Temporal phenomena in a range of disciplines are more naturally modelled in continuous-time than coerced into a discrete-time formulation. Differential systems form the mainstay of such modelling, in fields from physics to economics, geoscience to neuroscience. While powerful, these are fundamentally limited by their determinism. For the purposes of probabilistic inference, their extension to stochastic differential equations permits a continuous injection of noise and uncertainty into the system, the model, and its observation. This thesis considers Bayesian filtering for state and parameter estimation in general non-linear, non-Gaussian systems using these stochastic differential models. It identifies a number of challenges in this setting over and above those of discrete time, most notably the absence of a closed form transition density. These are addressed via a synergy of diverse work in numerical integration, particle filtering and high performance distributed computing, engineering novel solutions for this class of model. In an area where the default solution is linear discretisation, the first major contribution is the introduction of higher-order numerical schemes, particularly stochastic Runge-Kutta, for more efficient simulation of the system dynamics. Improved runtime performance is demonstrated on a number of problems, and compatibility of these integrators with conventional particle filtering and smoothing schemes discussed. Finding compatibility for the smoothing problem most lacking, the major theoretical contribution of the work is the introduction of two novel particle methods, the kernel forward-backward and kernel two-filter smoothers. By harnessing kernel density approximations in an importance sampling framework, these attain cancellation of the intractable transition density, ensuring applicability in continuous time. The use of kernel estimators is particularly amenable to parallelisation, and provides broader support for smooth densities than a sample-based representation alone, helping alleviate the well known issue of degeneracy in particle smoothers. Implementation of the methods for large-scale problems on high performance computing architectures is provided. Achieving improved temporal and spatial complexity, highly favourable runtime comparisons against conventional techniques are presented. Finally, attention turns to real world problems in the domain of Functional Magnetic Resonance Imaging (fMRI), first constructing a biologically motivated stochastic differential model of the neural and hemodynamic activity underlying the observed signal in fMRI. This model and the methodological advances of the work culminate in application to the deconvolution and effective connectivity problems in this domain.

Un système est défini par les entités et leurs interrelations dans un environnement qui est déterminé par une limite arbitraire. Les systèmes complexes présentent un comportement émergent sans un contrôleur central. Les systèmes concrets désignent ceux qui sont observables dans la réalité. Un modèle nous permet de comprendre, de contrôler et de prédire le comportement du système. Un modèle différentiel à partir d'un système pourrait être compris comme une sorte de loi physique sous-jacent représenté par l'un ou d'un ensemble d'équations différentielles. Ce travail vise à étudier et mettre en œuvre des méthodes pour effectuer la modélisation des systèmes automatisée par l'ordinateur. Cette thèse pourrait être divisée en trois étapes principales, ainsi: (1) le développement d'un solveur numérique automatisé par l'ordinateur pour les équations différentielles linéaires, partielles ou ordinaires, sur la base de la formulation de matrice pour une personnalisation propre de la méthode Ritz-Galerkin; (2) la proposition d'un schème de score d'adaptation qui bénéficie du solveur numérique développé pour guider l'évolution des modèles différentiels pour les systèmes complexes concrets; (3) une implémentation préliminaire d'une application de programmation génétique pour effectuer la modélisation des systèmes automatisée par l'ordinateur. Dans la première étape, il est montré comment le solveur proposé utilise les polynômes de Jacobi orthogonaux comme base complète pour la méthode de Galerkin et comment le solveur traite des conditions auxiliaires de plusieurs types. Solutions à approximations polynomiales sont ensuite réalisés pour plusieurs types des équations différentielles partielles linéaires, y compris les problèmes hyperboliques, paraboliques et elliptiques. Dans la deuxième étape, le schème de score d'adaptation proposé est conçu pour exploiter certaines caractéristiques du solveur proposé et d'effectuer l'approximation polynômiale par morceaux afin d'évaluer les individus différentiels à partir d'une population fournie par l'algorithme évolutionnaire. Enfin, une mise en œuvre préliminaire d'une application GP est présentée et certaines questions sont discutées afin de permettre une meilleure compréhension de la modélisation des systèmes automatisée par l'ordinateur. Indications pour certains sujets prometteurs pour la continuation de futures recherches sont également abordées dans ce travail, y compris la façon d'étendre ce travail à certaines classes d'équations différentielles partielles non-linéaires. / A system is defined by its entities and their interrelations in an environment which is determined by an arbitrary boundary. Complex systems exhibit emergent behaviour without a central controller. Concrete systems designate the ones observable in reality. A model allows us to understand, to control and to predict behaviour of the system. A differential model from a system could be understood as some sort of underlying physical law depicted by either one or a set of differential equations. This work aims to investigate and implement methods to perform computer-automated system modelling. This thesis could be divided into three main stages: (1) developments of a computer-automated numerical solver for linear differential equations, partial or ordinary, based on the matrix formulation for an own customization of the Ritz-Galerkin method; (2) proposition of a fitness evaluation scheme which benefits from the developed numerical solver to guide evolution of differential models for concrete complex systems; (3) preliminary implementations of a genetic programming application to perform computer-automated system modelling. In the first stage, it is shown how the proposed solver uses Jacobi orthogonal polynomials as a complete basis for the Galerkin method and how the solver deals with auxiliary conditions of several types. Polynomial approximate solutions are achieved for several types of linear partial differential equations, including hyperbolic, parabolic and elliptic problems. In the second stage, the proposed fitness evaluation scheme is developed to exploit some characteristics from the proposed solver and to perform piecewise polynomial approximations in order to evaluate differential individuals from a given evolutionary algorithm population. Finally, a preliminary implementation of a genetic programming application is presented and some issues are discussed to enable a better understanding of computer-automated system modelling. Indications for some promising subjects for future continuation researches are also addressed here, as how to expand this work to some classes of non-linear partial differential equations.