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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…
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are…
A (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d-1)n arbitrary functions of one variable. Given a PDE which does not pass the integrability test, the…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
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,…
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds…
We consider a general class of nonsmooth optimal control problems with partial differential equation (PDE) constraints, which are very challenging due to its nonsmooth objective functionals and the resulting high-dimensional and…
In this paper, we consider the problem of obtaining rational decay for a particular time-evolving fluid-structure model, the type of which has been considered in Chueshov and Ryzhkova (2013). In particular, this partial differential…
Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H\subseteq V^*$ $$ \left\{ \begin{align} &dX_t=A(t,X_t)dt+B(t,X_t)dW_t,\ t\in (0,T]\\\\& X_0=x\in H,…
Many scientific phenomena are modeled by Partial Differential Equations (PDEs). The development of data gathering tools along with the advances in machine learning (ML) techniques have raised opportunities for data-driven identification of…
Recent advances in operator learning have produced two distinct approaches for solving partial differential equations (PDEs): attention-based methods offering point-level adaptability but lacking spectral constraints, and spectral-based…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
We present an exact dimensional reduction for high-dimensional dynamical systems composed of $N$ identical dynamical units governed by quasi-linear ordinary differential equations (ODEs) of order $M$. In these systems, each unit follows a…
Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of Micro-Electro Mechanical Systems (MEMS) and is studied…
We formulate a class of nonlinear {evolution} partial differential equations (PDEs) as linear optimization problems on moments of positive measures supported on infinite-dimensional vector spaces. Using sums of squares (SOS) representations…
The use of deep learning methods for solving PDEs is a field in full expansion. In particular, Physical Informed Neural Networks, that implement a sampling of the physical domain and use a loss function that penalizes the violation of the…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…
We introduce PMODE (Partitioned Mixture Of Density Estimators), a general and modular framework for mixture modeling with both parametric and nonparametric components. PMODE builds mixtures by partitioning the data and fitting separate…
We give an exposition of recent results on regularity and Fredholm properties for first-order one-dimensional hyperbolic PDEs. We show that large classes of boundary operators cause an effect that smoothness increases with time. This…