4 papers · 1 filter
A Variational Framework for the Complexity of PDE Solutions
Juan Esteban Suarez Cardona, Holger Boche, Gitta Kutyniok
Partial Differential Equations (PDEs) are fundamental mathematical models for describing physical phenomena, yet most PDEs of practical interest require numerical approximations. T…
Hybrid Surrogate Models: Circumventing Gibbs Phenomenon for Partial Differential Equations with Finite Shock-Type Discontinuities
Juan-Esteban Suarez Cardona, Shashank Reddy, Michael Hecht
We introduce the concept of Hybrid Surrogate Models (HSMs) -- combining multivariate polynomials with Heavyside functions -- as approximates of functions with finitely many jump di…
Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
Juan-Esteban Suarez Cardona, Phil-Alexander Hofmann, Michael Hecht
We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials. Based on this polynomial surrogate model (PSM), we real…
Replacing Automatic Differentiation by Sobolev Cubatures fastens Physics Informed Neural Nets and strengthens their Approximation Power
Juan Esteban Suarez Cardona, Michael Hecht
We present a novel class of approximations for variational losses, being applicable for the training of physics-informed neural nets (PINNs). The loss formulation reflects classic…