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Parameter estimation and associated uncertainty quantification is an important problem in dynamical systems characterized by ordinary differential equation (ODE) models that are often nonlinear. Typically, such models have analytically…

Mathematical solvers use parametrized Optimization Problems (OPs) as inputs to yield optimal decisions. In many real-world settings, some of these parameters are unknown or uncertain. Recent research focuses on predicting the value of these…

Neural networks have the ability to serve as universal function approximators, but they are not interpretable and don't generalize well outside of their training region. Both of these issues are problematic when trying to apply standard…

We introduce an evidence-driven Bayesian formulation of physics-informed neural networks that enables automatic optimization of loss weights between PDE residuals, boundary conditions, and observational data. Unlike existing Bayesian PINN…

The linearised Laplace method for estimating model uncertainty has received renewed attention in the Bayesian deep learning community. The method provides reliable error bars and admits a closed-form expression for the model evidence,…

The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…

Large-scale linear models are ubiquitous throughout machine learning, with contemporary application as surrogate models for neural network uncertainty quantification; that is, the linearised Laplace method. Alas, the computational cost…

Markov chain Monte Carlo (MCMC) methods remain the mainstay of Bayesian estimation of structural equation models (SEM), though they often incur a high computational cost. We present a bespoke approximate Bayesian approach to SEM, drawing on…

Bayesian Neural Networks provide a principled framework for uncertainty quantification by modeling the posterior distribution of network parameters. However, exact posterior inference is computationally intractable, and widely used…

The problem of detecting the Out-of-Distribution (OoD) inputs is of paramount importance for Deep Neural Networks. It has been previously shown that even Deep Generative Models that allow estimating the density of the inputs may not be…

Bayesian probabilistic numerical methods for numerical integration offer significant advantages over their non-Bayesian counterparts: they can encode prior information about the integrand, and can quantify uncertainty over estimates of an…

Solving systems of ordinary differential equations (ODEs) is essential when it comes to understanding the behavior of dynamical systems. Yet, automated solving remains challenging, in particular for nonlinear systems. Computer algebra…

In this thesis, we disentangle the generalized Gauss-Newton and approximate inference for Bayesian deep learning. The generalized Gauss-Newton method is an optimization method that is used in several popular Bayesian deep learning…

Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we…

Neural additive models (NAMs) enhance the transparency of deep neural networks by handling input features in separate additive sub-networks. However, they lack inherent mechanisms that provide calibrated uncertainties and enable selection…

Variational approaches to approximate Bayesian inference provide very efficient means of performing parameter estimation and model selection. Among these, so-called variational-Laplace or VL schemes rely on Gaussian approximations to…

A widely used method to create a continuous representation of a discrete data-set is regression analysis. When the regression model is not based on a mathematical description of the physics underlying the data, heuristic techniques play a…

It is common practice to use Laplace approximations to compute marginal likelihoods in Bayesian versions of generalised linear models (GLM). Marginal likelihoods combined with model priors are then used in different search algorithms to…

In many fields of application, dynamic processes that evolve through time are well described by systems of ordinary differential equations (ODEs). The analytical solution of the ODEs is often not available and different methods have been…

Bayesian optimization is a highly efficient approach to optimizing objective functions which are expensive to query. These objectives are typically represented by Gaussian process (GP) surrogate models which are easy to optimize and support…