Amortized quadrature for posterior expectations in inverse problems

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.