Related papers: SODAs: Sparse Optimization for the Discovery of Di…
This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs…
The neural ordinary differential equation (ODE) framework has emerged as a powerful tool for developing accelerated surrogate models of complex physical systems governed by partial differential equations (PDEs). A popular approach for PDE…
Sparse Autoencoders (SAEs) have become an important tool in mechanistic interpretability, helping to analyze internal representations in both Large Language Models (LLMs) and Vision Transformers (ViTs). By decomposing polysemantic…
In this work (Part I), we study three time-discretization procedures of the Dynamical Low-Rank Approximation (DLRA) of high-dimensional stochastic differential equations (SDEs). Specifically, we consider the Dynamically Orthogonal (DO)…
In this paper, we study high-dimensional sparse Quadratic Discriminant Analysis (QDA) and aim to establish the optimal convergence rates for the classification error. Minimax lower bounds are established to demonstrate the necessity of…
In data-driven modeling of spatiotemporal phenomena careful consideration often needs to be made in capturing the dynamics of the high wavenumbers. This problem becomes especially challenging when the system of interest exhibits shocks or…
In this paper, some adaptive single-step methods like Trapezoid (TR), Implicit-mid point (IMP), Euler-backward (EB), and Radau IIA (Rad) methods are implemented in Maple to solve index-1 nonlinear Differential Algebraic Equations (DAEs).…
In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving…
Sparse autoencoders (SAEs) are a technique for sparse decomposition of neural network activations into human-interpretable features. However, current SAEs suffer from feature absorption, where specialized features capture instances of…
This paper studies the contraction property of time-varying differential-algebraic equation (DAE) systems by embedding them to higher-dimension ordinary differential equation (ODE) systems. The first result pertains to the equivalence of…
Statistical regression models whose mean functions are represented by ordinary differential equations (ODEs) can be used to describe phenomenons dynamical in nature, which are abundant in areas such as biology, climatology and genetics. The…
In scientific computation, it is often necessary to calculate higher-order derivatives of a function. Currently, two primary methods for higher-order automatic differentiation exist: symbolic differentiation and algorithmic automatic…
Ordinary differential equations (ODEs), via their induced flow maps, provide a powerful framework to parameterize invertible transformations for the purpose of representing complex probability distributions. While such models have achieved…
Discovering governing Partial Differential Equations (PDEs) from sparse and noisy data is a challenging issue in data-driven scientific computing. Conventional sparse regression methods often suffer from two major limitations: (i) the…
Identifying differential operators from data is essential for the mathematical modeling of complex physical and biological systems where massive datasets are available. These operators must be stable for accurate predictions for dynamics…
The $\Sigma$-method for structural analysis of a differential-algebraic equation (DAE) system produces offset vectors from which the sparsity pattern of a system Jacobian is derived. This pattern implies a block-triangular form (BTF) of the…
We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary…
Complex systems in physics, chemistry, and biology that evolve over time with inherent randomness are typically described by stochastic differential equations (SDEs). A fundamental challenge in science and engineering is to determine the…
Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities…
In radio frequency applications, electric circuits generate signals, which are amplitude modulated and/or frequency modulated. A mathematical modelling yields typically systems of differential algebraic equations (DAEs). A multivariate…