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Solving parametric partial differential equations (PDEs) presents significant challenges for data-driven methods due to the sensitivity of spatio-temporal dynamics to variations in PDE parameters. Machine learning approaches often struggle…
At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly,…
Neural network-based approaches have recently shown significant promise in solving partial differential equations (PDEs) in science and engineering, especially in scenarios featuring complex domains or incorporation of empirical data. One…
Machine learning for differential equations paves the way for computationally efficient alternatives to numerical solvers, with potentially broad impacts in science and engineering. Though current algorithms typically require simulated…
We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution…
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…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
The evolution of dynamical systems is generically governed by nonlinear partial differential equations (PDEs), whose solution, in a simulation framework, requires vast amounts of computational resources. In this work, we present a novel…
Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete candidate…
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,…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…
Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and…
Inverse problems involving partial differential equations (PDEs) with discontinuous coefficients are fundamental challenges in modeling complex spatiotemporal systems with heterogeneous structures and uncertain dynamics. Traditional…
Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at…
Many physical processes such as weather phenomena or fluid mechanics are governed by partial differential equations (PDEs). Modelling such dynamical systems using Neural Networks is an active research field. However, current methods are…
Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical…
Power systems are globally experiencing an unprecedented growth in size and complexity due to the advent of nonconventional generation and consumption technologies. To navigate computational complexity, power system dynamic models are often…
Learning the solution of partial differential equations (PDEs) with a neural network is an attractive alternative to traditional solvers due to its elegance, greater flexibility and the ease of incorporating observed data. However, training…
Stochastic differential equations (SDEs) are a ubiquitous modeling framework that finds applications in physics, biology, engineering, social science, and finance. Due to the availability of large-scale data sets, there is growing interest…
Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence,…