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Deep Learning Reduced Order Models (ROMs) are becoming increasingly popular as surrogate models for parametric partial differential equations (PDEs) due to their ability to handle high-dimensional data, approximate highly nonlinear…
Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning…
Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free…
We represent an integration algorithm combining the characteristics method and Hopf-Cole transformation. This algorithm allows one to partially integrate a large class of multidimensional systems of nonlinear Partial Differential Equations…
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…
This paper proposes the Proximal Iteratively REweighted (PIRE) algorithm for solving a general problem, which involves a large body of nonconvex sparse and structured sparse related problems. Comparing with previous iterative solvers for…
In this paper, we propose a distributed computing approach to solving large-scale robust stability problems on the simplex. Our approach is to formulate the robust stability problem as an optimization problem with polynomial variables and…
Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential…
We introduce a novel grid-independent model for learning partial differential equations (PDEs) from noisy and partial observations on irregular spatiotemporal grids. We propose a space-time continuous latent neural PDE model with an…
In the past, we have presented a systematic computational framework for analyzing self-similar and traveling wave dynamics in nonlinear partial differential equations (PDEs) by dynamically factoring out continuous symmetries such as…
We focus in this paper on the stochastic stabilization problems of PDEs by Levy noise. Sufficient conditions under which the perturbed systems decay exponentially with a general rate function are provided and some examples are constructed…
The objective of this paper is to report some computational results for the theory of DAE stability boundary, with the aim of advancing applications in power system voltage stability studies. Firstly, a new regularization transformation for…
We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for…
The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…
Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering…
We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling…
Partial Differential Equations (PDEs) describe several problems relevant to many fields of applied sciences, and their discrete counterparts typically involve the solution of sparse linear systems. In this context, we focus on the analysis…
We utilize extreme-learning machines for the prediction of partial differential equations (PDEs). Our method splits the state space into multiple windows that are predicted individually using a single model. Despite requiring only few data…
We present a unified theoretical framework for analyzing the stability and consistency of Physics-Informed Neural Networks (PINNs), grounded in operator coercivity, variational formulations, and non-asymptotic perturbation theory. PINNs…
Learning the evolutionary dynamics of Partial Differential Equations (PDEs) is critical in understanding dynamic systems, yet current methods insufficiently learn their representations. This is largely due to the multi-scale nature of the…