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