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Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method…
Transfer learning (TL) enables the transfer of knowledge gained in learning to perform one task (source) to a related but different task (target), hence addressing the expense of data acquisition and labeling, potential computational power…
Multiphysics problems such as multicomponent diffusion, phase transformations in multiphase systems and alloy solidification involve numerical solution of a coupled system of nonlinear partial differential equations (PDEs). Numerical…
Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modeling physical processes with rough spatio-temporal dynamics, such as turbulence flows, superconductors, and quantum dynamics. Although…
Physics-informed machine learning (PIML) has emerged as a promising alternative to conventional numerical methods for solving partial differential equations (PDEs). PIML models are increasingly built via deep neural networks (NNs) whose…
Machine learning methods have been successful in many areas, like image classification and natural language processing. However, it still needs to be determined how to apply ML to areas with mathematical constraints, like solving PDEs.…
Accurately reconstructing a global spatial field from sparse data has been a longstanding problem in several domains, such as Earth Sciences and Fluid Dynamics. Historically, scientists have approached this problem by employing complex…
Meshless methods are commonly used to determine numerical solutions to partial differential equations (PDEs) for problems involving free surfaces and/or complex geometries, approximating spatial derivatives at collocation points via local…
Time-dependent partial differential equations are a significant class of equations that describe the evolution of various physical phenomena over time. One of the open problems in scientific computing is predicting the behaviour of the…
Unlike many deterministic PDEs, stochastic equations are not amenable to the classical variational theory of Euler-Lagrange. In this paper, we show how self-dual variational calculus leads to solutions of various stochastic partial…
A spectral element method (SEM) is developed to solve polarized radiative transfer in multidimensional participating medium. The angular discretization is based on the discrete-ordinates approach, and the spatial discretization is conducted…
Deep learning neural networks have emerged as one of the most powerful classification tools for vision related applications. However, the computational and energy requirements associated with such deep nets can be quite high, and hence…
The high computational costs of video super-resolution (VSR) models hinder their deployment on resource-limited devices, (e.g., smartphones and drones). Existing VSR models contain considerable redundant filters, which drag down the…
Fast and accurate solutions of time-dependent partial differential equations (PDEs) are of pivotal interest to many research fields, including physics, engineering, and biology. Generally, implicit/semi-implicit schemes are preferred over…
Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To…
A Spectral Difference (SD) algorithm on tensor-product elements which solves the reacting compressible Navier-Stokes equations (NSE) is presented. The classical SD algorithm is shown to be unstable when a multispecies gas where…
Classical representation systems such as Fourier series, wavelets, and fixed dictionaries provide analytically tractable basis expansions, but they are not intrinsically adapted to the empirical structure of modern high-dimensional data.…
Vertical federated learning trains models from feature-partitioned datasets across multiple clients, who collaborate without sharing their local data. Standard approaches assume that all feature partitions are available during both training…
In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the…
Learning time-dependent partial differential equations (PDEs) that govern evolutionary observations is one of the core challenges for data-driven inference in many fields. In this work, we propose to capture the essential dynamics of…