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Autoregressive next-step prediction models have become the de-facto standard for building data-driven neural solvers to forecast time-dependent partial differential equations (PDEs). Denoise training that is closely related to diffusion…

Deep learning approaches for partial differential equations (PDEs) have received much attention in recent years due to their mesh-freeness and computational efficiency. However, most of the works so far have concentrated on time-dependent…

Neural surrogates for partial differential equations (PDEs) have become popular due to their potential to quickly simulate physics. With a few exceptions, neural surrogates generally treat the forward evolution of time-dependent PDEs as a…

Stiff systems of ordinary differential equations (ODEs) are pervasive in many science and engineering fields, yet standard neural ODE approaches struggle to learn them. This limitation is the main barrier to the widespread adoption of…

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…

The deep-learning-based least squares method has shown successful results in solving high-dimensional non-linear partial differential equations (PDEs). However, this method usually converges slowly. To speed up the convergence of this…

The concept of the path-dependent partial differential equation (PPDE) was first introduced in the context of path-dependent derivatives in financial markets. Its semilinear form was later identified as a non-Markovian backward stochastic…

In the present work, a multi-scale framework for neural network enhanced methods is proposed for approximation of function and solution of partial differential equations (PDEs). By introducing the multi-scale concept, the total solution of…

Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation…

A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these…

This work proposes a solution for the problem of training physics-informed networks under partial integro-differential equations. These equations require an infinite or a large number of neural evaluations to construct a single residual for…

Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems.…

With the immense computing power at our disposal, the numerical solution of partial differential equations (PDEs) is becoming a day-to-day task for modern computational scientists. However, the complexity of real-life problems is such that…

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…

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…

High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the…

In recent years, deep learning has been connected with optimal control as a way to define a notion of a continuous underlying learning problem. In this view, neural networks can be interpreted as a discretization of a parametric Ordinary…

Scientific machine learning is an emerging field that broadly describes the combination of scientific computing and machine learning to address challenges in science and engineering. Within the context of differential equations, this has…

In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could…

Deriving governing equations in Electromagnetic (EM) environment based on first principles can be quite tough when there are some unknown sources of noise and other uncertainties in the system. For nonlinear multiple-physics electromagnetic…