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In recent years, Physics-Informed Neural Networks (PINNs) have provided effcient data-driven methods for solving both forward and inverse problems of partial differential equations (PDEs). However, when addressing complex PDEs, PINNs face significant challenges in computational effciency and accuracy. In this study, we propose the Extended Mixed-Training Physics-Informed Neural Networks (X-MTPINNs), as illustrated in fig. 1., which effectively enhance the capability for solving nonlinear wave problems by integrating the domain decomposition technique of extended Physics-Informed Neural Networks (X-PINNs) with the mixed-training Physics-Informed Neural Networks (MTPINNs) framework. Compared to the classical PINNs model, the new model exhibits dual advantages: First, the mixed-training framework significantly improves convergence properties by optimizing the handling mechanism of initial and boundary conditions, achieving higher fitting accuracy for nonlinear wave solutions while reducing the computation time by approximately 40%. Second, the domain decomposition technique from X-PINNs strengthens the model’s representation capability for complex dynamical behaviors. Numerical experiments based on the Nonlinear Schrödinger Equation (NLSE) demonstrate that X-MTPINNs excel in solving two bright solitons, third-order rogue waves, and parameter inversion tasks, with prediction accuracy improved by one to two orders of magnitude over traditional PINNs. For inverse problems, the X-MTPINNs algorithm accurately identifies unknown parameters in the NLSE under noise-free, 2%, and 5% noisy conditions, addressing the complete failure of classical PINNs in parameter identification for NLSE under the studied scenarios, thereby exhibiting strong robustness.
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Keywords:
- PINNs /
- NLSE /
- X-MTPINNs /
- domain decomposition technique /
- parameters discovery
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