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Inverse problem or parameter estimation of ordinary differential equations (ODEs), the iterative process of minimizing the mismatch between model-predicted and experimental states by tuning the parameter values within an optimization…
We consider the problem of estimating parameters of stochastic differential equations (SDEs) with discrete-time observations that are either completely or partially observed. The transition density between two observations is generally…
In this report, we present and compare the results of an improved fractional and integer order partial differential equation (PDE)-based binarization scheme. The improved model incorporates a diffusion term in addition to the edge and…
Common techniques for the spatial discretisation of PDEs on a macroscale grid include finite difference, finite elements and finite volume methods. Such methods typically impose assumed microscale structures on the subgrid fields, so…
Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order…
We consider a shape optimization based method for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in an $L^p$-norm between…
We consider the numerical approximation of a general second order semi--linear parabolic stochastic partial differential equation (SPDE) driven by additive space-time noise. We introduce a new modified scheme using a linear functional of…
Parametric surrogate models for partial differential equations (PDEs) are a necessary component for many applications in the computational sciences, and convolutional neural networks (CNNs) have proved as an excellent tool to generate these…
Neural PDE surrogates are often deployed in data-limited or partially observed regimes where downstream decisions depend on calibrated uncertainty in addition to low prediction error. Existing approaches obtain uncertainty through ensemble…
In this work, we present an adjoint-based method for discovering the underlying governing partial differential equations (PDEs) given data. The idea is to consider a parameterized PDE in a general form and formulate a PDE-constrained…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…
We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [Appl. Numer. Math., 2017]), which we hereby refer to as the surrogate matrix methodology. This methodology is…
Partial Differential Equation (PDE)-based approaches have gained significant attention in image despeckling due to their strong capability to preserve structural details while suppressing noise. However, conventional second-order PDE models…
We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in $d\geq 1$ dimensions. Our main approach consists of…
A probability density function (pdf) encodes the entire stochastic knowledge about data distribution, where data may represent stochastic observations in robotics, transition state pairs in reinforcement learning or any other empirically…
We deal with the problem of parameter estimation in stochastic differential equations (SDEs) in a partially observed framework. We aim to design a method working for both elliptic and hypoelliptic SDEs, the latters being characterized by…
Solving partial differential equations (PDEs) is the canonical approach for understanding the behavior of physical systems. However, large scale solutions of PDEs using state of the art discretization techniques remains an expensive…
The need to rapidly solve PDEs in engineering design workflows has spurred the rise of neural surrogate models. In particular, neural operator models provide a discretization-invariant surrogate by retaining the infinite-dimensional,…
Optimization problems constrained by partial differential equations (PDEs) naturally arise in scientific computing, as those constraints often model physical systems or the simulation thereof. In an implicitly constrained approach, the…
Gradient-based optimization of engineering designs is limited by non-differentiable components in the typical computer-aided engineering (CAE) workflow, which calculates performance metrics from design parameters. While gradient-based…