Neural DJL solvers must pick the wave before solving it: a constrained energy objective recovers the solitary wave, while residual objectives lose the prescribed APE in one round

With the same neural trial space, the same constraint treatment and the same initial field, the choice of objective decides whether the solver converges to the physical solitary wave rather than merely to some field that satisfies the equation. Classical spectral solvers inherit selection from the fixed-APE structure of their iteration, whereas residual objectives vanish at every exact solution and therefore supply no energetic ranking among roots. From one KdV-derived start, the constrained Deep Ritz objective recovered the depression mode-1 wave and its multiplier converged to the value the equation requires, while the strong-form and 512-test-function weak-form objectives lost the APE in the first descent round and terminated far below the target.

Selection and resolution fall to different objectives: the energy objective picks the wave, and the residual objective resolves the selected wave to equation accuracy. The Ritz-recovered wave meets structure, speed and constraint checks but leaves a full residual above the acceptance threshold, because that residual is dominated by high-wavenumber error the energy barely penalizes. A residual refinement of the wave recovered from the consistent start, holding APE exact and differentiating the speed, lowered the residual below the acceptance threshold on both the optimization grid and a finer grid after 910 accepted steps and passed every acceptance item; the refined field's error against the spectral reference rose, indicating the improvement came from the spectral content of the error rather than its size.

Structurally excluding the mode-1 branch from the trial space lets the same representation compute mode-2 waves and quasi-steady cores on asymmetric stratifications. Classical solvers initialize from the leading linear eigenmode and cannot directly reach mode 2 on asymmetric ocean stratifications, where the mode-1 wave has lower energy at equal APE. Seven mode-2 waves and cores were computed on four stratifications; on the annual-mean South China Sea profile, the 350 m shelf core has a 29.9 m isopycnal displacement, 0.387 km half-width, a node at 132 m and lobes at 230 m and 68 m, matching the 230 m and 68 m lobes observed by Yang et al. at the same depth.

The results rest on one benchmark stratification under one fixed protocol: direct calculations at four amplitudes from the consistent start and continuation to a fifth. The random starts show the wave is reached from fields with no wave structure without measuring the size of its basin. All South China Sea results refer to an annual-mean fitted profile, the shelf-core comparison with observations is qualitative, and the APE is prescribed rather than fitted to observations.

The quasi-steady cores are not homogeneous-DJL solutions and lack an independent spectral reference, so their assessment relies on projected residuals, virial consistency and grid and domain checks. The radiation estimate neglects finite-amplitude feedback and errors shared by the core and truncation, and it is not validated against time-dependent simulation; the tails of the deep-water and smaller benchmark cores are unresolved, and their losses are conditional. The amplitude dependence of mode-1 residuals is unresolved against run-to-run scatter, and the dependence on stratification remains open.