In domains such as molecular and protein generation, physical systems exhibit inherent symmetries that are critical to model. Two main strategies have emerged for learning invariant distributions: designing equivariant network architectures and using data augmentation to approximate equivariance. While equivariant architectures preserve symmetry by design, they often involve greater complexity and pose optimization challenges. Data augmentation, on the other hand, offers flexibility but may fall short in fully capturing symmetries. Our framework enhances both approaches by reducing training variance and providing a provably lower-variance gradient estimator.

We present a neural network-based framework for solving the quantitative group testing (QGT) problem that achieves both high decoding accuracy and structural verifiability. In QGT, the objective is to identify a small subset of defective items among $N$ candidates using only $M \ll N$ pooled tests, each reporting the number of defectives in the tested subset. We train a multi-layer perceptron to map noisy measurement vectors to binary defect indicators, achieving accurate and robust recovery even under sparse, bounded perturbations. Beyond accuracy, we show that the trained network implicitly learns the underlying pooling structure that links items to tests, allowing this structure to be recovered directly from the network's Jacobian. This indicates that the model does not merely memorize training patterns but internalizes the true combinatorial relationships governing QGT. Our findings reveal that standard feedforward architectures can learn verifiable inverse mappings in structured combinatorial recovery problems.