Physics-informed method builds adaptive fluid simulations efficiently

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

Machine Learning

Summary

Simulating complex fluid flows often requires choosing a fixed mathematical form in advance, which can limit accuracy or efficiency. The authors propose a new method that gradually builds these mathematical models by adding pieces selectively as the simulation proceeds. They tested this approach on a fluid flow problem described by the Navier-Stokes equations and found it could achieve very low errors while using a manageable number of mathematical functions. This method balances precision, model size, and computation time better than some other approaches in this controlled test.

What this means in practice

Tested on simulated data.

Authors

Mykhailo Minin

Abstract

Physics-informed computational methods usually optimize parameters within a functional representation whose structure is fixed in advance. This work proposes a Physics-Informed Method of Group Data Handling (PI-GMDH), in which representations of coupled physical fields are progressively constructed during solution. Candidate functional directions are evaluated through the first variation of the complete physical and observational objective, introduced in packages, and followed by block-coordinate damped Gauss-Newton coefficient optimization. The framework is demonstrated with tensor-product Chebyshev functions on the incompressible Navier-Stokes equations using a two-dimensional time-dependent Taylor-Green benchmark. Under the tested configuration, adaptive PI-GMDH reached validation and held-out test losses of 5.299e-19 and 5.296e-19 with 204, 201, and 175 active functions for u, v, and p. Complete degree-by-degree and all-terms PI-GMDH variants, together with selected PINN and KAN reference configurations, are used to examine the effect of structural construction policy. The results show that, for this controlled synthetic benchmark, selective progressive construction can provide a favorable combination of accuracy, representation size, and wall-clock time. The comparison is illustrative rather than a claim of universal superiority over alternative physics-informed approaches.
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