MPINSDE: When Does Prior Knowledge Help a Neural SDE Extrapolate?

initial conditions outside its training data. MPINSDE builds prior knowledge into a neural SDE through a physics residual on the drift and a multiple-initial-condition (MIC) multiple-shooting curriculum, each reformulated for noise, so that a transition pseudo-likelihood replaces rollout error and the SDE is well-posed by construction. On a bistable double well standing in for a resettable, queryable process, we score recovery on the standard held-out-initial-condition region, of which only 2.2% of the samples are out of support, and extrapolation in a dedicated experiment where it is representable and unvisited. Most of the extrapolation gain is MIC's, and the physics residual alone is worse than the baseline on 11 of 12 accuracy measures. MPINSDE reduces the plain neural SDE's out-of-support error by +58% on every seed, level with MIC alone. Scored by our preregistered effect-size gate, the one-step contrast passes (+84%, $d_z = 1.12$) and the full-rollout contrast misses ($d_z = 0.91$). The residual's contribution is local rather than average. On leading windows over out-of-support samples MIC falls below the baseline (-6.1% to -1.6%) while MPINSDE gains (+8.8% to +14.5%), and across the physics-weight sweep under Euler–Maruyama it never loses to the baseline. In support, it improves upon the baseline on diffusion, calibration and invariant-law recovery, passing the gate on each. Rare events remain unsolved; every treated model underestimates the mean first-passage time, and the two MIC variants escape the well too early (-14% and -15% against -5% for the baseline). In its queryable regime, MPINSDE is a robust choice on this testbed, bounded by rare-event timing.