Error-Conditioned Generative Solvers for Ill-Posed Inverse Problems

Supervised conditional diffusion models solve inverse problems in a single sampling run with the measurement as conditioning, but never check whether the reconstruction reproduces the measurement under the forward operator. Plug-and-play posterior samplers enforce the measurement by optimizing a data-fidelity objective, which requires differentiating the operator and often the denoiser at every step, needs per-problem tuning, and is an unreliable proxy for reconstruction error when the operator has a large null space. We introduce error-conditioned generative solvers: at every sampling step the denoiser proposes a clean field, the forward operator is applied to it, and the resulting residual, together with a physics residual when governing equations are known, is fed back to the same denoiser as an input. Trained with the standard denoising loss, the network learns how to act on its own residual, with no backpropagation through the operator, denoiser, or sampling chain. We evaluate on six severely underdetermined problems: a new benchmark of three compressible flows reconstructed from an initial state and radiographs, together with compressed-sensing MRI, sparse-view CT and inverse scattering. Against the same conditional model at an equal number of network evaluations, residual feedback reduces reconstruction error by $6$–$40\%$. Our method attains the lowest reconstruction error of any posterior sampler we evaluate, while running $9$–$13\times$ faster than plug-and-play samplers. Its samples preserve shock structure that a lower-error regression model smears, and $85\%$ of its residual-driven correction lies in the operator's null space, where the measurement is blind.