Related papers: Inferring solutions of differential equations usin…
This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors…
Data-driven discovery of "hidden physics" -- i.e., machine learning of differential equation models underlying observed data -- has recently been approached by embedding the discovery problem into a Gaussian Process regression of spatial…
Reliability analysis aims at estimating the failure probability of an engineering system. It often requires multiple runs of a limit-state function, which usually relies on computationally intensive simulations. Traditionally, these…
The data-driven discovery of interpretable models approximating the underlying dynamics of a physical system has gained attraction in the past decade. Current approaches employ pre-specified functional forms or basis functions and often…
When modelling time series, it is common to decompose observed variation into a "signal" process, the process of interest, and "noise", representing nuisance factors that obfuscate the signal. To separate signal from noise, assumptions must…
The accurate numerical solution of partial differential equations is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here,…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
This paper explores the critical role of differentiation approaches for data-driven differential equation discovery. Accurate derivatives of the input data are essential for reliable algorithmic operation, particularly in real-world…
We develop a new Bayesian framework based on deep neural networks to be able to extrapolate in space-time using historical data and to quantify uncertainties arising from both noisy and gappy data in physical problems. Specifically, the…
Deep Gaussian Processes learn probabilistic data representations for supervised learning by cascading multiple Gaussian Processes. While this model family promises flexible predictive distributions, exact inference is not tractable.…
Robust physics (e.g., governing equations and laws) discovery is of great interest for many engineering fields and explainable machine learning. A critical challenge compared with general training is that the term and format of governing…
This paper addresses Bayesian inference related to partial differential equations (PDEs), particularly nonparametric regression constrained by PDEs. To effectively encode prior information, we propose a novel framework that learns a…
We present a deep learning framework for quantifying and propagating uncertainty in systems governed by non-linear differential equations using physics-informed neural networks. Specifically, we employ latent variable models to construct…
There is significant need for principled uncertainty reasoning in machine learning systems as they are increasingly deployed in safety-critical domains. A new approach with uncertainty-aware neural networks (NNs), based on learning…
A method to reconstruct fields, source strengths and physical parameters based on Gaussian process regression is presented for the case where data are known to fulfill a given linear differential equation with localized sources. The…
Equation learning aims to infer differential equation models from data. While a number of studies have shown that differential equation models can be successfully identified when the data are sufficiently detailed and corrupted with…
We algorithmically construct multi-output Gaussian process priors which satisfy linear differential equations. Our approach attempts to parametrize all solutions of the equations using Gr\"obner bases. If successful, a push forward Gaussian…
The data association problem is concerned with separating data coming from different generating processes, for example when data come from different data sources, contain significant noise, or exhibit multimodality. We present a fully…
In this work, we introduce a machine/deep learning methodology to solve parametric integrals. Besides classical machine learning approaches, we consider a differential learning framework that incorporates derivative information during…
We focus on variational inference in dynamical systems where the discrete time transition function (or evolution rule) is modelled by a Gaussian process. The dominant approach so far has been to use a factorised posterior distribution,…