We investigate energy supratransmission in a boundary-driven two-dimensional nonlinear Schrödinger equation with a central hole. Harmonic forcing with azimuthal modulation generates standing-wave states whose existence and stability depend on the driving amplitude, the inner radius, and the imposed azimuthal charge. Bifurcation analysis shows that small inner radii produce strongly confined states with higher destabilization thresholds, whereas larger radii yield broader profiles and smoother transitions between stable and unstable branches. The cubic--quintic and saturable models exhibit similar qualitative behaviour but differ quantitatively in their critical amplitudes and parameter dependence. A variational approximation captures the dependence of the critical drive on the azimuthal charge and nonlinear parameters, and clarifies how the nonlinear response shapes the stationary states near the turning point. Time-dependent simulations show that supratransmission occurs through the emission of localized pulses, with nonzero azimuthal charge triggering symmetry breaking and producing two-dimensional localized excitations (pearls). Isosurface plots provide a complementary view of the resulting radial and angular excursions. These results establish a quantitative framework for supratransmission in two-dimensional geometries and are relevant to driven nonlinear systems in optics, Bose--Einstein condensates, and structured media.

Gradient-based inversion of reaction-diffusion systems is typically approached via surrogate models or physics-informed neural networks (PINNs), while the most direct route, backpropagation through the PDE's structure itself, has largely been avoided. We pursue this direct route as a diagnostic probe, backpropagating a steady-state loss through unrolled Gray-Scott simulation to recover its parameters, with no surrogate or neural-network augmentation. Optimization fails to converge, and plotting the landscape directly locates the failure in its geometry -- flat plateaus with no gradient signal, bounded by sharp cliffs that align with bifurcation boundaries -- a structure that recurs across loss functions and is inherited however the gradients are routed to parameters. Reading this minimal setup as an ablation of PINN, we disentangle each component's role: with the neural network fixed, the residual loss is quadratic in the PDE parameters and yields a smooth landscape, so it alone already avoids the pathology, by implicitly encoding the full PDE dynamics across all initial conditions. The neural network, for its part, cannot repair an ill-posed parameter subspace, and so serves only to complete the observed data -- a division of labor not previously made explicit. These findings carry concrete design implications for PINN-type methods and a broader heuristic on when added dimensions actually help.

We present a rigorous theoretical framework for symmetry breaking and quantum irreversibility arising from stochastic Ito field reversal within a cubic-quintic nonlinear Schrodinger equation (CQ-NLSE) formalism. Starting from three physically motivated considerations, forward and backward nonlinear stochastic differential equations are derived via the Ito calculus. Kinematic time-reversal is shown to be fundamentally incompatible with the Ito stochastic structure, yielding the universal asymmetry-coupling parameter of 2/3. An energy-driven collapse operator proportional to the product of noise strength, local probability density, and excitation energy squared is introduced, amplifying the collapse in high-density, high-excitation regions. Exactly bright soliton solutions are obtained for a quasi-one-dimensional BEC of attractive Li-7 atoms, with forward and backward amplitude ratio of 1.870. Heat map analysis of the parameter planes reveals that the forward collapse operator grows monotonically in time while the backward counterpart decays, achieving a ratio approximately 1030, sharply distinguishing this framework from conventional symmetric collapse models.