Neural network-based method for exact solutions of Burgers-type equations

Obtaining exact solutions to nonlinear partial differential equations (NLPDEs) remains a fundamental challenge in mathematical physics, with traditional methods often limited by problem-specific constraints and tedious manual derivations. We propose an improved neural network-based symbolic computation method that overcomes these limitations for obtaining exact solutions of NLPDEs. Similar to other neural network-based symbolic methods, our approach employs neural network outputs as trial functions. By introducing different activation functions, new trial functions are derived. These trial functions incorporate the neural network’s weights and biases, thereby transforming the solution of Burgers-type equations into a problem of determining these parameters. As an extension of neural network-based methods, the improved method incorporates more diverse activation functions to generate more complex and rare exact solutions while reducing parameter complexity. Applying this framework, we successfully derive a rich set of exact solutions for one-dimensional, two-dimensional, and three-dimensional coupled Burgers equations. In addition, we draw 3D plots, curve plots, contour plots and heatmap plots to observe the characteristics and abundant dynamical behavior of these solutions. Each obtained exact solution is rigorously verified by substituting it back into the original governing equations. The results confirm that the equations are satisfied identically, ensuring that the solutions are exact. This work demonstrates that embedding symbolic computation within a flexible neural network architecture not only enriches the exact solution repertoire for Burgers-type equations, but also establishes a highly robust and universal analytical tool for decoding high-dimensional nonlinear dynamical systems.