Physics-informed neural networks-based solutions for the BBM-Burgers equation: forward and inverse modeling

Abstract In this work, we investigate the use of Physics-Informed Neural networks (PINNs) for solving both forward and inverse problems associated with the nonlinear Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation. This equation models nonlinear wave propagation with dissipative effects and poses considerable challenges for conventional numerical solvers, particularly under noisy or incomplete data. The proposed framework incorporates the governing physical laws directly into the loss function of the neural network, employs automatic differentiation for computing derivatives, and uses gradient-based optimization to train the model. To investigate the influence of activation functions on predictive performance, we compare three variants of the PINN framework using hyperbolic tangent, sine, and Gaussian exponential activations for data-driven solution approximation and parameter estimation. The results show that while all three activation functions can approximate solution dynamics, the $$\tanh (x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>tanh</mml:mo> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -based PINN consistently achieves higher accuracy and faster convergence in inverse settings. Furthermore, the framework exhibits robustness to noise and generalizes well across different network depths and neuron configurations. These findings confirm that the PINN approach provides accurate approximations to the BBM–Burgers equation in both forward and inverse scenarios. Overall, this study highlights the potential of PINNs as a powerful, mesh-free method for solving complex nonlinear PDEs across different fields.