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We propose a framework for surrogate modelling of spiking systems. These systems are often described by stiff differential equations with high-amplitude oscillations and multi-timescale dynamics, making surrogate models an attractive tool for system design and simulation. We parameterise the flow function of a spiking system using a recurrent neural network architecture, allowing for a direct continuous-time representation of the state trajectories. The spiking nature of the signals makes for a data-heavy and computationally hard training process; thus, we describe two methods to mitigate these difficulties. We demonstrate our framework on two conductance-based models of biological neurons, showing that we are able to train surrogate models which accurately replicate the spiking behaviour., QC 20240927

We consider the problem of approximating flow functions of continuous-time dynamical systems with inputs. It is well-known that continuous-time recurrent neural networks are universal approximators of this type of system. In this paper, we prove that an architecture based on discrete-time recurrent neural networks universally approximates flows of continuous-time dynamical systems with inputs. The required assumptions are shown to hold for systems whose dynamics are well-behaved ordinary differential equations and with practically relevant classes of input signals. This enables the use of off-the-shelf solutions for learning such flow functions in continuous-time from sampled trajectory data., QC 20240222Part of ISBN 979-8-3503-0124-3

Motivated by the need to localise faults along electrical power lines, this paper adopts a frequency-domain approach to parameter estimation for an infinite-dimensional linear dynamical system with one spatial variable. Since the time of the fault is unknown, and voltages and currents are measured at only one end of the line, distance information must be extracted from the post-fault transients. To properly account for high-frequency transient behaviour, the line dynamics is modelled directly by the Telegrapher's equation, rather than the more commonly used lumped-parameter approximations. First, the governing equations are non-dimensionalised to avoid ill-conditioning. A closed-form expression for the transfer function is then derived. Finally, nonlinear least-squares optimisation is employed to search for the fault location. Requirements on fault bandwidth, sensor bandwidth and simulation time-step are also presented. The result is a novel end-to-end algorithm for data generation and fault localisation, the effectiveness of which is demonstrated via simulation., Part of ISBN 9798350301243QC 20240222

We describe a recurrent neural network (RNN) based architecture to learn the flow function of a causal, time-invariant and continuous-time control system from trajectory data. By restricting the class of control inputs to piecewise constant functions, we show that learning the flow function is equivalent to learning the input-to-state map of a discrete-time dynamical system. This motivates the use of an RNN together with encoder and decoder networks which map the state of the system to the hidden state of the RNN and back. We show that the proposed architecture is able to approximate the flow function by exploiting the system's causality and time-invariance. The output of the learned flow function model can be queried at any time instant. We experimentally validate the proposed method using models of the Van der Pol and FitzHugh-Nagumo oscillators. In both cases, the results demonstrate that the architecture is able to closely reproduce the trajectories of these two systems. For the Van der Pol oscillator, we further show that the trained model generalises to the system's response with a prolonged prediction time horizon as well as control inputs outside the training distribution. For the FitzHugh-Nagumo oscillator, we show that the model accurately captures the input-dependent phenomena of excitability., QC 20240209Part of ISBN 9781713872344

Designing Luenberger observers for nonlinear systems involves the challenging task of transforming the state to an alternate coordinate system, possibly of higher dimensions, where the system is asymptotically stable and linear up to output injection. The observer then estimates the system's state in the original coordinates by inverting the transformation map. However, finding a suitable injective transformation whose inverse can be derived remains a primary challenge for general nonlinear systems. We propose a novel approach that uses supervised physics-informed neural networks to approximate both the transformation and its inverse. Our method exhibits superior generalization capabilities to contemporary methods and demonstrates robustness to both neural network's approximation errors and system uncertainties., Part of ISBN 979-8-3503-2806-6QC 20230922