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We link nonlinear manifold learning techniques for data analysis/compression with model reduction techniques for evolution equations with time scale separation. In particular, we demonstrate a `"nonlinear extension" of the POD-Galerkin…

This paper focuses on the identification of dynamical systems with tailor-made model structures, where neural networks are used to approximate uncertain components and domain knowledge is retained, if available. These model structures are…

Randomized neural networks (RNN) are a variation of neural networks in which the hidden-layer parameters are fixed to randomly assigned values and the output-layer parameters are obtained by solving a linear system by least squares. This…

This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and…

A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent…

The customizable nature of deep learning models have allowed them to be successful predictors in various disciplines. These models are often trained with respect to thousands or millions of instances for complicated problems, but the…

Modern generative machine learning models demonstrate surprising ability to create realistic outputs far beyond their training data, such as photorealistic artwork, accurate protein structures, or conversational text. These successes…

Many problems in the geophysical sciences demand the ability to calibrate the parameters and predict the time evolution of complex dynamical models using sequentially-collected data. Here we introduce a general methodology for the joint…

Solving and optimising Partial Differential Equations (PDEs) in geometrically parameterised domains often requires iterative methods, leading to high computational and time complexities. One potential solution is to learn a direct mapping…

Data-driven modeling and machine learning are widely used to model the behavior of dynamic systems. One application is the residual evaluation of technical systems where model predictions are compared with measurement data to create…

We present a method of parameter estimation for large class of nonlinear systems, namely those in which the state consists of output derivatives and the flow is linear in the parameter. The method, which solves for the unknown parameter by…

A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domain's boundary is described by a vector valued random field depending on a countable number of random variables…

State estimation that combines observational data with mathematical models is central to many applications and is commonly addressed through filtering methods, such as ensemble Kalman filters. In this article, we examine the signal-tracking…

Parameterized quantum circuits can be used as quantum neural networks and have the potential to outperform their classical counterparts when trained for addressing learning problems. To date, much of the results on their performance on…

The Discontinuous Galerkin time-domain method is well suited for adaptive algorithms to solve the time-domain Maxwell's equations and depends on robust and economically computable drivers. Adaptive algorithms utilize local indicators to…

This paper presents the construction of a particle filter, which incorporates elements inspired by genetic algorithms, in order to achieve accelerated adaptation of the estimated posterior distribution to changes in model parameters.…

Many types of physics-informed neural network models have been proposed in recent years as approaches for learning solutions to differential equations. When a particular task requires solving a differential equation at multiple…

Many machine learning algorithms have been developed under the assumption that data sets are already available in batch form. Yet in many application domains data is only available sequentially overtime via compute nodes in different…

We consider the discretization of a semilinear damped wave equation arising, for instance, in the modeling of gas transport in pipeline networks. For time invariant boundary data, the solutions of the problem are shown to converge…

In this work, we consider space-time goal-oriented a posteriori error estimation for parabolic problems. Temporal and spatial discretizations are based on Galerkin finite elements of continuous and discontinuous type. The main objectives…