Toward nonlinear representations with Gaussian-splat manifolds for physics-informed learning.

Conventional linear discretizations, including high-order schemes, often require prohibitively many degrees of freedom to resolve sharp, localized features, and the practical advantages of nonlinear, physics-informed representations over fixed linear spaces remain unclear. Here we introduce a compact nonlinear-manifold representation, based on Gaussian splatting, for solving partial differential equations with discontinuities and thin shear layers. Mobile, anisotropic Gaussian kernels are evolved or optimized to satisfy the governing equations: time-dependent problems through a projection-based scheme on the manifold, and steady forward and inverse problems through direct residual minimization. The representation offers closed-form spatial derivatives and a highly compact description of sharp features. Across smooth, shock-dominated and shear-layer benchmarks, it attains accuracy comparable to classical solvers for smooth flows while remaining markedly more compact in strongly nonlinear regimes, and it displays an approximate error invariance as the Reynolds number increases-departing from the error accumulation inherent to linear representations.