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Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the…
Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general…
Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to…
We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost…
Conservation laws in the form of elliptic and parabolic partial differential equations (PDEs) are fundamental to the modeling of many problems such as heat transfer and flow in porous media. Many of such PDEs are stochastic due to the…
We propose algorithms for solving high-dimensional Partial Differential Equations (PDEs) that combine a probabilistic interpretation of PDEs, through Feynman-Kac representation, with sparse interpolation. Monte-Carlo methods and…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
Multiphysics problems such as multicomponent diffusion, phase transformations in multiphase systems and alloy solidification involve numerical solution of a coupled system of nonlinear partial differential equations (PDEs). Numerical…
We present a framework for solving a broad class of ill-posed inverse problems governed by partial differential equations (PDEs), where the target coefficients of the forward operator are recovered through an iterative regularization scheme…
The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations…
Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and…
Enhancing neural networks with knowledge of physical equations has become an efficient way of solving various physics problems, from fluid flow to electromagnetism. Graph neural networks show promise in accurately representing irregularly…
Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods are best suited to regular rectangular grids, which leads to…
We present an initial implementation of a probabilistic PDE-constrained shape optimization algorithm. Our method is based on a novel probabilistic representation of the shape derivative, which is evaluated using Monte Carlo sampling; and…
Partial differential equations (PDEs) play a central role in describing many physical phenomena. Various scientific and engineering applications demand a versatile and differentiable PDE solver that can quickly generate solutions with…
Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods,…
We consider adaptive finite element methods for solving a multiscale system consisting of a macroscale model comprising a system of reaction-diffusion partial differential equations coupled to a microscale model comprising a system of…
We study semi-linear elliptic PDEs with polynomial non-linearity and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives,…