Overview
The Physics-Informed Method of Group Data Handling (PI-GMDH) is proposed as a computational approach addressing parameter optimization within functional representations. Unlike methods that fix functional structure beforehand, PI-GMDH progressively constructs representations of coupled physical fields concurrently with solution generation.
Research Context
Physics-informed computational methods typically involve optimizing parameters within a functional representation, the structure of which is predetermined. The presented work deviates from this common practice by introducing an adaptive construction process for these functional representations.
Approach
The PI-GMDH framework operates by progressively constructing representations for coupled physical fields. This construction involves evaluating candidate functional directions using the first variation of a combined physical and observational objective. These directions are introduced in packages. Subsequent to this evaluation, a block-coordinate damped Gauss-Newton algorithm is employed for coefficient optimization.
Demonstration and Configuration
The framework's application was demonstrated using tensor-product Chebyshev functions. The specific physical system chosen for this demonstration was the incompressible Navier-Stokes equations. A two-dimensional time-dependent Taylor-Green benchmark served as the test case for this application.
Variants and Reference Configurations
To investigate the impact of structural construction policy, the study examined several configurations:
- Complete degree-by-degree PI-GMDH variants.
- All-terms PI-GMDH variants.
- Selected Physics-Informed Neural Network (PINN) reference configurations.
- Selected Kolmogorov-Arnold Network (KAN) reference configurations.
Findings
Under the specific configuration tested with the Taylor-Green benchmark, the adaptive PI-GMDH method achieved the following results:
- Validation loss: $5.299 \times 10^{-19}$
- Held-out test loss: $5.296 \times 10^{-19}$
The method utilized a specific number of active functions for each field:
- u: 204 active functions
- v: 201 active functions
- p: 175 active functions
The comparative analysis, particularly for this controlled synthetic benchmark, indicated that selective progressive construction can offer a favorable combination of accuracy, representation size, and wall-clock time.
Why This Matters
The results from this controlled synthetic benchmark suggest that selective progressive construction can provide a favorable balance across accuracy, representation size, and computational time for certain physics-informed problems.
Key Limitations Mentioned by Researchers
The comparison is illustrative rather than a claim of universal superiority over alternative physics-informed approaches.