Related papers: A Latent space solver for PDE generalization
This paper develops one of the methods for study of nonlinear Partial Differential equations. We generalize Sato equation and represent the algorithm for construction of some classes of nonlinear Partial Differential Equations (PDE)…
This paper presents reproducing kernel Hilbert spaces method to obtain the numerical solution for partial differential equation constrained optimization problem.
In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…
In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the solution data can reveal the PDE operator depending on the…
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities.
Partial differential equation (PDE) is an important math tool in science and engineering. This paper experimentally demonstrates an optical neural PDE solver by leveraging the back-propagation-free on-photonic-chip training of…
Modeling complex dynamical systems with only partial knowledge of their physical mechanisms is a crucial problem across all scientific and engineering disciplines. Purely data-driven approaches, which only make use of an artificial neural…
We present Mechanistic PDE Networks -- a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in…
Using the rudiments of pde jets theory in a nonstandard setting, we first deepen and extend previous nonstandard existence results for generalized solutions of linear differential equations and second extend the previous results for linear…
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…
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.…
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…
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…
We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes,…
Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to…
The paper develops the method for construction of the families of particular solutions to the nonlinear Partial Differential Equations (PDE) without relation to the complete integrability. Method is based on the specific link between…
The generalization of neural networks is a central challenge in machine learning, especially concerning the performance under distributions that differ from training ones. Current methods, mainly based on the data-driven paradigm such as…
Partial differential equation (PDE)-constrained optimization arises in many scientific and engineering domains, such as energy systems, fluid dynamics and material design. In these problems, the decision variables (e.g., control inputs or…