Loss Landscape Diagnosis for Gradient-Based Gray-Scott System Inversion: Disentangling the Roles of PINN Components

Researchers at an undisclosed institution backpropagated a steady-state loss through unrolled Gray-Scott simulation to recover reaction-diffusion parameters, finding that optimization fails due to flat plateaus and sharp cliffs in the loss landscape that align with bifurcation boundaries. The study reveals that the residual loss in physics-informed neural networks (PINNs) alone produces a smooth, quadratic landscape by encoding full PDE dynamics, while the neural network component cannot fix ill-posed parameter subspaces and only completes observed data. These findings disentangle the distinct roles of PINN components and provide design implications for gradient-based PDE inversion methods.

arXiv:2606.11258v1 Announce Type: new Abstract: 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.