Bayesian inference for inverse problems involves computing expectations under posterior distributions -- e.g., posterior means, variances, or predictive quantities -- typically via Monte Carlo (MC) estimation. When the quantity of interest varies significantly under the posterior, accurate estimates demand many samples -- a cost often prohibitive for partial differential equation-constrained problems. To address this challenge, we introduce conditional neural control variates, a modular method that learns amortized control variates from joint model-data samples to reduce the variance of MC estimators. To scale to high-dimensional problems, we leverage Stein's identity to design an architecture based on an ensemble of hierarchical coupling layers with tractable Jacobian trace computation. Training requires: (i) samples from the joint distribution of unknown parameters and observed data; and (ii) the posterior score function, which can be computed from physics-based likelihood evaluations, neural operator surrogates, or learned generative models such as conditional normalizing flows. Once trained, the control variates generalize across observations without retraining. We validate our approach on stylized and partial differential equation-constrained Darcy flow inverse problems, demonstrating substantial variance reduction, even when the analytical score is replaced by a learned surrogate.

Neural Information Processing Systems http://nips.cc/

We identify mechanisms through which normalizing flows can simultaneously increase likelihood for all structured images.

The inverse of Sylvester flows can be easily computed using a fixed point iteration. The setup is identical to section C.1, where a single subflow is now either a residual block or a convolutional Sylvester flow transformation, with a leading actnorm layer [ Results are obtained by running models a single after random weight initialization. Additionally, the gated convolutions are replaced by denseblock layers.