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We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning…
The efficient solution of discretisations of coupled systems of partial differential equations (PDEs) is at the core of much of numerical simulation. Significant effort has been expended on scalable algorithms to precondition Krylov…
Recent advances in deep learning have inspired numerous works on data-driven solutions to partial differential equation (PDE) problems. These neural PDE solvers can often be much faster than their numerical counterparts; however, each…
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…
Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields. Neural networks are well-suited for PDE parameter estimation due to their…
This work concerns the exponential stabilization of underactuated linear homogeneous systems of m parabolic partial differential equations (PDEs) in cascade (reaction-diffusion systems), where only the first state is controlled either…
We further develop a new framework, called PDE Acceleration, by applying it to calculus of variations problems defined for general functions on $\mathbb{R}^n$, obtaining efficient numerical algorithms to solve the resulting class of…
Partial differential equations (PDEs) are widely used for the description of physical and engineering phenomena. Some key parameters involved in PDEs, which represent certain physical properties with important scientific interpretations,…
Embedding physical knowledge into neural network (NN) training has been a hot topic. However, when facing the complex real-world, most of the existing methods still strongly rely on the quantity and quality of observation data. Furthermore,…
A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a…
Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental…
We address a physics-informed neural network based on the concept of random projections for the numerical solution of IVPs of nonlinear ODEs in linear-implicit form and index-1 DAEs, which may also arise from the spatial discretization of…
A finite element discretization using a method of lines approached is proposed for approximately solving the Poisson-Nernst-Planck (PNP) equations. This discretization scheme enforces positivity of the computed solutions, corresponding to…
Neural networks have been applied to control problems, typically by combining data, differential equation residuals, and objective costs in the training loss or by incorporating auxiliary architectural components. Instead, we propose a…
We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable…
In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational…
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection…
We consider constrained bilinear optimal control of second-order linear evolution partial differential equations (PDEs) with a reaction term on the half line, where control arises as a time-dependent reaction coefficient and constraints are…