Can we determine summands from their partial sums?

1. By calculating how many $s_i$ are zero, you can determine how many $\lambda_i$ are zero. If there are any, they will be creating repeated entries $s_i$ throughout, but in a systematic manner where one can eliminate their effect; indeed, if there are $k$ zeroes, then there will be $k$ extra duplicates of every entry. For simplicity, suppose $\lambda_1 > 0$.

2. Now, indeed, $\lambda_1 = s_2$.

3. We proceed by induction. Suppose we have identified $\lambda_1,\ldots,\lambda_j$ and calculated the partial sums consisting of only $\lambda_1,\ldots,\lambda_j$, such as $\lambda_1+\lambda_3$. Then the smallest remaining partial sum must be $\lambda_{j+1}$. Proof: Otherwise the smallest remaining partial sum would have to be a sum with at least one unknown $\lambda_m$ that is (by definition) not yet in the list of known partial sums, which would imply that $\lambda_m$ is smaller than the smallest remaining partial sum; a contradiction.

4. Now that $\lambda_{j+1}$ is also known, consider the known list of partial sums to include all sums of $\lambda_1,\ldots,\lambda_{j+1}$.

Actually, separate treatment of steps 1 and 2 is not necessary; one only has to initialize the list of known partial sums with the empty sum $s_1=0$.

This method probably works even if you are missing entires from the list of partial sums, as long as all the missing entries are strictly greater than $\lambda_n$.