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Many computer vision and image processing problems can be posed as solving partial differential equations (PDEs). However, designing PDE system usually requires high mathematical skills and good insight into the problems. In this paper, we…

Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…

We develop an \textit{a posteriori} error analysis for a numerical estimate of the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This quantity…

Firedrake is a new tool for automating the numerical solution of partial differential equations. Firedrake adopts the domain-specific language for the finite element method of the FEniCS project, but with a pure Python runtime-only…

The physical world is governed by the laws of physics, often represented in form of nonlinear partial differential equations (PDEs). Unfortunately, solution of PDEs is non-trivial and often involves significant computational time. With…

This paper resolves a common complexity issue in the Bethe approximation of statistical physics and the Belief Propagation (BP) algorithm of artificial intelligence. The Bethe approximation and the BP algorithm are heuristic methods for…

This paper proposes the Nerual Energy Descent (NED) via neural network evolution equations for a wide class of deep learning problems. We show that deep learning can be reformulated as the evolution of network parameters in an evolution…

We formulate and analyze a goal-oriented adaptive finite element method for a symmetric linear elliptic partial differential equation (PDE) that can simultaneously deal with multiple linear goal functionals. In each step of the algorithm,…

The main objective of exams consists in performing an assessment of students' expertise on a specific subject. Such expertise, also referred to as skill or knowledge level, can then be leveraged in different ways (e.g., to assign a grade to…

We propose a novel Skew Gradient Embedding (SGE) framework for systematically reformulating thermodynamically consistent partial differential equation (PDE) models-capturing both reversible and irreversible processes-as generalized gradient…

Initial boundary value problems arise commonly in applications with engineering and natural systems governed by nonlinear partial differential equations (PDEs). Operator learning is an emerging field for solving these equations by using a…

Partial differential equation (PDE) is an important math tool in science and engineering. This paper experimentally demonstrates an optical neural PDE solver by leveraging the back-propagation-free on-photonic-chip training of…

Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE…

This paper is concerned with the finite element discretization of the data driven approach according to arXiv:1510.04232 for the solution of PDEs with a material law arising from measurement data. To simplify the setting, we focus on a…

We introduce ART, a new corpus-level autoencoding approach for training dense retrieval models that does not require any labeled training data. Dense retrieval is a central challenge for open-domain tasks, such as Open QA, where…

Recent advances in foundational models have yielded reasoning systems capable of achieving a gold-medal standard at the International Mathematical Olympiad. The transition from competition-level problem-solving to professional research,…

Repository-level fault localization (FL) and automated program repair (APR) require an agent to identify the relevant code units across files, follow call and data dependencies, and generate a valid patch. Existing graph-based systems…

Defect depth quantification in additively manufactured (AM) components remains a significant challenge for non-destructive testing (NDT). This study proposes a Pixel-wise Quantitative Thermography Neural Network (PQT-Net) to address this…

Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…

This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…