Structure Preserving PINNs through Chebychev-Type Loss Functions
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We introduce a class of structure-preserving physics-informed neural networks (SP-PINNs) that leverage Chebyshev scalarization to balance multiple training objectives in learning solutions to nonlinear dispersive PDEs. By appropriately weighting equation residuals and boundary conditions, the method promotes dynamics that inherently respect the underlying Hamiltonian structure. We provide a rigorous convergence analysis, establishing uniform error bounds under standard regularity assumptions. Applied to the Korteweg-de Vries, Camassa-Holm, and Zakharov-Kuznetsov equations, the approach yields solutions that accurately capture long-time behavior and exhibit emergent Hamiltonian conservation.