Active Learning guided Gaussian Process Regressor for Non-Linear PDE Modeling for environmental simulations

In this research, we present a novel method for solving non-linear, time-dependent partial differential equations (PDEs) using active learning combined with Gaussian Process Regression (GPR). PDEs like Burger’s equation describe fluid and traffic flow, incorporating congestion and shock waves, whereas the Allen-Cahn equation contributes to environmental control by predicting pollutant diffusion. Other examples include the Stefan equation, which models heat transfer in materials, crucial for efficient energy storage, and the Korteweg-de Vries equation that eases waterway management through wave propagation modeling. Dense discretisation grids are necessary for traditional numerical techniques to these equations, which are computationally demanding. Our method reduces the requirement for big datasets and computer resources by using active learning to select the most informative data points intelligently. Solving these PDEs with machine learning instead of more conventional techniques is essential since this allows for quicker, more scalable solutions for intricate systems. By merging GPR with active learning, we anticipate the numerical solutions with high accuracy while decreasing the amount of labeled samples. The findings demonstrate that, in comparison to conventional numerical methods, the use of active learning in conjunction with the GPR approach greatly reduces the data needs. Accurately predicting the numerical answer, it presents a viable option for energy-related applications and environmental modelling.