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The past few years have witnessed the great success of Diffusion models~(DMs) in generating high-fidelity samples in generative modeling tasks. A major limitation of the DM is its notoriously slow sampling procedure which normally requires…
This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
We develop a simple and reliable analytical model that allows describing the electromagnetic response of all-dielectric metasurfaces consisting of a single-layer array of high-permittivity spherical particles. By combining Mie theory with a…
Inverse problems are ubiquitous in nature, arising in almost all areas of science and engineering ranging from geophysics and climate science to astrophysics and biomechanics. One of the central challenges in solving inverse problems is…
In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the…
The first step towards applying isogeometric analysis techniques to solve PDE problems on a given domain consists in generating an analysis-suitable mapping operator between parametric and physical domains with one or several patches from…
Electrical Impedance Tomography (EIT) is a non-invasive imaging technique that reconstructs conductivity distributions within a body from boundary measurements. However, EIT reconstruction is hindered by its ill-posed nonlinear inverse…
In this study, we present a novel computational framework that integrates the finite volume method with graph neural networks to address the challenges in Physics-Informed Neural Networks(PINNs). Our approach leverages the flexibility of…
Electrostatic interactions between dielectric objects are complex and of a many-body nature, owing to induced surface bound charge. We present a collection of techniques to simulate dynamical dielectric objects. We calculate the surface…
We present a spectrally accurate embedded boundary method for solving linear, inhomogeneous, elliptic partial differential equations (PDE) in general smooth geometries, focusing in this manuscript on the Poisson, modified Helmholtz, and…
In the realm of LiDAR-based perception, significant strides have been made, yet domain generalization remains a substantial challenge. The performance often deteriorates when models are applied to unfamiliar datasets with different LiDAR…
3D Gaussian splatting (3DGS) has shown its detailed expressive ability and highly efficient rendering speed in the novel view synthesis (NVS) task. The application to inverse rendering still faces several challenges, as the discrete nature…
We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes,…
In this paper, a generalized macromodeling approach is presented to simulate complex electromagnetic (EM) surfaces consisting of unit cells with connected conductors. Macromodels of each unit cell are produced by applying the equivalence…
This paper presents novel methodologies for the numerical simulation of scattering of elastic waves by both closed and open surfaces in three-dimensional space. The proposed approach utilizes new integral formulations as well as an…
In this paper we analyze a space-time unfitted finite element method for the discretization of scalar surface partial differential equations on evolving surfaces. For higher order approximations of the evolving surface we use the technique…
A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in three dimensions is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a…
We propose a general framework for conditional sampling in PDE-based inverse problems, targeting the recovery of whole solutions from extremely sparse or noisy measurements. This is accomplished by a function-space diffusion model and…
Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen…