Recovering Physical Parameters from Fragmented Observations via Exact Distributed Spline Merging

Summary

The paper applies the classical additivity of fixed-basis ridge-regression sufficient statistics to tensor-product B-spline field reconstruction: each data holder computes a local Gram matrix and moment vector, and summing them yields a solution mathematically identical to centralized fitting via one matrix solve, with no iterative synchronization. The author carefully scopes this exactness to fixed-feature squared-error objectives and disclaims extension to jointly trained multilayer networks. A physics-recovery pipeline then fits a differentiable spline field to distributed observations, extracts finite-difference derivatives, and recovers PDE parameters by linear regression. On synthetic diffusion and wave equations driven by analytically evolved Fourier modes (10,000 noiseless observations), the diffusion coefficient is recovered to 0.11% and wave speed to 0.12% error, while a 5,000+5,000 spatial split reproduces the centralized estimator with delta-max below 1e-15 (Table 1); a single noisy, sparse trial (1,000 observations, sigma=0.02) degrades wave-speed error to 8.6%. On 41 years of NOAA OISST V2 data over the tropical Pacific, the two-way merge exactly reproduces centralized in-sample RMSE (0.639 degrees C). Limitations: the identity itself is textbook material, so novelty lies mainly in the application; synthetic tests are noiseless and easy; the real-data experiment partitions an already-interpolated gridded product and performs no parameter recovery; there is no uncertainty quantification and no comparison against competing distributed methods; code is promised only upon acceptance.