Related papers: A Framework for Solving Parabolic Partial Differen…
Solving high-dimensional parabolic partial differential equations (PDEs) with deep learning methods is often computationally and memory intensive, primarily due to the need for automatic differentiation (AD) to compute large Hessian…
We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show…
We deal with a class of semilinear parabolic PDEs on the space of continuous functions that arise, for example, as Kolmogorov equations associated to the infinite-dimensional lifting of path-dependent SDEs. We investigate existence of…
We introduce a convergent finite difference method for solving the optimal transportation problem on the sphere. The method applies to both the traditional squared geodesic cost (arising in mesh generation) and a logarithmic cost (arising…
We prove existence and uniqueness results for solutions to a class of optimal transportation problems with infinitely many marginals, supported on the real line. We also provide a characterization of the solution with an explicit formula.…
Spectral methods for solving partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) often use Fourier or polynomial spectral expansions on either uniform and non-uniform grids. However, while very widely…
We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial…
In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational…
In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already…
We present a method for optimal coordination of multiple vehicle teams when multiple endpoint configurations are equally desirable, such as seen in the autonomous assembly of formation flight. The individual vehicles' positions in the…
This work is concerned with solving high-dimensional Fokker-Planck equations with the novel perspective that solving the PDE can be reduced to independent instances of density estimation tasks based on the trajectories sampled from its…
This investigation is motivated by the problem of optimal design of cooling elements in modern battery systems. We consider a simple model of two-dimensional steady-state heat conduction described by elliptic partial differential equations…
We investigate regularity and a priori estimates for Fokker-Planck and Hamilton-Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $s\in(1/2,1)$. As for Fokker-Planck equations, we establish…
Shape optimization with constraints given by partial differential equations (PDE) is a highly developed field of optimization theory. The elegant adjoint formalism allows to compute shape gradients at the computational cost of a further PDE…
We consider existence and uniqueness for several examples of linear parabolic equations formulated on moving hypersurfaces. Specifically, we study in turn a surface heat equation, an equation posed on a bulk domain, a novel coupled…
We propose to solve polynomial hyperbolic partial differential equations (PDEs) with convex optimization. This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution,…
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…
Many systems of partial differential equations have been proposed as simplified representations of complex collective behaviours in large networks of neurons. In this survey, we briefly discuss their derivations and then review the…
CASL-HJX is a computational framework designed for solving deterministic and stochastic Hamilton-Jacobi equations in two spatial dimensions. It provides a flexible and efficient approach to modeling front propagation problems, optimal…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…