dynoNet: A neural network architecture for learning dynamical systems

Abstract

This article introduces a network architecture, called dynoNet, utilizing linear dynamical operators as elementary building blocks. Owing to the dynamical nature of these blocks, dynoNet networks are tailored for sequence modeling and system identification purposes. The back-propagation behavior of the linear dynamical operator with respect to both its parameters and its input sequence is defined. This enables end-to-end training of structured networks containing linear dynamical operators and other differentiable units, exploiting existing deep learning software. Examples show the effectiveness of the proposed approach on well-known system identification benchmarks. K E Y W O R D Smachine learning, neural networks, system identification INTRODUCTION ContributionThis article introduces dynoNet, a neural network architecture tailored for sequence modeling and dynamical system learning (a.k.a. system identification). The network is designed to process time series of arbitrary length and contains causal linear time-invariant (LTI) dynamical operators as building blocks. These LTI layers are parametrized in terms of rational transfer functions, and thus apply infinite impulse response (IIR) filtering to their input sequence. In the dynoNet architecture, the LTI layers are combined with static (i.e., memoryless) nonlinearities which can be either elementary activation functions applied channelwise; fully connected feed-forward neural networks; or other differentiable operators (e.g., polynomials). Both the LTI and the static layers defining a dynoNet are in general multi-input multi-output (MIMO) and can be interconnected in an arbitrary fashion.Overall, the dynoNet architecture can represent rich classes of nonlinear, causal dynamical relations. Moreover, dynoNet networks can be trained end-to-end by plain back-propagation using standard deep learning (DL) software. Technically, this is achieved by introducing the LTI dynamical layer as a differentiable operator, endowed with a well-defined forward and backward behavior and thus compatible with reverse-mode automatic differentiation. 1 Special care is taken to devise closed-form expressions for the forward and backward operations that are convenient from a computational perspective.A software implementation of the linear dynamical operator based on the PyTorch DL framework 2 has been developed and is available in the GitHub repository https://github.com/forgi86/dynonet.git.

Flexible-Joint Manipulator Trajectory Tracking with Learned Two-Stage Model employing One-Step Future Prediction

“…The second part is based on LTI dynamical operators, as implemented in the novel dynoNet 1 . LTI operators were shown to be efficient [2] when learning the complex causal non-linear dynamics, while being suitable for an end-to-end backpropagation. These properties are advantageous for learning the dynamics of the flexible-joint manipulator.…”

“…The architecture of a branch is as follows: first the LTI block with a = 2 and b = 2, which outputs 14 features (motivated to represent a rough position + velocity approximation). a and b define the polynomial order for the denominator and nominator of a rational transfer function (see [2] for details). The LTI block is followed by two fully connected layers.…”

“…The LTI layers are parameterized in terms of rational transfer functions, and thus apply infinite impulse response (IIR) filtering to the input. Stacking this hardwired linear structure in multiple layers together with fully connected layers was shown [2] to approximate the complex non-linear dynamics. By providing the forward dynamics estimation obtained from the FIN, we allow the model to take into account the predicted future in which we would execute the next command as it is.…”

“…Neural networks (NNs) are known for their ability to generalize and model complex non-linear relations. We present In particular, as a part of our model we use the novel dynoNet [2], which resembles the features of RNN [3] and 1-D convolution [4]. The LTI layers are specifically designed for sequence modeling and system identification, successfully approximating complex non-linear causal dynamics, while being differentiable and suitable for backpropagation.…”

On the adaptation of recurrent neural networks for system identification

“…Other contributions propose new DL-based model structures explicitly conceived for SI purposes. For instance, [7] introduces a novel neural architecture which includes linear transfer functions as elementary building blocks, while [8] proposes model structures based on Koopman operator theory [9], and lift non-linear statespace dynamics to a higher-dimensional space where the dynamics are linear. The mapping from the original to the higher-dimensional space is learned using DL tools.…”