A new hybrid approach for solving partial differential equations: Combining Physics-Informed Neural Networks with Cat-and-Mouse based Optimization

Abstract

Partial differential equations (PDEs) are essential for modeling a wide range of physical phenomena. Physics-Informed Neural Networks (PINNs) offer a promising numerical framework for solving PDEs, but their performance often depends on the choice of optimization strategy and network configuration. In this study, we propose a hybrid PINN with a Cat and Mouse-based Optimizer (CMBO) to enhance optimization effectiveness and improve accuracy across elliptic, parabolic, and hyperbolic PDEs. CMBO utilizes a cat and mouse interaction mechanism to effectively balance exploration and exploitation, improving parameter initialization and guiding the optimization process toward favorable regions of the parameter space. Extensive experiments were conducted under varying scenarios, including different numbers of hidden layers (3, 5, 7) and neurons per layer (10, 30, 50). The proposed PINN-CMBO was systematically evaluated against state-of-the-art optimization methods, including PINN Adam, PINN L-BFGS, PINN Adam L-BFGS, and PINN PSO, across a diverse set of PDE categories. Experimental results revealed that PINN CMBO consistently achieved superior performance, recording the lowest loss values among all methods within fewer iteration. For parabolic and hyperbolic PDEs, PINN CMBO achieved an impressive minimum loss value, significantly outperforming PINN Adam, PINN L-BFGS, PINN Adam L-BFGS, and PINN PSO. Similar improvements were observed in elliptic and parabolic PDEs, where PINN-CMBO demonstrated unparalleled accuracy and stability across all tested network configurations. The integration of CMBO into PINN enabled efficient parameter initialization, driving a substantial reduction in the loss function compared to conventional PINN approaches. By guiding the training process toward optimal regions of the parameter space, PINN-CMBO not only accelerates convergence but also enhances overall performance. These findings establish PINN-CMBO as a highly effective framework for solving complex PDE problems, surpassing existing methods in terms of accuracy and stability.