  In this article, we consider algebras $\mathcal{A}$ of non-formal
pseudodifferential operators over $S^1$ which contain $C^\infty(S^1),$
understood as multiplication operators. We apply a construction of Chern-Weil
type forms in order to get $2k-$closed cocycles. For $k=1,$ we obtain a cocycle
on the algebra of (maybe non classical) pseudodifferential operators with the
same cohomology class as the Schwinger cocycle on the algebra of Classical
pseudodifferential operators, previously extended and studied by the author on
algebras of the same type. We also prove non-triviality in Hochschild
cohomology for $k=2.$
