A physics-informed B-spline reconstruction of flow data

Continuous B-spline models provide compact, differentiable representations of discrete flow snapshots, but data-only fitting does not control consistency with prescribed governing equations. We introduce physics-informed B-spline reconstruction (PI-BSR), a post hoc method that optimizes a fixed tensor-product B-spline representation of a completed flow trajectory. PI-BSR combines data fidelity with soft residuals for the governing partial differential equations, initial and boundary conditions, and integral balances over spatial subdomains. A shared-face construction ensures algebraic cancellation of internal fluxes, while exact sparse residual-Jacobian-transpose actions provide coefficient gradients without forming dense Jacobians. In a one-dimensional convection–diffusion problem, the selected strong-form model reduces the midpoint PDE MSE by 84.8% and the analytical relative L 2 field error by 8.1% relative to data-only MFA. With the strong-form weight fixed, adding the integral-balance term further reduces the regional q = 4 balance MSE by 28.8% and the post-selection whole-domain balance RMS by 15.7%. In a controlled Burgers test, forced observations are reconstructed using a prescribed unforced operator. The resulting subdomain-balance improvement is modest, but consistent with the deliberate model mismatch. In a lid-driven cavity, adding the integral-balance term at fixed strong-form weights reduces the regional q = 4 x - and y -momentum RMS residuals by 72.9% and 92.1%, respectively, with modest increases in velocity error. Overall, PI-BSR improves targeted equation and balance diagnostics at fixed spline capacity while preserving a compact, continuously differentiable representation for downstream analysis.