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A method for approximating sixth-order ordinary differential equations is proposed, which utilizes a deep learning feedforward artificial neural network, referred to as a neural solver. The efficacy of this unsupervised machine learning…

We propose a novel type of nonlinear solver acceleration for systems of nonlinear partial differential equations (PDEs) that is based on online/adaptive learning. It is applied in the context of multiphase flow in porous media. The proposed…

Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require…

We present a unified convergence theory for gradient-based training of neural network methods for partial differential equations (PDEs), covering both physics-informed neural networks (PINNs) and the Deep Ritz method. For linear PDEs, we…

The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary…

In certain practical engineering applications, there is an urgent need to perform repetitive solving of partial differential equations (PDEs) in a short period. This paper primarily considers three scenarios requiring extensive repetitive…

Solving nonlinear optimal control problems is a challenging task, particularly for high-dimensional problems. We propose algorithms for model-based policy iterations to solve nonlinear optimal control problems with convergence guarantees.…

The traditional limitations of neural networks in reliably generalizing beyond the convex hulls of their training data present a significant problem for computational physics, in which one often wishes to solve PDEs in regimes far beyond…

In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the solution data can reveal the PDE operator depending on the…

Partial Differential Equations (PDEs) have long been recognized as powerful tools for image processing and analysis, providing a framework to model and exploit structural and geometric properties inherent in visual data. Over the years,…

Recent progress in scientific machine learning (SciML) has opened up the possibility of training novel neural network architectures that solve complex partial differential equations (PDEs). Several (nearly data free) approaches have been…

Probabilistic solvers for ordinary differential equations (ODEs) provide efficient quantification of numerical uncertainty associated with simulation of dynamical systems. Their convergence rates have been established by a growing body of…

Numerical solutions of partial differential equations (PDEs) require expensive simulations, limiting their application in design optimization, model-based control, and large-scale inverse problems. Surrogate modeling techniques seek to…

Although deep models have been widely explored in solving partial differential equations (PDEs), previous works are primarily limited to data only with up to tens of thousands of mesh points, far from the million-point scale required by…

Deep neural networks (DNN) have been used to model nonlinear relations between physical quantities. Those DNNs are embedded in physical systems described by partial differential equations (PDE) and trained by minimizing a loss function that…

When solving partial differential equations (PDEs), classical numerical methods often require fine mesh grids and small time stepping to meet stability, consistency, and convergence conditions, leading to high computational cost. Recently,…

PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning…

Continuous-depth neural networks, such as the Neural Ordinary Differential Equations (ODEs), have aroused a great deal of interest from the communities of machine learning and data science in recent years, which bridge the connection…

Recent advances in deep learning makes solving parabolic partial differential equations (PDEs) in high dimensional spaces possible via forward-backward stochastic differential equation (FBSDE) formulations. The implementation of most…

We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we…