From Learning Operators to Inferring Operators: ORIA for Bayesian Reasoning in Function Space

Scientific inference requires reasoning about unknown systems from limited observations and using uncertainty to guide further discovery. We introduce ORIA, a continuum framework for Bayesian operator inference in function space. Trained under priors over operators, ORIA conditions on a few input–output function pairs to produce joint posterior predictive distributions for unseen systems, without test-time optimization. Its continuum architecture supports prediction across discretizations, while its joint predictive distribution captures uncertainty and dependencies across entire response fields. This enables uncertainty-aware prediction of global field-structure-dependent physical quantities and guides the selection of informative experiments. Comparisons with exact Bayesian references assess posterior fidelity, and experiments on diverse physical systems demonstrate up to an order of magnitude lower relative error than baselines. Posterior-guided measurement selection reduces average prediction error by 13–18% relative to random selection. We further characterize how reliability depends on the training prior and how prior revision can improve transfer. ORIA paves a promising path toward efficient foundation models for scientific computing that know what they do not know.