9 research outputs found
Improving physics-informed DeepONets with hard constraints
Current physics-informed (standard or operator) neural networks still rely on
accurately learning the initial conditions of the system they are solving. In
contrast, standard numerical methods evolve such initial conditions without
needing to learn these. In this study, we propose to improve current
physics-informed deep learning strategies such that initial conditions do not
need to be learned and are represented exactly in the predicted solution.
Moreover, this method guarantees that when a DeepONet is applied multiple times
to time step a solution, the resulting function is continuous.Comment: 15 pages, 5 figures, 4 tables; release versio
Lowest dimensional example on non-universality of generalized In\"on\"u-Wigner contractions
We prove that there exists just one pair of complex four-dimensional Lie
algebras such that a well-defined contraction among them is not equivalent to a
generalized IW-contraction (or to a one-parametric subgroup degeneration in
conventional algebraic terms). Over the field of real numbers, this pair of
algebras is split into two pairs with the same contracted algebra. The example
we constructed demonstrates that even in the dimension four generalized
IW-contractions are not sufficient for realizing all possible contractions, and
this is the lowest dimension in which generalized IW-contractions are not
universal. Moreover, this is also the first example of nonexistence of
generalized IW-contraction for the case when the contracted algebra is not
characteristically nilpotent and, therefore, admits nontrivial diagonal
derivations. The lower bound (equal to three) of nonnegative integer parameter
exponents which are sufficient to realize all generalized IW-contractions of
four-dimensional Lie algebras is also found.Comment: 15 pages, extended versio
Lie-orthogonal operators
Basic properties of Lie-orthogonal operators on a finite-dimensional Lie
algebra are studied. In particular, the center, the radical and the components
of the ascending central series prove to be invariant with respect to any
Lie-orthogonal operator. Over an algebraically closed field of characteristic
0, only solvable Lie algebras with solvability degree not greater than two
admit Lie-orthogonal operators whose all eigenvalues differ from 1 and -1. The
main result of the paper is that Lie-orthogonal operators on a simple Lie
algebra are exhausted by the trivial ones. This allows us to give the complete
description of Lie-orthogonal operators for semi-simple and reductive algebras,
as well as a preliminary description of Lie-orthogonal operators on Lie
algebras with nontrivial Levi-Mal'tsev decomposition. The sets of
Lie-orthogonal operators of some classes of Lie algebras (Heisenberg algebras,
almost Abelian algebras, etc.) are directly computed. In particular, it appears
that the group formed by the equivalence classes of Lie-orthogonal operators on
a Heisenberg algebra is isomorphic to the standard symplectic group of an
appropriate dimension.Comment: 17 pages, minor improvements have been mad
Equivalence of diagonal contractions to generalized IW-contractions with integer exponents
AbstractWe present a simple and rigorous proof of the claim by Weimar-Woods [E. Weimar-Woods, Contractions, generalized Inönü–Wigner contractions and deformations of finite-dimensional Lie algebras, Rev. Math. Phys. 12 (2000) 1505–1529.] that any diagonal contraction (e.g., a generalized Inönü–Wigner contraction) is equivalent to a generalized Inönü–Wigner contraction with integer parameter powers