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Tensor networks provide compact and scalable representations of high-dimensional data, enabling efficient computation in fields such as quantum physics, numerical partial differential equations (PDEs), and machine learning. This paper…
We are interested in the numerical solution of coupled nonlinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard…
Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations (PDEs). Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs…
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…
Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are…
Homogeneous polynomial dynamical systems (HPDSs), which can be equivalently represented by tensors, are essential for modeling higher-order networked systems, including ecological networks, chemical reactions, and multi-agent robotic…
We develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the…
Numerous complex real-world systems, such as those in biological, ecological, and social networks, exhibit higher-order interactions that are often modeled using polynomial dynamical systems or homogeneous polynomial dynamical systems…
Time-dependent Partial Differential Equations with given initial conditions are considered in this paper. New differentiation techniques of the unknown solution with respect to time variable are proposed. It is shown that the proposed…
We develop a novel deep learning technique, termed Deep Orthogonal Decomposition (DOD), for dimensionality reduction and reduced order modeling of parameter dependent partial differential equations. The approach consists in the construction…
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years,…
Dynamic mode decomposition (DMD) has become a powerful data-driven method for analyzing the spatiotemporal dynamics of complex, high-dimensional systems. However, conventional DMD methods are limited to matrix-based formulations, which…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…
Predicting outcomes and planning interactions with the physical world are long-standing goals for machine learning. A variety of such tasks involves continuous physical systems, which can be described by partial differential equations…
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…
Model reduction plays a critical role in system control, with established methods such as balanced truncation widely used for linear systems. However, extending these methods to nonlinear settings, particularly polynomial dynamical systems…