Physics‐Guided Neural Regression Modeling of Energy‐Balance Proxies for Hydraulic Fracture Propagation in 1D–3D

ABSTRACT Energy‐balance surrogates can support early hydraulic‐fracture screening before fully coupled simulations are run. This paper defines a physics‐guided neural regression benchmark for controlled energy‐proxy learning across increasing input dimensionality. We generate samples with fracture length , fracture width , pressure gradient , and fluid viscosity with numerical extrema, compute , add noise, and train on . Inputs are (1D), with fixed (2D), and (3D); features are standardized, with target standardization only for 1D/2D. The 1D/2D networks use 4256 ReLU layers, and the 3D network uses 3 128 Swish layers selected from a limited practical sweep. On noise‐free proxy test targets after noisy‐target training, MSEs are 0.0031 (1D), 1.4580 (2D), and 10917.8261 (3D), with NRMSE 0.0004/0.0069/0.0918 and 1.0000/1.0000/0.9916. KGD‐ and PKN‐inspired analytical trends are used as reference controls to separate the proposed attenuation‐driven energy proxy from classical opening‐dominated fracture scalings. Their poor agreement with the proxy shows that the benchmark is a distinct surrogate task rather than a reduced KGD/PKN solution. The result is a reproducible benchmark for assessing dimensional scaling, training stability, and surrogate accuracy in hydraulic‐fracture energy modeling before extension to coupled elasticity–fluid‐flow validation.