We study finite-horizon continuous-time policy evaluation from discrete closed-loop trajectories under time-inhomogeneous dynamics. The target value surface solves a backward parabolic equation, but the Bellman baseline obtained from one-step recursion is only first-order in the grid width. We estimate the time-dependent generator from multi-step transitions using moment-matching coefficients that cancel lower-order truncation terms, and combine the resulting surrogate with backward regression. The main theory gives an end-to-end decomposition into generator misspecification, projection error, pooling bias, finite-sample error, and start-up error, together with a decision-frequency regime map explaining when higher-order gains should be visible. Across calibration studies, four-scale benchmarks, feature and start-up ablations, and gain-mismatch stress tests, the second-order estimator consistently improves on the Bellman baseline and remains stable in the regime where the theory predicts visible gains. These results position high-order generator regression as an interpretable continuous-time policy-evaluation method with a clear operating region.

Trump administration'unlocks' 140MILLION barrels of precious Iranian oil with major policy change to fight back against'hoarding' China... here's what it means for your wallet Buffy the Vampire Slayer star Nicholas Brendon dead at 54 as'heartbroken' family reveal cause of death Joseph Duggar's wife Kendra is arrested for allegedly endangering welfare of a minor as he faces new charges Behind closed doors, the Duggar family's next nightmare began long before Joseph's arrest: Insiders reveal what they knew and how they plan to recover America is about to be torn apart by a financial tsunami - and it's not just an oil crisis to fear. However, it seems not every corner of Britain is quite so beautiful - as a survey has revealed the least scenic locations. Voters on the Scenic Or Not survey awarded the top spot to Basingstoke's Newbury Road. This unappealing location received the lowest possible score, with just one out of 10 for'scenicness'. And while Cornwall might be renowned for its beautiful scenery, a rather less attractive part of the county - the Electricity Station in Landulph - joins Basingstoke at the bottom of the pile.

I'm standing in front of a quantum computer built out of atoms and light at the UK's National Quantum Computing Centre on the outskirts of Oxford. On a laboratory table, a complex matrix of mirrors and lenses surrounds a Rubik's Cube-size cell where 100 cesium atoms are suspended in grid formation by a carefully manipulated laser beam. The cesium atom setup is so compact that I could pick it up, carry it out of the lab, and put it on the backseat of my car to take home. I'd be unlikely to get very far, though.

Simulation plays a central role in scientific discovery. In many applications, the bottleneck is no longer running a simulator; it is choosing among large families of plausible simulators, each corresponding to different forward models/hypotheses consistent with observations. Over large model families, classical Bayesian workflows for model selection are impractical. Furthermore, amortized model selection methods typically hard-code a fixed model prior or complexity penalty at training time, requiring users to commit to a particular parsimony assumption before seeing the data. We introduce PRISM, a simulation-based encoder-decoder that infers a joint posterior over both discrete model structures and associated continuous parameters, while enabling test-time control of model complexity via a tunable model prior that the network is conditioned on. We show that PRISM scales to families with combinatorially many (up to billions) of model instantiations on a synthetic symbolic regression task. As a scientific application, we evaluate PRISM on biophysical modeling for diffusion MRI data, showing the ability to perform model selection across several multi-compartment models, on both synthetic and in vivo neuroimaging data.

This is the Picard rank one analogue to Proposition 5.3 in the main text. The number in each dimension is given in Table 1.

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science.