502 research outputs found

    Bayesian Learning and Predictability in a Stochastic Nonlinear Dynamical Model

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    Bayesian inference methods are applied within a Bayesian hierarchical modelling framework to the problems of joint state and parameter estimation, and of state forecasting. We explore and demonstrate the ideas in the context of a simple nonlinear marine biogeochemical model. A novel approach is proposed to the formulation of the stochastic process model, in which ecophysiological properties of plankton communities are represented by autoregressive stochastic processes. This approach captures the effects of changes in plankton communities over time, and it allows the incorporation of literature metadata on individual species into prior distributions for process model parameters. The approach is applied to a case study at Ocean Station Papa, using Particle Markov chain Monte Carlo computational techniques. The results suggest that, by drawing on objective prior information, it is possible to extract useful information about model state and a subset of parameters, and even to make useful long-term forecasts, based on sparse and noisy observations

    PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equations

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    We present a new category of physics-informed neural networks called physics informed variational embedding generative adversarial network (PI-VEGAN), that effectively tackles the forward, inverse, and mixed problems of stochastic differential equations. In these scenarios, the governing equations are known, but only a limited number of sensor measurements of the system parameters are available. We integrate the governing physical laws into PI-VEGAN with automatic differentiation, while introducing a variational encoder for approximating the latent variables of the actual distribution of the measurements. These latent variables are integrated into the generator to facilitate accurate learning of the characteristics of the stochastic partial equations. Our model consists of three components, namely the encoder, generator, and discriminator, each of which is updated alternatively employing the stochastic gradient descent algorithm. We evaluate the effectiveness of PI-VEGAN in addressing forward, inverse, and mixed problems that require the concurrent calculation of system parameters and solutions. Numerical results demonstrate that the proposed method achieves satisfactory stability and accuracy in comparison with the previous physics-informed generative adversarial network (PI-WGAN).Comment: 23 page

    UQ and AI: data fusion, inverse identification, and multiscale uncertainty propagation in aerospace components

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    A key requirement for engineering designs is that they offer good performance across a range of uncertain conditions while exhibiting an admissibly low probability of failure. In order to design components that offer good performance across a range of uncertain conditions, it is necessary to take account of the effect of the uncertainties associated with a candidate design. Uncertainty Quantification (UQ) methods are statistical methods that may be used to quantify the effect of the uncertainties inherent in a system on its performance. This thesis expands the envelope of UQ methods for the design of aerospace components, supporting the integration of UQ methods in product development by addressing four industrial challenges. Firstly, a method for propagating uncertainty through computational models in a hierachy of scales is described that is based on probabilistic equivalence and Non-Intrusive Polynomial Chaos (NIPC). This problem is relevant to the design of aerospace components as the computational models used to evaluate candidate designs are typically multiscale. This method was then extended to develop a formulation for inverse identification, where the probability distributions for the material properties of a coupon are deduced from measurements of its response. We demonstrate how probabilistic equivalence and the Maximum Entropy Principle (MEP) may be used to leverage data from simulations with scarce experimental data- with the intention of making this stage of product design less expensive and time consuming. The third contribution of this thesis is to develop two novel meta-modelling strategies to promote the wider exploration of the design space during the conceptual design phase. Design Space Exploration (DSE) in this phase is crucial as decisions made at the early, conceptual stages of an aircraft design can restrict the range of alternative designs available at later stages in the design process, despite limited quantitative knowledge of the interaction between requirements being available at this stage. A histogram interpolation algorithm is presented that allows the designer to interactively explore the design space with a model-free formulation, while a meta-model based on Knowledge Based Neural Networks (KBaNNs) is proposed in which the outputs of a high-level, inexpensive computer code are informed by the outputs of a neural network, in this way addressing the criticism of neural networks that they are purely data-driven and operate as black boxes. The final challenge addressed by this thesis is how to iteratively improve a meta-model by expanding the dataset used to train it. Given the reliance of UQ methods on meta-models this is an important challenge. This thesis proposes an adaptive learning algorithm for Support Vector Machine (SVM) metamodels, which are used to approximate an unknown function. In particular, we apply the adaptive learning algorithm to test cases in reliability analysis.Open Acces

    A Bayesian approach to data-driven discovery of nonlinear dynamic equations

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    Dynamic equations parameterized by differential equations are used to represent a variety of real-world processes. The equations used to describe these processes are generally derived based on physical principles and a scientific understanding of the process. Statisticians have embedded these physically-inspired differential equations into a probabilistic framework, providing uncertainty quantification to parameter estimates and model specification. These statistical models typically rely on a predefined differential equation or class of models to represent the dynamics of the system. Recently, methods have been developed to discover the governing equation of complex systems. However, these approaches rarely account for uncertainty in the discovered equations, and when uncertainty is accounted for, it is not for the complete system. This dissertation begins with a statistical model for the seasonal temperature cycle over North America, where the dynamics of the system are parameterized by a specified functional form. The model highlights how the seasonal cycle is changing in space and time, motivating the need to better understand the driving mechanisms of such systems. Then, a statistical approach to data-driven discovery is proposed, where uncertainty is incorporated throughout the complete modeling process. The novelty of the approach is the dynamics are treated as a random process, which has not be considered previously in the data-driven discovery literature. The proposed approach sits at the junction between the statistical approach of incorporating dynamic equations in a probabilistic framework and the data-driven discovery methods proposed in computer science, physics, and applied mathematics. The proposed method is put into context within the broader literature, highlighting its contribution to the field of data-driven discovery.Includes bibliographical references
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