Uncertainty in the solution of an inverse problem and in the tasks performed on it is quantified by posterior expectations, each an average of an integrand over $M$ posterior samples. While designed quadratures improve on the $O(M^{-1/2})$ error of Monte-Carlo estimation, they solve an optimization problem, often against the posterior density, for every new observation, which can be computationally costly. To address this limitation, we introduce the quadrature field, a set-equivariant network that maps an observation and its $M$ posterior samples to an $M$-node signed-weight quadrature in one forward pass. Trained once on a family of posteriors to minimize the worst-case integration error over a class of functions, it serves any observation, any $M$ and any integrand in that class with no further optimization. We show that, with high probability and up to a computable slack, the resulting quadrature is never worse than the Monte-Carlo estimate built from the same samples. We validate the quadrature field on closed-form and on learned posteriors, one constrained by a partial differential equation, where it improves on the Monte-Carlo estimate in median at every node count, often by orders of magnitude.

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