KdV solitons no longer retrained per parameter: scaled TW-PINN trains once and keeps two solitons apart after collision

When the dispersion parameter changes, KdV soliton solutions no longer need a separately trained physics-informed neural network for each value: the scaled TW-PINN trains once on the normalized equation and recovers solutions for different amplitudes and dispersion parameters through inverse scaling. Earlier physics-informed neural networks for KdV generally had to be retrained when the dispersion parameter changed, and long-time prediction beyond the training time domain tended to lose accuracy. In the one-soliton experiments, four amplitude and dispersion combinations shared a single normalized training run; at times when the scaled coordinates already lay outside the training domain, the scaled PINN-CL with conservation-law loss noticeably underestimated the peak and oscillated, while the scaled TW-PINN kept the soliton profile.

A two-soliton solution cannot be represented by a single coordinate, so the network uses two traveling-wave coordinates and trains in two steps: first it learns each soliton's profile and propagation speed, then it uses those speeds to build coordinates for the full solution including the nonlinear interaction. With single-step training, any two distinct speed values form an invertible linear transformation of the inputs, and the hidden layers can still fit the same solution, so the true propagation speeds are not uniquely identified. Step one trains on a short time interval around the initial time, and the learned propagation speeds have a mean absolute error of 3.77e-03; step two uses the exact speeds to isolate speed-estimation error and compares three sampling strategies over ten independent runs each on the training domain.

Collocation points are no longer placed only uniformly over the whole domain or by residual size; soliton-informed sampling (SIS) adds points along the soliton trajectories predicted in step one, concentrating training where the solution varies most. Residual-based adaptive refinement depends on the current network prediction, repeatedly evaluates candidate points, and does not explicitly enforce balanced sampling around each soliton. On the two-soliton training domain, SIS gives an L2 error of 4.18e-03 and an L∞ error of 4.23e-02, both below RAR's 8.03e-03 and 7.85e-02, with lower phase-shift errors for both solitons; the model using only globally uniform sampling failed to recover the two solitons' separation after the collision.

The two-soliton model passes the traveling-wave coordinates through a tanh mapping that compresses them locally, so the solution in the original variables can still be reconstructed when scaled coordinates lie far beyond the training domain. Without the tanh mapping, the model predicts soliton positions reasonably well but shows profile discrepancies and spurious oscillations outside the soliton regions. With a dispersion parameter that expands the range of scaled coordinates and an evaluation domain ten times the training domain along each coordinate, the model with the tanh mapping kept its maximum spatial error relatively small over the whole evaluation interval, while the model without it grew steadily with extrapolation time.

The demonstrated scope is limited to one-, two- and three-soliton solutions of the KdV equation with initial conditions taken from exact soliton profiles; the two-soliton step two uses exact propagation speeds so that the traveling-wave coordinates and sampling strategy are examined on their own. The authors propose extending the work to more general multi-soliton configurations, initial conditions beyond exact soliton profiles, and more complex nonlinear dispersive systems.

The two-soliton step two uses exact propagation speeds, so the reported errors do not include the effect of step-one speed estimation; the three-soliton experiment only indicates that the same training procedure extends to that case, and the authors describe its relevance to general multi-soliton problems as potential.