Related papers: Bayesian Inference for PDE-based Inverse Problems …

Inverse analysis, such as model calibration, often suffers from a lack of informative data in complex real-world scenarios. The standard remedy, designing new experimental setups, is often costly and time-consuming, while readily available…

Numerical simulations of physical systems exhibit discrepancies arising from unmodeled physics and idealizations, as well as numerical approximation errors stemming from discretization and solver tolerances. This article reviews techniques…

In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which…

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…

Neural networks have emerged as promising tools for solving partial differential equations (PDEs), particularly through the application of neural operators. Training neural operators typically requires a large amount of training data to…

Solutions to inverse problems that are ill-conditioned or ill-posed may have significant intrinsic uncertainty. Unfortunately, analysing and quantifying this uncertainty is very challenging, particularly in high-dimensional problems. As a…

Out-of-distribution (OOD) generalization has long been a challenging problem that remains largely unsolved. Gaussian processes (GP), as popular probabilistic model classes, especially in the small data regime, presume strong OOD…

Recent advances in reconstruction methods for inverse problems leverage powerful data-driven models, e.g., deep neural networks. These techniques have demonstrated state-of-the-art performances for several imaging tasks, but they often do…

Inverse optimization (IO) is used to estimate unknown parameters of an optimization model from observed decisions. In the data-driven context, the estimated parameters are inherently uncertain, yet quantifying this uncertainty has received…

We propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers. The approach is of Bayesian type, and it exploits a heavy-tailed t distribution for data noise…

In this work, a method for obtaining pixel-wise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal…

The Bayesian approach has proved to be a coherent approach to handle ill posed Inverse problems. However, the Bayesian calculations need either an optimization or an integral calculation. The maximum a posteriori (MAP) estimation requires…

In this paper, we present a novel methodology for automatic adaptive weighting of Bayesian Physics-Informed Neural Networks (BPINNs), and we demonstrate that this makes it possible to robustly address multi-objective and multi-scale…

Scientific machine learning has been successfully applied to inverse problems and PDE discovery in computational physics. One caveat concerning current methods is the need for large amounts of ("clean") data, in order to characterize the…

Probabilistic programming systems enable users to encode model structure and naturally reason about uncertainties, which can be leveraged towards improved Bayesian optimization (BO) methods. Here we present a probabilistic program embedding…

The combination of ordinary differential equations and neural networks, i.e., neural ordinary differential equations (Neural ODE), has been widely studied from various angles. However, deciphering the numerical integration in Neural ODE is…

Inference in the presence of outliers is an important field of research as outliers are ubiquitous and may arise across a variety of problems and domains. Bayesian optimization is method that heavily relies on probabilistic inference. This…

Symbolic regression with polynomial neural networks and polynomial neural ordinary differential equations (ODEs) are two recent and powerful approaches for equation recovery of many science and engineering problems. However, these methods…

Identifying a low-dimensional informed parameter subspace offers a viable path to alleviating the dimensionality challenge in the sampled-based solution to large-scale Bayesian inverse problems. This paper introduces a novel gradient-based…

Geoscientists use observed data to estimate properties of the Earth's interior. This often requires non-linear inverse problems to be solved and uncertainties to be estimated. Bayesian inference solves inverse problems under a probabilistic…