Physics-Informed Method of Group Data Handling: Adaptive Construction of Functional Representations with an Application to the Navier-Stokes Equations

1. Introduction

• The central structural question is whether governing equations can determine which functional components constitute an approximation, rather than only which parameter values minimize its objective.Many approaches choose the general form before optimization begins.Source: §1 · p. 1

1.1 Relation to Existing Approaches

• PI-GMDH applies GMDH’s self-organizing principle to physical-field representations, using governing equations to help decide which functional components become active.Classical GMDH generates, fits, evaluates, and selectively retains candidate models as complexity develops.Source: §1 · p. 1
• Candidate functions are evaluated as perturbation directions through the first variation of the complete coupled physics-informed objective, rather than generated from local residual subproblems or solution snapshots.This distinguishes the selection mechanism from residual-driven basis enrichment and connects it to function-space greedy approximation while propagating candidates through governing operators and coupled residuals.Source: §1.1 · p. 2
• Chebyshev approximation and physics-informed spectral representations are not claimed as novel; Chebyshev functions provide a differentiable candidate family for studying structural construction.The framework requires differentiable physical residuals and candidate directions with evaluable operator responses; the study uses a two-dimensional time-dependent Taylor-Green benchmark.Source: §1.1 · p. 2
• PI-GMDH couples progressive construction of functional representations with physics-informed coefficient optimization, evaluating candidate components through the first variation of the complete coupled objective.Selected components are introduced as packages before block-coordinate coefficient reoptimization, allowing coupled fields to develop different active representations.Source: §1.1 · p. 2
• The study positions its contribution as integrating package-wise construction of coupled field representations with first-variation selection, based on a literature review rather than an absolute priority claim.The Taylor-Green study demonstrates the mechanism under controlled conditions, not a comprehensive comparison across systems, data regimes, or selection criteria.Source: §1.1 · p. 3

2.1 Physical problem, functional representation, and packages

• Each field is represented by an active set from a functional family, which can be extended by packages before the enlarged representation is reoptimized.Package construction may follow a hierarchy, add one candidate at a time, introduce the remaining candidate space, or use a physics-guided criterion.Source: §2.1 · pp. 3–4

2.2 Physics-guided candidate response and adaptive package construction

• Each candidate’s first-order response spans the complete coupled residual; normalized alignment q and per-point interaction m guide state-dependent screening and bounded package growth.The screening threshold, package-size rule, and hierarchy-exploration rule are implementation parameters, not defining properties of PI-GMDH.Source: §2.2 · p. 4

2.3 Coefficient optimization

• After structural extension, damped Gauss-Newton updates estimate coefficients field by field while holding the other field representations fixed.The candidate response used before activation becomes a Jacobian column for coefficient correction after activation.Source: §2.3 · p. 5

2.4 Package-construction policies

• The framework compares adaptive subsets, next-degree shells, one-term additions, and all remaining candidates as package policies; these variants isolate policy effects and are not classical GMDH algorithms.Source: §2.4 · p. 5

• The application considers an incompressible velocity field u(x,t) and pressure p(x,t).Source: §3.1 · p. 5
• The governing setup assumes constant kinematic viscosity ν.Source: §3.1 · p. 5

For current approximations ^u and ^p,

• Observation mismatches can be included as residuals, alongside other observational, boundary, or initial-condition residuals.Source: For current approximations ^u and ^p · p. 6

3.2 Residual response to candidate functions

• A velocity candidate generally affects multiple residual blocks, including observation, incompressibility, and momentum responses.Source: §3.2 · p. 6
• Pressure candidates leave incompressibility unchanged; without pressure observations or another pressure constraint, pressure is determined only through spatial derivatives.The transformation p(x,t) → p(x,t)+g(t) leaves the pressure gradient and momentum residuals unchanged.Source: §3.2 · p. 6

3.3 Two-dimensional specialization

• In the 2D specialization, the residual ordering includes observed u and v mismatches, divergence, and x- and y-momentum residuals; candidate relevance changes as field approximations evolve.Source: §3.3 · p. 6

4.1 Analytical benchmark

• The benchmark uses a time-dependent Taylor-Green solution on the full spatial cell and maps its coordinates to the Chebyshev interval.Training, validation, and test points are sampled independently; pressure and pressure gradients are not direct training targets.Source: §4.1 · p. 7, §4.2 · p. 7

4.3 Chebyshev realization

• Separate active sets represent u, v, and p with tensor-product Chebyshev functions, whose derivatives are evaluated analytically.The complete three-variable basis grows by total degree.Source: §4.3 · p. 7

4.4 PI-GMDH package variants

• Adaptive packages select eligible candidate functions using q and m, then introduce a bounded package before coefficient reoptimization.The reported runs use q ≥ 0.05 and a 256-term cap.Source: §4.4 · pp. 7–8
• Structural-growth ablations compare single-function additions, complete degree shells, and a one-shot all-terms extension using the same candidate family and physical objective.The complete-shell benchmark run was capped at Dmax = 13; the all-terms realization extends through degree 20.Source: §4.4 · p. 8

4.5 Coefficient optimization and implementation

• After each structural extension, coefficients are updated by alternating block-coordinate damped Gauss-Newton optimization.The retained sequence is u → p → v → p, with pressure updated twice because velocity updates change the coupled momentum residual.Source: §4.5 · p. 8

4.6 External reference models

• Selected PINN and KAN runs are illustrative references, not estimates of their classes’ best attainable performance or total tuning cost.No exhaustive architecture or hyperparameter search was performed, and the reported timings exclude unmeasured exploratory tuning.Source: §4.6 · p. 8

5.1 Full-cell validation trajectories

• Adaptive PI-GMDH continued reducing both velocity errors, while reference models and nonadaptive variants leveled off at substantially higher values over the recorded trajectories.The pressure-gradient errors are reconstructed through coupled momentum residuals, not supplied observational targets.Source: §5.1 · p. 8, Figure 1 · p. 9, Figure 2 · p. 9

5.2 Package granularity in a large-sample structural-growth experiment

• In the large-sample experiment, one-term growth reached the approximately 10^-19 loss regime with nearly a complete degree-13 representation but remained unfinished, while complete-shell growth reached degree 20 sooner.The comparison exposes a trade-off between structural selectivity and the overhead of repeated coefficient reoptimization.Source: §5.2 · pp. 9–10, Figure 3 · p. 10

5.3 Validation and held-out test evaluation

• 5.299×10^-19 validation loss and 5.296×10^-19 test loss were attained by adaptive PI-GMDH using 204, 201, and 175 active functions for u, v, and p.The held-out test loss was more than twelve orders of magnitude below the all-terms realization’s 2.722×10^-6, using 580 active functions overall versus 5,313.Source: §5.3 · pp. 10–11, Table 1 · p. 10

5.4 Representation size and structural efficiency

• Adaptive PI-GMDH used 580 active functions overall, about 10.9% of the all-terms coefficient count and 34.5% of the complete degree-13 count.The capped degree-by-degree run outperformed the all-terms representation on total test objective, but it does not establish degree-20 performance.Source: §5.4 · p. 11, §5.3 · p. 11

6. Discussion

• One-term growth accumulated 1,663 active functions before interruption within degree 13 and reached the same order of training loss as completed degree growth through degree 20, but at greater wall-clock cost.This experiment used a different, approximately million-point sampling protocol and is intended to illustrate structural-growth behavior, not to merge with the full-cell validation/test comparison.Source: §6 · p. 12
• Adaptive PI-GMDH reached a held-out test loss of 5.296×10^-19 with 580 active functions, versus 2.722×10^-6 for the 5,313-coefficient all-terms model and 5.776×10^-8 for the Dmax = 13 degree-growth model.The comparison associates the observed separation with construction and optimization path, while noting that alternative policies might reach comparable accuracy with more time or reoptimization stages.Source: §6 · p. 12
• Held-out pressure-gradient RMSEs show that pressure-gradient information can be recovered through coupling to observed velocities, although the pressure-gauge ambiguity remains.Pressure and its gradients are not direct observational targets in this benchmark.Source: §6 · p. 12
• The package threshold, size, hierarchy exploration, and balance between structural extension and coefficient relaxation are control parameters, motivating online package-policy adaptation as future work.The proposed options include contextual-bandit, reinforcement-learning, or simpler adaptive controllers.Source: §6 · p. 12
• The evidence is limited to a single smooth analytical flow with dense, noise-free velocity observations; same-domain held-out points do not establish continuous-domain error bounds or performance in other regimes.The selected PINN and KAN references were not exhaustively tuned, so the comparison does not establish intrinsic superiority over those model classes.Source: §6 · p. 13

7. Conclusions

• PI-GMDH progressively constructs a functional approximation by adding packages, then optimizing their coefficients against the complete physical and observational objective.Its adaptive package strategy evaluates inactive directions through the first variation; residual alignment and interaction magnitude inform relevance, and residual-response vectors also form coefficient-optimization Jacobian columns.Source: §7 · p. 13
• Adaptive PI-GMDH reached validation loss 5.299×10^-19 and held-out test loss 5.296×10^-19 with 204/201/175 active functions for u/v/p, compared with test losses 5.776×10^-8 for Dmax = 13 degree growth and 2.722×10^-6 for all-terms degree 20.The conclusion describes the result as a favorable accuracy–time trade-off for this synthetic problem under the tested configurations, not evidence of universal superiority.Source: §7 · p. 13
• The conclusions limit the evidence to the tested synthetic benchmark and call for evaluation with sparse or noisy data, other differential systems, and broader package-policy controls.The comparison does not establish that physics-guided scoring alone caused the improvement or that competing model classes cannot perform as well or better.Source: §7 · p. 13

Funding

• The research was partially supported by the U.S. Department of State under Grant No. NSEPEUR23002-1019.Source: Funding · p. 14

Declaration of Generative AI and AI-Assisted Technologies

• OpenAI Codex assisted with software development, debugging, and refinement, while OpenAI GPT assisted with manuscript preparation; the author determined the methodology, experiments, results assessment, and conclusions.The author states that AI-assisted content was reviewed and that they take responsibility for the final work.Source: Declaration of Generative AI and… · p. 14

Data and Code Availability

• The PI-GMDH implementation and numerical-experiment code are public in the PIMGDH GitHub repository.The manuscript and repository describe the benchmark definitions and experimental settings needed to reproduce the calculations.Source: Data and Code Availability · p. 14