Recovering Governing Dynamics from Distributed Observations via Exact Spline Merging

• Each data holder sends local fixed-basis spline sufficient statistics, not raw observations, and a one-shot merge reproduces centralized fitting without iterative synchronization.The exact guarantee applies to recovering field coefficients in the fixed-feature squared-error setting.Source: §1 · pp. 1–2
• Prior model-merging methods address different settings, while federated scientific machine learning requires iterative training and field reconstruction or equation-discovery methods typically assume centralized data.The introduction situates the fixed-feature aggregation approach alongside earlier work on ridge-regression statistics and one-shot federated aggregation.Source: §1 · p. 1
• For MNIST, merged spline reconstructions match centralized fits with worst-case δmax < 10^-13 across all ten digits when all holders contribute.The figure also illustrates degradation when a holder is absent, with distinctive middle features affected more than simpler shapes.Source: §1 · p. 1, Figure 1 · p. 2
• Physical-parameter estimation from derivatives of the reconstructed field is approximate, unlike the exact aggregation guarantee for field coefficients.The parameters include diffusivity, wave speed, and viscosity.Source: §1 · p. 2
• The pipeline recovers governing parameters to sub-percent accuracy for linear PDEs and 5% for nonlinear Burgers, while distributed merging causes no degradation versus centralized fitting.The paper also reports a systematic noise sensitivity analysis.Source: §1 · p. 2

2 BACKGROUND

• Multiple holders share a scalar field’s coordinate system, fixed basis, coefficient ordering, and regularized objective to recover pooled-fit coefficients and estimate governing-equation parameters.A holder may represent a spatial region, institutional partition, or class-conditional subset.Source: §2 · p. 3

2.1 B-SPLINE BASIS FUNCTIONS

• Cubic B-splines on a uniform grid have compact support over four adjacent knot intervals, are C2-continuous across interior knots, and form a partition of unity.The representation can express nonlinear dependence on x while remaining linear in its coefficients.Source: §2.1 · p. 3

2.2 FITTING AS LINEAR REGRESSION

• With positive ridge regularization, squared-error fitting has a strictly convex objective and a unique global minimum, even when the feature matrix is rank-deficient.The resulting normal equations determine the fitted coefficients.Source: §2.2 · p. 3

2.3 GRAM MATRIX, MOMENT VECTOR, AND SUFFICIENT STATISTICS

• For fixed-basis ridge fitting, the Gram matrix and moment vector suffice to compute the coefficient estimate, with summary size independent of observation count.Dense summaries can exceed raw-data size when K is large relative to N; exploiting sparsity or structure may be needed for a bandwidth advantage.Source: §2.3 · p. 3, §2.4 · p. 4

2.4 COMPOSABILITY: FROM SUFFICIENCY TO EXACT DISTRIBUTED MERGING

• Each data holder can merge local Gram matrices and moment vectors to recover exactly the centralized fit, without sharing raw observations or iterative synchronization.The merge requires a single matrix solve and extends to any number of holders.Source: §2.4 · p. 4
• A data merge uses local moment vectors, while a prediction-matching merge uses locally fitted coefficients; the two coincide when local fits are at their unregularized optima.The experiments use the data merge.Source: §2.4 · p. 4
• The guarantee applies to fixed feature maps with quadratic objectives, not exact merging of fully trainable multilayer networks whose changing features invalidate prior Gram matrices.It also does not remove field-approximation, derivative-estimation, or physical-regression identifiability errors.Source: §2.4 · p. 4

3 METHOD

• A fixed feature vector makes 1D curves, 2D surfaces, and 3D spatiotemporal fields use the same Gram-merge algebra.The method specifies field representation, basis geometry, and parameter recovery.Source: §3 · p. 4

3.1 TENSOR-PRODUCT SPATIOTEMPORAL FIELDS

• Tensor-product fields combine cubic spatial B-splines with temporal B-splines or Fourier harmonics, flattening coefficients into a feature-vector model to which Gram merging applies.For two spatial dimensions and one temporal dimension, the feature matrix has one column per tensor-product basis feature.Source: §3.1 · p. 5

3.2 BASIS GEOMETRY AND MERGE STATISTICS

• Compactly supported B-splines yield banded univariate Gram blocks and zero interactions for nonoverlapping supports, while global Fourier features generally yield non-sparse Gram matrices.Exact statistic aggregation is demonstrated for spline and hybrid representations; basis support alone does not explain approximate averaging, and controlled comparisons of basis families are left to future work.Source: §3.2 · p. 5

3.3 PHYSICAL PARAMETER RECOVERY

• Central finite differences on the fitted field provide derivatives for ordinary least-squares recovery of PDE parameters that enter linearly.Recovery depends on field approximation, derivative computation, and conditioning; nearly zero Laplacians make diffusion recovery unreliable, and finite-difference error creates noisy features whose errors may be correlated with the response.Source: §3.3 · pp. 5–6
• Spatial fragmentation can make local spline Gram matrices rank-deficient, but exact unregularized local minimizers still yield the same merged statistics.Composability also holds with a globally shared quadratic penalty when the summed Gram matrix plus penalty matrix is positive definite.Source: §3.3 · p. 6

4.1 DIFFUSION EQUATION

• Diffusion recovered D = 0.04994 with 0.12% relative error from 10,000 observations; centralized and merged fields were visually indistinguishable.The fit used a 3D tensor-product B-spline field with 1,331 basis functions. The figure’s error panels show field-approximation error, not merge error.Source: §4.1 · p. 6

4.2 WAVE EQUATION

• Wave speed was recovered as cw,rec = 0.9991 with 0.09% error from 10,000 observations, and distributed merging matched centralized fitting to δmax < 10−12.The wave field conserves energy; centralized and merged reconstructions were visually indistinguishable.Source: §4.2 · p. 6, §4.1 · p. 6

4.3 VISCOUS BURGERS EQUATION (NONLINEAR)

• Burgers recovered κrec = 0.1050 with 5.0% error from 5,000 observations, while merging reproduced centralized fitting to δmax < 2 × 10−11.The nonlinear u ux feature had higher parameter error in this configuration, and representation and derivative-estimation errors were not isolated. Refining the reference grid from 80×60 to 320×240 reduced reference-field recovery error from 0.89% to 0.06%.Source: §4.3 · p. 6

4.4 HEAT EQUATION WITH SPATIALLY VARYING SOURCE

• For heat with a spatially varying source, Drec = 0.02977 had 0.76% relative error from 10,000 observations, and merging matched centralized fitting to δmax < 10−12.The periodic source was S(x, y) = 0.3 + 0.2 cos x + 0.1 sin 2y; the solution used an exact modal formula.Source: §4.4 · p. 6

4.5 NOISE SENSITIVITY AND ROBUSTNESS

• Across 90 diffusion and wave noise conditions, merging reproduced centralized estimates to δmax < 10−11; mean relative parameter error was below 1% at 3,000+ observations with up to 1% noise.High noise systematically attenuated diffusion estimates; increasing observation count with fixed unnormalized ridge penalty also changes effective regularization. Table 1 summarizes merge results across all four PDEs on balanced regional draws; separate full-domain draws explain slight estimate differences.Source: §4.5 · p. 7, Table 1 · p. 7

4.6 REAL DATA: NOAA SEA-SURFACE TEMPERATURE

• On NOAA OI SST V2 data, merged and centralized fits both had in-sample RMSE 0.639°C; held-out spatial and temporal RMSEs were 0.643°C and 0.663°C, respectively.The product covered December 1981 to January 2023 and approximately 4.41 million ocean grid-month values. The merge reproduced centralized reconstruction RMSEs in both held-out tests; the data are a gridded analysis product, not raw sensor records.Source: §4.6 · p. 7

5 CONCLUSION

• Across four PDEs, merging caused zero degradation: prediction-space discrepancies stayed below 2×10^-11; linear-PDE parameter recovery was sub-percent, Burgers recovery was 5%, and NOAA SST reconstruction RMSEs matched centralized fitting.The conclusion reports these results for synthetic experiments and real NOAA sea-surface temperature observations.Source: §5 · p. 8
• For fixed-feature squared-error fitting, holders communicate only Gram matrices and moment vectors; limitations include tensor-product scaling (P = kq), finite-difference derivatives, and requiring known PDE structure.The protocol merges composable sufficient statistics without sharing raw observations.Source: §5 · p. 8

REPRODUCIBILITY STATEMENT

• Experiments used fixed random seeds; linear PDE data came from analytic Fourier-mode evolution, while Burgers used dealiased pseudospectral RK4 with N=256 and ∆t=0.001.Experiment descriptions report key hyperparameters; supplementary code provides full specifications, and the experiment code is available at the repository linked in the abstract.Source: REPRODUCIBILITY STATEMENT · p. 8

ETHICS STATEMENT

• The work involved no human subjects or private data; NOAA SST data is public, and the framework shares sufficient statistics rather than individual records but makes no formal privacy guarantee.Privacy leakage through shared statistics was not analyzed.Source: ETHICS STATEMENT · p. 8

AI USE STATEMENT

• The author designed and conducted the research and takes responsibility for the final work; Claude and ChatGPT assisted with debugging, literature search, prose editing, and mathematical cross-checks.The author also performed primary-source verification of citations.Source: AI USE STATEMENT · p. 8

A EXPERIMENTAL CONFIGURATION AND ADDITIONAL RESULTS

• Experiments used InKAN v0.4.2 with a piecewise polynomial basis, PyTorch float64, and NVIDIA H100 80GB GPUs; released code runs on CPU.Table 2 summarizes the experimental configuration and identifies P as total coefficients and Neval as the FD derivative grid.Source: A EXPERIMENTAL CONFIGURATION AND… · p. 11, Table 2 · p. 11
• The merged field matches centralized fitting in Burgers (δmax < 2 × 10^-11) and heat-with-source (δmax < 10^-12).Source: Figure 4 · p. 11, Figure 5 · p. 12

B NOISE SENSITIVITY STUDY

• In all 90 conditions, distributed merging reproduced the centralized estimator (δmax < 10−11), and all wave conditions produced valid c c2w ≥0.Table 3 reports mean ± std relative error over five independent seeds, with noise levels defined relative to a fixed reference-grid field standard deviation.Source: Table 3 · p. 12
• At 5% noise with 1K observations, diffusion estimates had mean signed bias −0.003, consistent with errors-in-variables attenuation under high noise.The bias analysis describes diffusion estimates as systematically attenuated under high noise.Source: B NOISE SENSITIVITY STUDY · p. 12
• Wave-speed recovery is more robust at high observation counts because temporal smoothing benefits both utt and the Laplacian used by the estimator.The wave-speed estimator depends on utt and the Laplacian.Source: B NOISE SENSITIVITY STUDY · p. 12

C DISCUSSION

• Exact merging applies to fixed-feature models with a quadratic objective, not independently trained multilayer networks; it extends to shared quadratic penalties when P_s G_s + λR is positive definite.Changing an earlier network layer invalidates later-layer Gram matrices.Source: C DISCUSSION · p. 12
• Basis support constrains Gram sparsity, but averaging also depends on function space, coverage, regularization, and conditioning; controlled basis-family comparison remains future work.Source: C DISCUSSION · p. 12
• Coefficient count grows exponentially with input dimension, while dense Gram storage is O(P^2) and solving is O(P^3); dense summaries exceeded raw-data size in synthetic experiments.Communication savings require sparsity exploitation, which was not implemented. Finite-difference derivatives limit physics-recovery accuracy; the NOAA experiment uses an already-interpolated product, so real-data physical-parameter inference remains unvalidated, and privacy leakage from shared statistics is unanalyzed.Source: C DISCUSSION · p. 13
• Next steps include analytic B-spline derivatives, periodic basis functions, vector-field extensions, and quantitative comparison with federated scientific machine-learning methods (Zhang et al., 2025).Source: C DISCUSSION · p. 13
• Synthetic PDE reconstructions from centralized fitting and Gram merging are identical (δmax < 10^-12); NOAA residuals show reconstruction error, not merged-versus-centralized disagreement.The synthetic figure covers centralized and merged fits; the NOAA figure compares the reconstruction with the interpolated product.Source: Figure 2 · p. 14, Figure 3 · p. 15