MS20.2: Physics-enhanced machine learning and data-driven nonlinear dynamics

This talk aims to introduce a statistical approach to identify the stochastic dynamic system behind a measured time series data. The method is formulated as a maximum likelihood problem, where we estimate the parameters of the dynamical system generating the data by solving the corresponding optimisation problem. This flexible formulation allows the identification of stochastic dynamical systems even through noisy measurements. We demonstrate the method on synthetic datasets generated by stochastic nonlinear oscillators and investigate statistical properties of the estimators through numerical experiments.

The Gaussian process latent force model (GPLFM) has been shown to be an effective and practical method for a number of tasks within dynamics, for example, joint estimation of inputs and states or, more recently, in the recovery of nonlinear restoring forces for system identification. One possible limitation of the GPLFM is that the estimated force is modelled a priori as a Gaussian process, hence is a stationary process. This work extends the model to account for non-stationary force estimation, whether an external or internal forcing, by means of a deep GPLFM formulation.

The quasipotential serves as a natural extension of the potential function in nongradient systems, and it enables predicting the maximum likelihood transition paths, transition rates and expected exit times of metastable dynamical states. In this work, the combination of a neural network and a sparse regression identification algorithm is employed to perform the symbolic discovery of quasipotential functions. The data-driven approach is benchmarked in an archetypal system with know exact quasipotential and applied to the dynamics of a nanomechanical resonator.

We present a data-driven, non-intrusive framework with embedded uncertainty quantification to build interpretable reduced-order models (ROMs) using variational autoencoders and variational identification of nonlinear dynamics.

This paper presents an innovative data-driven model order reduction approach for complex dynamical system through a piecewise artificial neural network architecture, which incorporates a state space-based attention mechanism, designed to enhance both predictive performance and interpretability. The proposed framework combines multiple sub-networks, each characterized by distinct architecture (hyper-parameters), to capture diverse patterns and features within the data. The attention mechanism at the culmination of the architecture further enriches the model’s interpretability by dynamically indicating which sub-network is most representative for a given input, offering an insightful decomposition of the system’s phase space. Our framework can be seen as a generalization of the concepts presented by Rewienski et al. [1], evolving the idea to encompass a broader range of models and complex scenarios. The novelty of our approach lies in its ability to learn directly from data both the model order reduction and the phase space partition, rather than relying only on physics knowledge. To demonstrate the efficacy and interpretive capabilities of our framework, we apply it to a network system of 20 Kuramoto oscillators. The results showcase not only good performance in terms of efficiency but also an enhanced ability to glean meaningful insights from the model’s predictions. This balance of high performance and interpretability positions our piecewise approach as a promising tool for applications where understanding the “why” behind predictions is as crucial as the predictions themselves.

To minimize the mismatch between a physical system and its digital twin (constituted by a model), model updating can be applied. In this work, an efficient, online model updating method is employed to update interpretable parameter values of nonlinear dynamics models. This method uses inverse mapping models that map a selected set of measured features to a set of parameter values. To do so, the inverse mapping model is constituted by a neural network that is trained based on simulated (training) data using supervisory learning. Here, the set of selected features is obtained by employing (joint) mutual information-based feature selection. To illustrate the method, it is applied to a high-tech industrial use case for which measurements are collected from a wire bonder. Here, it is shown that the updated models have higher predictive capacity than a reference model of which parameters are manually determined.