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Physics-Guided Deep Learning for Spatiotemporal Forecasting
Published in Anuj Karpatne, Ramakrishnan Kannan, Vipin Kumar, Knowledge-Guided Machine Learning, 2023
Rui Wang, Robin Walters, Rose Yu
The key to building equivariant networks is that the composition of equivariant functions is equivariant. Hence, if the maps between layers of a neural network are equivariant, then the whole network will be equivariant. Note that both the linear maps and activation functions must be equivariant. An important consequence of this principle is that the hidden layers must also carry a G-action.
Characteristic matrix functions for delay differential equations with symmetry
Published in Dynamical Systems, 2023
Equivariant Pyragas control [9] adapts the Pyragas feedback scheme so that the feedback term vanishes on a periodic orbit with a specific spatio-temporal pattern. More precisely, suppose that (38) is equivariant with respect to a compact symmetry group ;(38) has a periodic solution with minimal period p>0; is a discrete wave and is a spatio-temporal symmetry of , i.e. for some .
On the equivariance properties of self-adjoint matrices
Published in Dynamical Systems, 2020
Michael Dellnitz, Bennet Gebken, Raphael Gerlach, Stefan Klus
Moreover, if in addition the entire function f is -equivariant, then symmetry-related bifurcations of the system will be governed by rather than , and this leads to phenomena which would generically be unexpected if only is taken into account. This would apply, for instance, to numerical discretizations of the cubic or quintic Ginzburg–Landau equation on a -symmetric spatial domain [2,7]. Thus, from an abstract point of view our results are strongly related to the notion of hidden symmetries which has been introduced in connection with the occurrence of unexpected bifurcations in partial differential equations with Neumann boundary conditions [8,13]. We will illustrate this fact by several examples in the following sections.
Estimation after Selection from Uniform Populations under an Asymmetric Loss Function
Published in American Journal of Mathematical and Management Sciences, 2019
Suppose that be a given scale-equivariant estimator of θS, where , and is a real-valued function defined on . Define the estimator , where andis as defined in Lemma 1. Then, the estimatorimproves upon the estimatorunder the ASE loss function (1) provided that, and there is a strict inequality for some.