Physics-Informed Neural Networks for Error-Corrected Finite Difference Energy Balance in 1D Hydraulic Fracture Propagation

Accurate 1D energy balance modeling in hydraulic fracture propagation is crucial for optimizing treatment designs and mitigating numerical errors. Conventional Finite Difference Methods (FDM) are widely used but suffer from truncation errors, numerical instability, and high computational costs. This study introduces a 1D Physics-Informed Neural Network (PINN)-enhanced finite difference framework to correct numerical errors in real time while maintaining computational efficiency. The proposed model integrates energy balance equations into the loss function, ensuring physical consistency and improved predictive accuracy. A synthetic dataset of N = 1000 samples was generated, covering fracture length L $\in$ [0.1, 10] meters, fracture width W $\in$ [0.01, 1] meters, pressure gradient P $\in$ [1, 50] MPa/m, and fluid viscosity $\mu \in$ [0.001, 0.1] Pa·s. The PINN architecture consists of five hidden layers with 512 neurons each, utilizing the Swish activation function and dropout regularization (10%). Training is performed using the Adam optimizer with a decaying learning rate $\alpha_0 = 0.0003 e^{-0.9t}$ Performance evaluation demonstrates a significant MSE reduction from 264064.79 (CD) to 18.93 (PINN), with a generalization error decrease of 90.21%. The 1D PINN-augmented model successfully minimizes numerical artifacts, suppresses oscillatory behavior near fracture tips, and enhances computational stability. This study provides a foundation for extending PINN-enhanced FDM frameworks to higher-dimensional cases, including 2D simulations for fracture network modeling and 3D fully coupled fluid-solid interaction systems. Future research will focus on adaptive training strategies and scalability to complex multi-physics environments.