Given a forward mapping Φ : R N → R M , the region {x ∈ R N , ∥Φ(x) -y∥ 2 ≤ ε}, where y ∈ R M is a given vector and ε ≥ 0 is a perturbation amplitude, represents the set of all possible inputs x that could have produced the measurement y within an acceptable error margin. This set reflects the inherent uncertainty or indeterminacy in recovering the true input x solely from the noisy observation y, which is a key challenge in inverse problems. In this work, we develop a numerical algorithm called Jackpot (Jacobian Kernel Projection Optimization) which approximates this set with a low-dimensional adversarial manifold. The proposed algorithm leverages automatic differentation, allowing it to handle complex, high dimensional mappings such as those found when dealing with dynamical systems or neural networks. We demonstrate the effectiveness of our algorithm on various challenging large-scale, non-linear problems including parameter identification in dynamical systems and blind image deblurring. The algorithm is integrated within the Python package deepinv.