Related papers: Towards information optimal simulation of partial …

We consider a simple neural field model in which the state variable is dendritic voltage, and in which somas form a continuous one-dimensional layer. This neural field model with dendritic processing is formulated as an integro-differential…

We present a computational technique for modeling the evolution of dynamical systems in a reduced basis, with a focus on the challenging problem of modeling partially-observed partial differential equations (PDEs) on high-dimensional…

We introduce a new approach for designing numerical schemes for stochastic differential equations (SDEs). The approach, which we have called direction and norm decomposition method, proposes to approximate the required solution $X_t$ by…

Differential equations and numerical methods are extensively used to model various real-world phenomena in science and engineering. With modern developments, we aim to find the underlying differential equation from a single observation of…

The measured spatiotemporal response of various physical processes is utilized to infer the governing partial differential equations (PDEs). We propose SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE), a technique…

This paper investigates Gaussian Markov random field approximations to nonstationary Gaussian fields using graph representations of stochastic partial differential equations. We establish approximation error guarantees building on the…

In image reconstruction, an accurate quantification of uncertainty is of great importance for informed decision making. Here, the Bayesian approach to inverse problems can be used: the image is represented through a random function that…

Physics-informed neural networks (PINNs) have successfully addressed various computational physics problems based on partial differential equations (PDEs). However, while tackling issues related to irregularities like singularities and…

Variational inference techniques based on inducing variables provide an elegant framework for scalable posterior estimation in Gaussian process (GP) models. Besides enabling scalability, one of their main advantages over sparse…

Smooth random Gaussian functions play an important role in mathematical physics, a main example being the random plane wave model conjectured by Berry to give a universal description of high-energy eigenfunctions of the Laplacian on generic…

In this paper, we consider a nonlinear PDE system governed by a parabolic heat equation coupled in a nonlinear way with a hyperbolic momentum equation describing the behavior of a displacement field coupled with a nonlinear elliptic…

Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. Many SPDEs that appear in such applications include non-globally monotone nonlinearities.…

This work presents a data-driven approach to the identification of spatial and temporal truncation errors for linear and nonlinear discretization schemes of Partial Differential Equations (PDEs). Motivated by the central role of truncation…

In many areas of science and engineering, discovering the governing differential equations from the noisy experimental data is an essential challenge. It is also a critical step in understanding the physical phenomena and prediction of the…

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…

Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…

All discretized numerical models contain modelling errors - this reality is amplified when reduced-order models are used. The ability to accurately approximate modelling errors informs statistics on model confidence and improves…

We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning. We suppose that the phenomenon of interest can be modeled by a parametrized partial…

In this paper, we consider a zero-order stochastic oracle model of estimating definite integrals. In this model, integral estimation methods may query an oracle function for a fixed number of noisy values of the integrand function and use…

Nonlinear (systems of) ordinary differential equations (ODEs) are common tools in the analysis of complex one-dimensional dynamic systems. In this paper we propose a smoothing approach regularized by a quasilinearized ODE-based penalty in…