Highly-scalable, Physics-Informed GANs for Learning Solutions of Stochastic PDEs

In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.

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Stochastic PDEs, random fields and exact mean-values

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8121/aba655 ◽

2020 ◽

Vol 53 (40) ◽

pp. 405002

Author(s):

Manuel O Cáceres

Keyword(s):

Random Fields ◽

Stochastic Pdes ◽

Mean Values

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The infinitesimal generator of the stochastic Burgers equation

Probability Theory and Related Fields ◽

10.1007/s00440-020-00996-5 ◽

2020 ◽

Vol 178 (3-4) ◽

pp. 1067-1124

Author(s):

Massimiliano Gubinelli ◽

Nicolas Perkowski

Keyword(s):

Burgers Equation ◽

Martingale Problem ◽

Uniqueness Result ◽

Stochastic Pdes ◽

Martingale Approach ◽

Infinite Dimensional ◽

Cylinder Test ◽

Stochastic Burgers Equation ◽

Trivial Intersection ◽

Backward Equation

Abstract We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for “energy solutions” of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427–471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially $$L^2$$ L 2 -ergodic, and that the stochastic Burgers equation on the real line is ergodic.

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