Moving-Latent Neural PDE Surrogates

Neural surrogates offer data-driven alternatives to computationally expensive numerical solvers of partial differential equations (PDEs) by learning to approximate PDE solution operators directly from simulated trajectories. Among various approaches, latent-space surrogates further reduce the cost of long-horizon prediction by evolving compact representations instead of dense physical fields. However, existing approaches typically evolve latent features on fixed spatial or static abstract supports, limiting their ability to adapt where representational capacity is allocated as the solution moves and deforms. We introduce Moving-Latent Neural PDE Surrogates (MoLa-PDE), where latent features evolve together with their spatial coordinates, allowing a compact latent representation to adapt not only what information is stored but also where it is represented as the PDE dynamics evolve. Making such a moving representation effective requires both modeling interactions on evolving spatial supports and learning how the support itself should deform. To address the former, we develop a geometry-structured continuous operator that explicitly combines feature-dependent interactions with local neighborhood geometry through localized polynomial–Fourier bases. To address the latter, we introduce a structured latent motion mechanism that infers potential fields from global interactions and local differential features, and converts them into support deformation through a Hodge-inspired construction. Experiments across controlled synthetic problems and diverse PDE benchmarks show that the proposed moving-latent representation consistently improves prediction accuracy over fixed-support latent surrogates, while revealing interpretable latent-motion patterns that reflect evolving spatial structures.