator. From the three different random seeds, we choose the model that achieves the best validation loss, which results in an RMSE of 0 . 0327 , a PSNR of 5 . 99 , and SSIM of 0 . 9796 on the test set.

Fig. 18 shows how the encoder learns an almost linear relationship between the actual configuration of the system and the predicted latent space representation. Furthermore, we notice that both the ground-truth and the learned potential energy are convex and exhibit a global minimum at q = 0m .

We compare the performance of P-satI-D , D+FF , and P-satI-D+FF controllers based on the CON model in Fig. 19. For the P-satI-D controller, we choose the control gains K p = 10 , K i = 10 , K d = 5 , υ = 1 . The D+FF controller uses K d = 3 . 5 . Finally, the P-satI-D+FF is configured with K p = 2 , K i = 0 . 3 , K d = 3 . 5 , υ = 1 . The results show that the P-satI-D+FF controller exhibits thanks to its feedforward term no overshooting and a faster response time than the pure feedback controller P-satI-D . The high accuracy of the feedfoward term can be seen from the performance of the D+FF controller, that only exhibits relatively small steady-state error. Adding small proportional and integral feedback actions in the P-satI-D+FF controller keeps the compliance high while removing the steady-state error and reducing the response time.

Finally, we visualize the behavior of the P-satI-D+FF controller as a sequence of stills in Fig. 20.

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E.2 Latent-space control of a two segment PCC soft robot

E.2.1 Potential energy landscape

When leveraging (learned) dynamical models for setpoint regulation, it is essential to accurately estimate the potential energy as this dictates the efficacy of the feedforward terms. Therefore, we qualitatively evaluate the potential energy landscape of the CON latent dynamic model.

In Fig. 21(a), we can see how CON contains a single, isolated, and globally asymptotically stable equilibrium as proven in Appendix A.2 and Section 2, respectively.

Furthermore, we want to verify that the learned potential corresponds to the actual potential energy of the simulated system. An autonomous continuum soft robot with the tip pointing downwards in a straight configuration exhibits an isolated, globally asymptotically stable equilibrium at q = 0 (i.e., zero strains) [42]. For this purpose, we can compare the learned potential energy field in Fig. 21(b) with the ground-truth potential energy field in Fig. 21(c). We confirm, based on Fig. 21(b), that, indeed, the learned potential also has its minimum close to/at q = 0 . Although the field is shaped slightly differently, the potential forces are clearly pointing inwards towards the global attractor.

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For the control experiments, we train instances of the MECH-NODE and CON-M models with latent dimension n z = 2 and with neural network weights initialized with three different random seeds. For MECH-NODE , we choose the model with the lowest validation loss (seed 0 ).

For the CON network, we found that model-based control does not perform as well when the latent stiffness Γ w (as visualized in Fig. 21(a)) is significantly larger along one of the Eigenvectors than along the other one. Therefore, we evaluate the Eigenvalues of the learned stiffness matrix in W -coordinates after training: λ 1 , 2 (Γ w ) . Particularly, we choose the seed that minimizes the normalized standard deviation of the Eigenvalues

E.2.3 Additional control results

Additional results for the P-satI-D feedback controller based on the MECH-NODE and CON models are provided in Fig. 23, Fig. 24, respectively and for the P-satI-D+FF controller based on the CON model in Fig. 25. Sequences of stills for the CON P-satI-D+FF controller are provided in Fig. 22.

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F Extended discussion on future applications and limitations

F.1 Systems for which we would expect the proposed method to work

Mechanical systems with continuous dynamics, dissipation, and a single, attractive equilibrium point. The proposed method is a very good fit for mechanical systems with continuous dynamics, dissipation, and a single, attractive equilibrium point. In this case, the real system and the latent dynamics share the energetic structure and stability guarantees. Examples of such systems include many soft robots, deformable objects with dominant elastic behavior, and other mechanical structures with elasticity.

Local modeling of (mechanical) systems that do not meet the global assumptions. Even if the global assumptions of the proposed method are not met, the method can still be applied to model the local behavior around a local asymptotic equilibrium point of the system (i.e., in the case of multi-stability). For example, the method could be used to model the behavior of a robotic leg locally in contact with the ground, a cobot's interaction with its environment, etc.

F.2 Systems for which we could envision the proposed method to work under (minor) modifications

Mechanical systems without dissipation. The proposed method would currently not work well for mechanical systems without any dissipation, as (a) the original system will likely not have a globally asymptotically stable equilibrium point, and more importantly, (b) we currently force the damping learned in latent space to be positive definite. However, these systems are not common in practice as friction and other dissipation mechanisms are omnipresent, and the proposed method can learn very small damping values (e.g., the mass-spring+friction system). A possible remedy could be to relax the positive definiteness of the damping matrix in the latent space, allowing for zero damping. This would allow the method to work for systems without dissipation, such as conservative systems. Examples of such systems include a mass-spring system without damping, the n-body problem, etc.

Systems with discontinuous dynamics. The proposed method might underperform for systems with highly discontinuous dynamics, such as systems with impacts, friction, or other discontinuities. In these cases, the latent dynamics might not capture the real system's behavior accurately, and the control performance of feedforward + feedback will very likely be worse than pure feedback. Again, the method should be able to capture local behavior well. A possible remedy for learning global dynamics could be to augment the latent dynamics with additional terms that capture the discontinuities, such as contact and friction models (e.g., stick-slip friction).

Systems with multiple equilibrium points. The original system having multiple equilibria conflicts with the stability assumptions underlying the proposed CON latent dynamics. In this case, as, for example, seen on the pendulum+friction and double pendulum + friction results, the method might work locally but will not be able to capture the global behavior of the system. A possible remedy could be to relax the global stability assumptions of the CON network. For example, the latent dynamics could be learned in the original coordinates of CON while allowing W also to be negative definite. This would allow the system to have multiple equilibria &attractors. Examples of such systems include a robotic arm under gravity, pendula under gravity, etc.

Systems with periodic behavior. The proposed method will likely not work well for systems with periodic behavior, as they do not have a single, attractive equilibrium point. Examples of such systems include a massspring system with a periodic external force, a pendulum with a periodic external force, some chemical reactions, etc. Again, it is likely possible to apply the presented method to learning a local behavior (i.e., not completing the full orbit). A possible remedy could be to augment the latent dynamics with additional terms that capture the periodic behavior, such as substituting the harmonic oscillators with Van der Pol oscillators to establish a limit cycle or a supercritical Hopf bifurcation.

F.3 Systems for which we would not expect the proposed method to work

Nonholonomic systems. The proposed method likely would not work well for nonholonomic systems, as both structure (e.g., physical constraints) and stability characteristics would not be shared between the real system and the latent dynamics. Examples of such systems include vehicles, a ball rolling on a surface, and many mobile robots.

Partially observable and non-markovian systems. As the CON dynamics are evaluated based on the latent position and velocity encoded by the observation of the current time step and the observation-space velocity, we implicitly assume that the system is (a) fully observable and (b) fulfills the Markov property. This assumption might not hold for partially observable systems, such as systems with hidden states or systems with delayed observations. Examples of such cases include settings where the system is partially occluded or in situations without sufficient (camera) perspectives covering the system. Furthermore, time-dependent material properties, such as viscoelasticity or hysteresis, that are present and significant in some soft robots and deformable objects are not captured by the method in its current formulation.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors introduce Coupled Oscillator Networks (CONs), extending the idea of coRNNs but with the difference that the generalized force is symmetric, enabling the definition of a potential energy expression which can be exploited for energy shaping based control methods. It is proven, that this model is asymptotically stable and, given an input, input-to-state-stable. It is shown that CONs provide similar performance compared to other state of the art methods. Then, the proposed network is used together with a variational autoencoder to learn a low-dimensional representation of the dynamics of a soft robot. Finally, a PID controller with an energy shaping part is designed to steer the soft robot to a given desired state. The main contribution is the new network structure which allows us to exploit the potential energy expression.\n\nSTRENGTHS:\n- It was a pleasure to read the article as it is nicely written, and all steps are explained in detail and clarity.\n- Computing the solution of the CON by splitting the dynamics into a linear part (where the solution can be computed in closed form) and a nonlinear part is an original idea.\n- The new structure of the proposed network that allows us to exploit energy-based control methods open doors for many interesting applications and might be a significant contribution.\n\nWEAKNESSES:\n- Theorem 1 seems to be trivial as it elaborates on the (nonlinear) coupling of passive systems (damped mass-spring systems) which is always passive since it does not “produce” any energy and thus, is inherently stable. (Same for Theorem 2). In this sense, the theoretical contribution seems limited. Is there anything I missed?\n- The statement in Line 266 “PID controller has several well-known drawbacks, such […] steady-state errors (in case the integral gain is chosen to be zero)” is misleading as in this case we would say it’s a PD controller.\n- The main contribution seems to be the new network structure that allows to exploit the potential energy for energy shaping methods. The VAE and controller are existing methods. Thus, a more detailed elaboration on the performance and limitation of CONs would be beneficial as it is evaluated for soft robotics data sets only.\n\nQUESTIONS:\n- Can you add the inference time for the different methods in Table 1? If that’s not possible, could you give some general comments on the inference time of CONs?\n- Figure 3 visualizes that the controller with feed-forward part leads to heavy oscillations in the systems. Is that due to a poorly tuned controller or do you see the reason in the CON model?\n\nUPDATE: New score after discussion\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n- This paper proposes Coupled Oscillator Networks (CONs); Networks consisting of coupled one-dimensional dampened harmonic oscillators, coupled through a neuron-like connection.\n- It is shown that under some constraints, the unforced coupled oscillators have a single, globally stable equilibrium and the forced system is globally ISS stable.\n- As the CON has no closed-form solution, the authors propose to split the solution in one decoupled linear, analytically solvable term and a residual non-linear term which is. For this system, a closed-form approximation is derived (CON-CFA).\n- High dimensional observations are encoded via a VAE into a compressed latent space that is trained to reconstruct future observations that are predicted via a CON latent dynamics model. This system is trained jointly.\n- “CONs are an ideal fit for learning latent dynamics as they guarantee that the latent states stay bounded.”\n- The approaches (CON with two different latent dimensions,S and M, and CON-CFA) are evaluated on a simulated soft-robots dataset and benchmarked against other latent-space dynamics methods based on neural ODEs ore autoregressive models.\n- When predicting system state, CON-M performs on par with sota methods, CON-S and CON-CFA perform almost as good, while cheaper computationally.\n- For control, potential-shaping strategies are combined with PID, resulting in a latent-space control law. A forcing decoder is then trained to to predict control inputs. This approach strongly outperforms a simple latent space PID controller.\n\nSTRENGTHS:\n- Very interesting and (to my knowledge) novel approach of controlling robots in a latent dynamics space, modeled through a network of coupled oscillators with provable stability guarantees.\n- Extensive proofs and detailed descriptions of implementation.\n- Promising experimental evaluation.\n- Very concise and complete presentation of their approach.\n\nWEAKNESSES:\n- Limited experimental evaluation on a single simulated soft robot.\n- It is unclear how the obtained results generalize to other robots, in particular with less smooth contact dynamics.\n\nQUESTIONS:\n- Have you tried the method on other (non-soft) robots? Do you have any intuition on how your method would perform on environments with many contact forces?\n- Have you considered non-physical systems?\n\nLIMITATIONS:\nMany limitations of the method are clearly stated.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-08-01"}